On-Orbit Geometric Calibration and Accuracy Validation of the Jilin1-KF01B Wide-Field Camera

Abstract

On-orbit geometric calibration is key to improving the geometric positioning accuracy of high-resolution optical remote sensing satellite data. Grouped calibration with geometric consistency (GCGC) is proposed in this paper for the Jilin1-KF01B satellite, which is the world’s first satellite capable of providing 150-km swath width and 0.5-m resolution data. To ensure the geometric accuracy of high-resolution image data, the GCGC method conducts grouped calibration of the time delay integration charge-coupled device (TDI CCD). Each group independently calibrates the exterior orientation elements to address the multi-time synchronization issues between imaging processing system (IPS). An additional inter-chip geometric positioning consistency constraint is used to enhance geometric positioning consistency in the overlapping areas between adjacent CCDs. By combining image simulation techniques associated with spectral bands, the calibrated panchromatic data are used to generate simulated multispectral reference band image as control data, thereby enhancing the geometric alignment consistency between panchromatic and multispectral data. Experimental results show that the average seamless stitching accuracy of the basic products after calibration is better than 0.6 pixels, the positioning accuracy without ground control points(GCPs) is better than 20 m, the band-to-band registration accuracy is better than 0.3 pixels, the average geometric alignment consistency between panchromatic and multispectral data are better than 0.25 multispectral pixels, the geometric accuracy with GCPs is better than 2.1 m, and the geometric alignment consistency accuracy of multi-temporal data are better than 2 m. The GCGC method significantly improves the quality of image data from the Jilin1-KF01B satellite and provide important references and practical experience for the geometric calibration of other large-swath high-resolution remote sensing satellites.

1. Introduction

On 3 July, 2021, at 10:51 AM, the Jilin1-KF01B satellite was successfully launched from the Taiyuan Satellite Launch Center, Xinzhou, China. The Jilin1-KF01B sub-meter wide-field camera (Jilin1-KF01B WF camera) features a design with a large aperture, wide field of view, long focal length, and an off-axis three-mirror-anastigmat optical system. It is capable of providing push-broom images with a panchromatic resolution of 0.5 m, a multispectral resolution of 2 m, and a swath width greater than 150 km.

Table 1. Main parameters of the Jilin1-KF01B WF camera.

Jilin1-KF01B WF Camera	Parameter
Camera Type	Linear Array TDI CCD Push-Broom Camera
Detector Mosaicking Method	Mechanical Staggered
Optical System	Off-Axis Three-Mirror-Anastigmat Optical System
Quantization Bit Depth	12 bit
Swath Width	150 km
SNR	>100:1
Spectral Range	Panchromatic	450 nm~800 nm
	Blue	450 nm~510 nm
	Green	519 nm~580 nm
	Red	630 nm~690 nm
	Near-Infrared (NIR)	770 nm~895 nm
Spatial Resolution of Data Products	Panchromatic	0.5 m
	Multispectral	2 m
MTF	Panchromatic	>0.16
	Multispectral	>0.28

provides detailed information about the Jilin1-KF01B WF camera.

High-precision geometric positioning is fundamental to maximizing the performance and value of high-resolution satellites [1,2]. On-orbit geometric calibration is a crucial step in enhancing the geometric performance of high-resolution remote sensing satellites and is also a necessary procedure for satellite geometric correction processing [3]. Before the launch of the Jilin1-KF01B satellite, strict laboratory calibration was conducted on the onboard payloads. The two-dimensional high-precision interior orientation element calibration method for the linear array camera improved the reprojection error of the interior orientation element calibration in the laboratory to 0.34 pixels [4]. However, due to environmental changes during satellite launch and subsequent orbital operations, the status of the onboard measurement devices changed, rendering the laboratory calibration parameters inadequate to represent the satellite’s true on-orbit state. This discrepancy leads to a decline in the geometric positioning accuracy of optical images [5]. Therefore, using photogrammetric methods to precisely calibrate the interior and exterior orientation elements of the imaging system is crucial for providing accurate geometric imaging parameters, which are essential for the high-precision geometric processing of optical remote sensing images [2].

Geometric calibration of satellite imagery is crucial for ensuring geometric quality and positioning accuracy of the images. To improve the geometric calibration accuracy of satellite imagery, extensive research has been conducted and demonstrated the applicability of the look-angle model for high-resolution satellite geometric calibration both theoretically and experimentally [6]. By applying on-orbit geometric calibration, the geolocation accuracy without ground control points (GCPs) of international satellites is shown in the

Table 2. Geometric positioning accuracy without GCPs after on-orbit geometric calibration.

Country	Satellite	Geolocation Accuracy without GCPs
France	SPOT [7,8,9]	50 m
	Pléiades 1A [10]	10 m
	Pléiades 1B [10,11]	4.5 m
	Pléiades Neo [12]	5 m
America	IKONOS [1,13,14]	12 m
	OrbView-3 [15]	10 m
	GeoEye-1 [16,17]	4 m
	WorldView-1 [18]	4 m
	WorldView-2 [17]	4 m
	WorldView-3 [19]	4 m
China	ZY3-01 [20]	20 m
	ZY3-02 [21]	10 m
	ZY-1 02C [22]	100 m
	Gaofen-4 [23,24]	1 pixel
	Gaofen-5 [25]	60 m
	Gaofen-6 [26]	3 pixels
	Luojia-1 01 [27]	650 m
	Zhuhai-1 [28]	1.5 pixels
	GaoFen-14 [29]	1.24 m

. The results indicate that in-orbit geometric calibration can significantly improve the geometric positioning accuracy of optical remote sensing satellite data.

The Jilin1-KF01B WF camera’s focal plane utilizes an array of multiple TDI CCDs mechanically staggered and stitched together, achieving an imaging swath width of 150 km. Adjacent CCDs capture images of the same ground target at different times, with varying satellite attitudes and viewing angles, making seamless stitching between adjacent CCDs a challenging task. To better compensate for image motion, multiple imaging processing system (IPS) are used, allowing for precise and independent control of TDI CCD timing across different fields of view. However, this design also introduces varying errors among the detectors within the different IPS. Additionally, the high-precision mapping requirements for wide-swath satellites demand greater geometric alignment consistency between panchromatic and multispectral data, necessitating further improvement through on-orbit geometric calibration. Consequently, traditional on-orbit geometric calibration methods are difficult to apply directly. Grouped calibration with geometric consistency (GCGC) method is proposed to address these challenges and achieve high-precision geometric positioning for the Jilin-1 KF01B WF camera data.

The main contributions of this study are as follows:

• To achieve high-precision geometric positioning for satellites with large field-of-view and mechanically staggered multiple TDI CCDs, this paper proposes the Grouped Calibration with Geometric Consistency (GCGC) method. GCGC addresses the multi-timing synchronization issues among IPS by calibrating the TDI CCDs in groups, with each group undergoing independent exterior orientation element calibration.
• The GCGC method improves the geometric positioning consistency between images from adjacent CCDs through an additional inter-chip geometric positioning consistency constraint. The GCGC method makes use of image simulation technology based on spectral band correlation [30], which enhances the geometric alignment consistency between panchromatic and multispectral data.
• Through long-term, widely distributed on-orbit data, comprehensive geometric accuracy evaluation experiments were conducted from multiple perspectives. These evaluations provide important references and practical experience for the geometric calibration of other large-swath, high-resolution remote sensing satellites.

The rest of this paper is organized as follows. Section 2 introduces the on-orbit geometric calibration model of the Jilin1-KF01B WF camera, and provides a detailed explanation of the GCGC method and the practical processing workflow. Section 3 presents the on-orbit geometric calibration experimental results of the Jilin1-KF01B WF camera based on the GCGC method, and provides a detailed evaluation of the geometric accuracy of the data using on-orbit data. Section 4 summarizes the study’s conclusions based on detailed analysis.

2. Methods

2.1. Sub-Meter Wide-Field Camera On-Orbit Geometric Calibration Model

The Jilin1-KF01B WF camera employs a line-push-broom imaging method. Its rigorous geometric positioning model conforms to the central projection collinearity equation. Therefore, based on the collinearity equation and the principles of optical satellite push-broom imaging [31], and incorporating auxiliary data such as GPS measurements and satellite attitude measurements during the imaging process, a rigorous geometric positioning model has been constructed, as shown in Equation (1).

$$\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]={\left[\begin{array}{c}{X}_s\\ Y_s\\ Z_s\end{array}\right]}_t+m{\left(R_{J2000}^{WGS84}{R}_{body}^{J2000}\right)}_t\left[\begin{array}{c}{D}_x\\ D_y\\ D_z\end{array}+\begin{array}{c}{d}_x\\ d_y\\ d_z\end{array}+R_{camera}^{body}\left[\begin{array}{c}x-x_0\\ y-y_0\\ f\end{array}\right]\right]$$

(1)

In the above collinearity equation, ${\left[\begin{array}{ccc}{X}_s& Y_s& Z_s\end{array}\right]}_t^T$ represents the position vector of the GPS phase center at time t in the WGS84 coordinate system, ${\left(R_{J2000}^{WGS84}\right)}_t$ donates the rotation matrix from the J2000 coordinate system to the WGS84 coordinate system at time t, ${\left(R_{body}^{J2000}\right)}_t$ donates the rotation matrix from the satellite body coordinate system to the J2000 coordinate system at time t, ${\left[\begin{array}{ccc}{D}_x& D_y& D_z\end{array}\right]}^T$ is the coordinate of the GPS phase center in the satellite body coordinate system, ${\left[\begin{array}{ccc}{d}_x& d_y& d_z\end{array}\right]}^T$ is the translation from the camera coordinate system origin to the satellite body coordinate system, $R_{camera}^{body}$ donates the rotation matrix from the camera coordinate system to the satellite body coordinate system. In Equation ${\left[\begin{array}{ccc}x-x_0& y-y_0& -f\end{array}\right]}^T$, ${\left(\begin{array}{cc}x& y\end{array}\right)}^T$ represents the coordinates of the current image point, ${\left(\begin{array}{cc}{x}_0& y_0\end{array}\right)}^T$ is the coordinates of the camera’s principal point. $f$ is the camera’s focal length. ${\left[\begin{array}{ccc}X& Y& Z\end{array}\right]}^T$ represents the position vector of the point ${\left(\begin{array}{cc}x& y\end{array}\right)}^T$ in the WGS84 coordinate system.

The Jilin1-KF01B satellite, equipped with a sub-meter wide-field camera, has a complex imaging chain, leading to multiple geometric errors. These include camera installation errors, timing measurement errors, satellite attitude observation errors, GPS measurement errors, GPS eccentricity errors, image point errors due to internal camera distortions, and random errors introduced by the satellite’s dynamic motion during imaging. Current studies indicate that these errors are difficult to fully isolate. Therefore, on-orbit geometric calibration is divided into calibration of exterior orientation elements and interior orientation elements [32].

The essence of exterior orientation element calibration is to establish a compensation model for equipment installation errors and systematic errors during the attitude and trajectory measurement process. This model considers the characteristic of satellite imaging being dynamic, specifically accounting for attitude drift during motion. Consequently, a bias matrix model that incorporates temporal characteristics is used as the exterior orientation element calibration model, as shown in Equation (2). The sub-meter wide-field camera achieves extensive data coverage through the use of multiple stitched detectors, each covering areas with different ground speeds. This requires precise electronic image shift matching and row timing matching. The sub-meter wide-field camera utilizes multiple IPS to conduct fine row timing and image shift matching. Therefore, in the geometric calibration process, the entire camera cannot use a traditional uniform compensation model calibration strategy. Instead, it should calibrate the exterior orientation element compensation matrices for the detectors of different IPS separately.

$$R_u^v=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]=\left[\begin{array}{ccc}\cos ({\phi }_u+{\phi }_{v}t)& 0& \sin ({\phi }_u+{\phi }_{v}t)\\ 0& 1& 0\\ -\sin ({\phi }_u+{\phi }_{v}t)& 0& \cos ({\phi }_u+{\phi }_{v}t)\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos ({\omega }_u+{\omega }_{v}t)& -\sin ({\omega }_u+{\omega }_{v}t)\\ 0& \sin ({\omega }_u+{\omega }_{v}t)& \cos ({\omega }_u+{\omega }_{v}t)\end{array}\right]\left[\begin{array}{ccc}\cos {\kappa }_u& -\sin {\kappa }_u& 0\\ \sin {\kappa }_u& \cos {\kappa }_u& 0\\ 0& 0& 1\end{array}\right]$$

(2)

The essence of the interior orientation element calibration model is to recover the true pointing direction of the imaging sensors in the satellite coordinate system. Therefore, the look-angle model can be used to express the interior orientation elements. The look-angle model, as depicted in Figure 1, indicates that the pointing angle of a pixel in the camera coordinate system can be represented by a specific formula, as shown in Equation (3).

$$\begin{array}{l}\tan {\psi }_x={\displaystyle \frac{x}{f}}\\ \tan {\psi }_y={\displaystyle \frac{y}{f}}\end{array}$$

(3)

This paper utilizes a polynomial-based sensor look-angle model, which involves fitting a polynomial to the look-angle of each sensor element on each CCD within the camera’s coordinate system. Based on this look-angle model, and incorporating a bias matrix model that accounts for temporal characteristics, a rigorous geometric model for the sub-meter wide-field camera is developed. This model is detailed in Equations (4) and (5).

$$\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]={\left[\begin{array}{c}{X}_s\\ Y_s\\ Z_s\end{array}\right]}_t+m{\left(R_{J2000}^{WGS84}{R}_{body}^{J2000}\right)}_t{R}_u^v{R}_{camera}^{body}\left[\begin{array}{c}\tan {\psi }_x\\ \tan {\psi }_y\\ 1\end{array}\right]$$

(4)

$$\left\{\begin{array}{c}\tan {\psi }_x=a_0+a_{1}s+a_2{s}^2+\cdots a_i{s}^i\\ \tan {\psi }_y=b_0+b_{1}s+b_2{s}^2+\cdots b_i{s}^i\end{array}\right.$$

(5)

2.2. GCGC Method

The imaging chain of the sub-meter wide-field camera on the Jilin1-KF01B satellite is relatively complex, with various types of geometric errors that are difficult to completely separate. Although system errors within and outside the camera can be compensated for through the calibration of interior and exterior orientation elements, there is a strong correlation between these elements. If solved simultaneously, it is challenging to obtain reliable results. Therefore, a step-by-step iterative strategy is needed to solve the on-orbit geometric calibration parameters for the interior and exterior orientation elements. Additionally, a high-precision dense matching algorithm is employed to acquire densely and evenly distributed control points, thereby increasing the number of observation equations in the calibration model and improving the accuracy of the on-orbit geometric calibration parameter solution.

As shown in Figure 2, the Jilin1-KF01B WF camera employs a mechanical staggered arrangement of multiple TDI CCDs, achieving an imaging swath width of 150 km. Given the characteristics of independent control of TDI CCD timing by multiple IPS, as well as the practical data processing and application requirements, the wide-field camera is logically divided into six groups of cameras, each with a 25 km swath width, designated as PMS01-06 (Panchromatic and Multispectral Sensor). Within each logical camera group, the TDI CCDs are controlled by the same IPS for uniform image timing. In the GCGC method, the on-orbit geometric calibration is conducted separately for each logical camera unit.

The GCGC method focuses on two key aspects of geometric positioning consistency. One is the geometric positioning consistency in the overlapping areas between adjacent CCDs, and the other is the geometric positioning consistency between panchromatic data and multispectral data. The mechanical staggered arrangement of the TDI CCDs means that adjacent CCDs capture images of the same ground target under different times, different satellite attitudes, and viewing angles. Additionally, the overlapping regions between adjacent CCDs exhibit different distortion coefficients within the optical system. These factors significantly increase the difficulty of seamless internal field-of-view stitching between images from adjacent CCDs. To address the issue of misalignment in the stitching of images from the same region captured by adjacent CCDs, an on-orbit geometric calibration method with additional inter-chip geometric positioning consistency constraints is employed. The image captured by adjacent CCDs under different conditions should have consistent object-space geographic coordinates for the same region. By incorporating geometric positioning consistency constraints between image segments into the in-orbit geometric calibration model, the accuracy of interior orientation element calibration coefficients between adjacent CCDs is enhanced, enabling seamless stitching of images between adjacent CCDs. Based on this constraint, GCGC will perform dense matching of the same region’s image captured by adjacent CCDs during the calibration process to obtain inter-chip tie points. However, the image coordinates of tie points on different ground objects do not exhibit a uniform translation relationship but instead dynamically change with terrain fluctuations. Taking mountainous areas as an example (as shown in the Figure 3), it can be seen that the image row coordinate offset changes dynamically over a large range. Therefore, the GCGC algorithm designs a tie point matching method based on geometric positioning constraints for connection point extraction. Therefore, the GCGC algorithm designs a tie point matching method based on geometric positioning constraints for connection point extraction. As shown in Figure 4, first, a rigorous geometric model is established for each detector to solve for the conversion between image point coordinates and ground point coordinates, as well as the reverse conversion from ground point coordinates to image point coordinates. Then, based on geometric constraints, the algorithm extracts the overlapping region between adjacent detectors by selecting a series of points along the edges of the detectors. For each point, a rigorous geometric positioning model is established. By combining Digital Elevation Model (DEM) data, the coordinate transformation between adjacent detectors is realized according to $\left(x_L y_L\right)\to \left(X Y Z\right)\to \left(x_R^0 y_R^0\right)$, where $x_L$ and $y_L$ are the column and row coordinates of the image point in the image coordinate system of detector 1, and X, Y, and Z are the object space coordinate system corresponding to the image point. The $x_R^0$ and $y_R^0$ are the calculated column and row coordinates of the image point in the image coordinate system of detector 2. The mean value of the results for the point set is taken to obtain the along-track misalignment and cross-track overlap, thus extracting the overlapping region images. The feature points are extracted from the overlapping region images of detector 1, and these feature points are projected onto the overlapping region images of detector 2 through a rigorous geometric model to obtain the initial positioning coordinates. Finally, tie points are obtained through Fourier matching and least squares matching.

These tie points’ geographic coordinates are then calculated using rigorous geometric models of CCD respectively. These geographic coordinates are biased calculated values; therefore, based on the geometric positioning consistency condition, the average of the coordinates is taken as the object-space geographic coordinate to participate in the calibration coefficient calculation. These geographic coordinates are biased calculated values; therefore, based on the geometric positioning consistency condition, the coordinates ${\left[\begin{array}{ccc}{X}_i& Y_i& Z_i\end{array}\right]}^T={\left[\begin{array}{ccc}{\displaystyle \frac{X_{ccd(i)}+X_{ccd(i+1)}}{2}}& {\displaystyle \frac{Y_{ccd(i)}+X_{ccd(i+1)}}{2}}& {\displaystyle \frac{Z_{ccd(i)}+Z_{ccd(i+1)}}{2}}\end{array}\right]}^T$ is taken as the object-space geographic coordinate to participate in the calibration coefficient calculation.

Panchromatic and multispectral data share the same compensation model for installation errors, attitude, and orbit measurement errors, allowing them to be compensated using a unified set of exterior orientation calibration parameters. However, since different spectral bands are arranged at different positions on the focal plane, they each have independent interior orientation elements that require separate calibration. The panchromatic achieves its interior orientation calibration parameters and exterior orientation calibration parameters through absolute calibration. Considering that different spectral bands with the same resolution capture images of the same ground object from different imaging angles and attitudes due to their different positions on the focal plane, it is challenging to achieve high spectral band registration accuracy if each band is independently calibrated with control data. Therefore, the red spectral band can be selected as the reference band for absolute calibration with control data, while the other multispectral bands can be calibrated relative to the reference band using high-precision dense matching to obtain control points for solving the interior orientation calibration parameters.

Traditionally, multispectral reference band calibration has often used the same calibration strategy as the panchromatic band, where calibration is performed with control data. However, the resolution of multispectral data are often much lower than that of control data, and there is typically a time discrepancy between the acquisition of multispectral data and control data. Factors such as changes in ground objects and variations in solar elevation angle can lead to significant grayscale differences between images, making it difficult to match the multispectral reference band with control data, thereby reducing the number of matching points and the accuracy of the calibration parameters. Furthermore, the calibration process typically does not account for the geometric positioning consistency between the panchromatic and multispectral data. Poor geometric consistency between panchromatic and multispectral data can lead to color overflow issues in subsequent fusion products, necessitating additional matching between panchromatic and multispectral data during the fusion process. This increases the difficulty of fusion processing and places higher demands on the texture of the ground objects in the data, making the process time-consuming and labor-intensive.

To address these issues, the already calibrated panchromatic data can be used as control data for the interior parameter calibration of the multispectral reference band. This approach resolves the time discrepancy between multispectral data and traditional digital orthophoto map (DOM) control data, while also improving the geometric positioning consistency between panchromatic and multispectral data. However, differences in spectral response intensity, illumination intensity, and resolution between the panchromatic and multispectral reference band data pose challenges in extracting high-precision dense control points. To overcome this, the calibration algorithm proposed in this study uses simulated reference band data generated by the spectral correlation imaging simulation technique [30] based on the calibrated panchromatic data as control data for the interior parameter calibration of the multispectral reference band.

The detailed comparison of panchromatic data, original red band data, and simulated red (S-Red) band data at the same sampling resolution for three typical land cover types—urban areas, farmland, and mountainous regions—is shown in Figure 5. The comparison reveals that the original red band data and simulated red band data are closer in terms of gray-level distribution and clarity. This characteristic can increase the number and accuracy of matching points between the two, thereby enhancing calibration precision. A quantitative analysis of the three groups of images was conducted using four metrics: mean square error (MSE), histogram correlation (HC), blur value [33,34], and energy concentration [35]. MSE and HC are used to assess the gray-level distribution between the red band and panchromatic band, as well as between the red band and simulated red band. A smaller MSE indicates a closer gray-level distribution, while the HC closer to 1 signifies a higher correlation. The concentration blur value of and energy concentration are used to evaluate data clarity, the closer the values, the more similar the clarity. As shown in

Table 3. Statistical table of similarity evaluation parameters for typical land cover images.

Land Cover Types	Comparison Group	MSE	HC	Blur Value			Energy Concentration
				PAN	Red	S-Red	PAN	Red	S-Red
Urban areas	Red and PAN	0.0121	0.9630	0.340	0.781	0.767	0.963	0.087	0.127
	Red and S-Red	0.0017	0.9870
Farmland areas	Red and PAN	0.0103	0.8960	0.395	0.520	0.587	0.869	0.538	0.431
	Red and S-Red	0.0015	0.9837
Mountainous areas	Red and PAN	0.0283	0.9043	0.326	0.479	0.526	0.951	0.626	0.501
	Red and S-Red	0.0022	0.9868

, the gray-level distribution and clarity of the red band and simulated red band are relatively close, whereas there are significant differences between the red band and the panchromatic band.

2.3. GCGC Process

The on-orbit geometric calibration process for the Jilin1-KF01B WF camera is shown in Figure 6 and Figure 7.

Absolute geometric calibration process for the panchromatic band using a Stepwise iterative method with additional inter-chip geometric positioning consistency constraints. The algorithm flow is shown in Algorithm 1:

Algorithm 1 Absolute geometric calibration process for the panchromatic band

Input:   Orthorectified image data (DOM) of the calibration field;   Digital Elevation Model (DEM) of the calibration field;   Panchromatic data to be calibrated; Output:   Panchromatic on-orbit geometric calibration parameters; 1.    Initial Model Construction: Use laboratory geometric calibration coefficients to construct the initial on-orbit geometric calibration model; 2.    Calibration Region Selection and GCPs extraction: Select a calibration region within the image. Apply an image matching algorithm to perform high-precision dense matching between the image to be calibrated and control data, obtaining N control points uniformly distributed across the region. The control points have WGS84 geocentric Cartesian coordinates ${\left[\begin{array}{ccc}{X}_i& Y_i& Z_i\end{array}\right]}^T$ and corresponding image coordinates ${\left[\begin{array}{cc}{x}_i& y_i\end{array}\right]}^T$, $i=1,2,3,\dots ,N$; 3.    Matching Adjacent CCDs: For the same target region captured by adjacent CCDs within the calibration region, perform high-precision dense matching using an image matching algorithm, obtaining M control points with WGS84 geocentric Cartesian coordinates ${\left[\begin{array}{ccc}{X}_i& Y_i& Z_i\end{array}\right]}^T$ and corresponding image coordinates ${\left[\begin{array}{cc}(x_{Li}& y_{Li}\end{array}),\begin{array}{cc}(x_{Ri}& y_{Ri}\end{array})\right]}^T$, $i=1,2,3,\dots ,M$; 4.    Equation Transformation: Transform Equation (4) into Equation (6) $${\left(R_{J2000}^{WGS84}{R}_{body}^{J2000}\right)}_t^{-1}{\left[\begin{array}{c}X-X_s\\ Y-Y_s\\ Z-Z_s\end{array}\right]}_t=m{R}_u^v{R}_{camera}^{body}\left[\begin{array}{c}\tan {\psi }_x\\ \tan {\psi }_y\\ 1\end{array}\right]$$ (6) 5.    Simultaneously, let ${\left[\begin{array}{c}\tilde{X}\\ \tilde{Y}\\ \tilde{Z}\end{array}\right]}_t={\left(R_{J2000}^{WGS84}{R}_{body}^{J2000}\right)}_t^{-1}{\left[\begin{array}{c}X-X_s\\ Y-Y_s\\ Z-Z_s\end{array}\right]}_t$ and $\left[\begin{array}{c}\tilde{x}\\ \tilde{y}\\ \tilde{z}\end{array}\right]=R_{camera}^{body}\left[\begin{array}{c}\tan {\psi }_x\\ \tan {\psi }_y\\ 1\end{array}\right]$ in combination with Equation (2), derive Equation (7): $$\left\{\begin{array}{c}{f}_x={\displaystyle \frac{a_1\tilde{\mathrm{x}}+a_2\tilde{y}+a_3\tilde{z}}{c_1\tilde{x}+c_2\tilde{y}+c_3\tilde{z}}}-{\displaystyle \frac{\tilde{X}}{\tilde{Z}}}\\ f_y={\displaystyle \frac{b_1\tilde{x}+b_2\tilde{y}+b_3\tilde{z}}{c_1\tilde{x}+c_2\tilde{y}+c_3\tilde{z}}}-{\displaystyle \frac{\tilde{Y}}{\tilde{Z}}}\end{array}\right.$$ (7) 6.    Exterior Calibration Coefficient Estimation: Treat the current interior calibration coefficients as “true values.” Linearize Equation (7) based on the on-orbit geometric calibration model, establish the error equations, solve for the exterior calibration coefficients ${\left[\begin{array}{cccc}{\phi }_u& {\omega }_u& {\kappa }_u& \begin{array}{cc}{\phi }_v& {\omega }_v\end{array}\end{array}\right]}^T$, and update the geometric calibration model; 7.    Interior Calibration Coefficient Estimation: Treat the estimated exterior calibration coefficients as “true values.” Linearize Equation (7) based on the on-orbit geometric calibration model, establish the error equations, solve for the interior calibration coefficients, and update the geometric calibration model; 8.    Iterative Calculation: Repeat steps (6) and (7) until the calibration coefficients for the interior and exterior orientation elements converge and stabilize, at which point the iterative calculation can stop; 9.    Model Parameter Update: Use the calculated calibration coefficients for the interior and exterior orientation elements to update the geometric positioning model parameters for the panchromatic band.

Geometric calibration process for the multispectral reference band using a stepwise iterative method with additional inter-chip geometric positioning consistency constraints. The algorithm flow is shown in Algorithm 2:

Algorithm 2 Geometric calibration process for the multispectral reference band

Input:   Panchromatic on-orbit geometric calibration parameters;   Panchromatic data;   Digital Elevation Model (DEM) of the calibration field;   Multispectral reference band data to be calibrated; Output:   Multispectral reference band on-orbit geometric calibration parameters 1.    Reference Band Selection: Choose one band from the blue, green, or red bands as the reference band for multispectral calibration. In this paper, the red band is selected as the reference; 2.    Simulated Data Generation: Based on the selected reference band, use the already calibrated panchromatic image data to generate simulated reference band data using a band correlation method; 3.    Geometric Correction: Perform geometric correction on the simulated reference band data to obtain the control data for the simulated reference band. 4.    Initial Model Construction: Construct the initial on-orbit geometric calibration model; 5.    Calibration Region Selection and GCPs extraction: Select a calibration region within the image. Apply an image matching algorithm to perform high-precision dense matching between the image to be calibrated and the control data for the simulated reference band, obtaining N control points with WGS84 geocentric Cartesian coordinates ${\left[\begin{array}{ccc}{X}_i& Y_i& Z_i\end{array}\right]}^T$ and corresponding image coordinates ${\left[\begin{array}{cc}{x}_i& y_i\end{array}\right]}^T$, $i=1,2,3,\dots ,N$; 6.    Matching Adjacent CCDs: For the same target region captured by adjacent CCDs within the calibration region, perform high-precision dense matching using an image matching algorithm, obtaining M control points with WGS84 geocentric Cartesian coordinates ${\left[\begin{array}{ccc}{X}_i& Y_i& Z_i\end{array}\right]}^T$ and corresponding image coordinates ${\left[\begin{array}{cc}(x_{Li}& y_{Li}\end{array}),\begin{array}{cc}(x_{Ri}& y_{Ri}\end{array})\right]}^T$, $i=1,2,3,\dots ,M$; 7.    Interior Calibration Coefficient Estimation: Treat the exterior calibration coefficients from the panchromatic band calibration as "true values." Use the on-orbit geometric calibration model to solve for the interior calibration coefficients and update the geometric calibration model; 8.    Iterative Calculation: Repeat step (7) until the calibration coefficients for the interior orientation elements converge and stabilize, at which point the iterative calculation can stop; 9.    Model Parameter Update: Use the calculated interior orientation element calibration coefficients to update the geometric positioning model parameters for the corresponding multispectral reference band.

The relative geometric calibration process for non-reference multispectral bands follows similar steps as for the reference band. During this process, use the already calibrated reference band data as control data and sequentially execute steps (4) through (9).

3. Results and Discussion

3.1. Experimental Data

To verify the correctness of the calibration model and method presented in this paper, an on-orbit geometric calibration experiment was conducted on the Jilin1-KF01B WF camera. The experimental data for on-orbit geometric calibration is located in the South Asian region (34.05°N–34.27°N, 64.13°E–66.57°E), and the data acquisition information is shown in the “On-orbit Geometric Calibration” section of

Table 4. Experimental data information.

Experiment	ID	Imaging Time	Image Angle (°)
			Roll	Pitch	Yaw
On-orbit Geometric Calibration	1	2021-07-28	0.00	0.00	3.21
Basic Products Geometric Accuracy	1	2021-08-17	3.00	−0.18	3.40
	2	2021-09-02	0.00	0.00	3.07
	3	2021-09-12	0.00	0.00	3.06
	4	2021-12-23	3.00	−0.17	3.17
	5	2022-02-18	−3.00	0.17	3.17
	6	2022-03-06	3.00	−0.15	2.93
	7	2022-04-09	2.99	−0.18	3.52
	8	2022-05-28	1.81	−0.12	3.69
	9	2022-08-20	−3.00	0.16	3.13
	10	2022-11-17	−0.14	0.01	3.27
	11	2022-12-31	0.00	0.00	3.29
	12	2023-01-20	−2.99	0.18	3.39
	13	2023-03-29	3.00	−0.17	3.21
	14	2023-07-05	−1.00	0.06	3.28
	15	2023-08-16	−3.00	0.17	3.17
	16	2023-10-27	3.00	−0.16	3.08
	17	2024-01-28	−0.50	0.03	3.20
	18	2024-02-17	3.00	−0.18	3.36
	19	2024-05-13	2.99	−0.18	3.41
	20	2024-06-09	1.20	−0.07	3.23

and the calibration data illustrated in Figure 8. The on-orbit geometric calibration coefficients obtained were used for basic data production. The geometric accuracy of the basic data was comprehensively evaluated through experiments on geometric accuracy, positioning accuracy with GCPs, and geometric consistency across multiple temporal datasets. The relevant data information of geometric accuracy is provided in “Basic Products Geometric Accuracy” section of Table 4. The data acquisition period spans from 17 August 2021, to 9 June 2024, and the geographic coordinates of the data center point are shown in

Table A1. The statistics of positioning accuracy without GCPs and the band-to-band registration accuracy.

ID	Group ID	Central Longitude and Latitude		Positioning Accuracy without GCPs (meters)		Band-to-Band Registration Accuracy after Calibration (pixels)
		Longitude	Latitude	Before Calibration	After Calibration	Red-Blue		Red-Green		Red-NIR
						Cross-Track	Along-Track	Cross-Track	Along-Track	Cross-Track	Along-Track
1	PMS01	−49.8695	−25.7476	925.17	6.83	0.058	0.061	0.053	0.056	0.194	0.216
	PMS02	−49.2812	−24.3115	915.24	12.91	0.061	0.064	0.046	0.046	0.156	0.171
	PMS03	−49.0342	−24.3603	900.86	16.17	0.057	0.060	0.045	0.047	0.165	0.181
	PMS04	−48.7376	−24.1929	891.28	18.12	0.061	0.066	0.049	0.052	0.186	0.212
	PMS05	−48.4869	−24.2407	931.90	17.09	0.064	0.070	0.047	0.050	0.165	0.183
	PMS06	−48.2339	−24.2853	912.43	17.05	0.042	0.044	0.029	0.029	0.113	0.119
2	PMS01	−101.7210	37.3772	903.52	7.06	0.027	0.027	0.025	0.026	0.101	0.109
	PMS02	−101.4400	37.3290	922.18	4.33	0.030	0.030	0.027	0.027	0.109	0.114
	PMS03	−101.1600	37.2804	928.43	7.06	0.028	0.027	0.025	0.025	0.111	0.114
	PMS04	−100.8800	37.2311	921.17	8.03	0.029	0.028	0.026	0.025	0.108	0.109
	PMS05	−100.6000	37.1810	904.76	8.49	0.031	0.030	0.028	0.027	0.113	0.117
	PMS06	−100.3210	37.1305	913.42	8.29	0.030	0.028	0.026	0.023	0.077	0.071
3	PMS01	111.7500	38.2888	891.24	14.03	0.068	0.072	0.059	0.062	0.166	0.167
	PMS02	112.0360	38.2393	907.50	19.80	0.062	0.064	0.051	0.051	0.152	0.150
	PMS03	112.3220	38.1901	878.65	23.69	0.045	0.048	0.038	0.038	0.163	0.176
	PMS04	112.6070	38.1402	896.13	21.34	0.052	0.054	0.043	0.043	0.168	0.163
	PMS05	112.8930	38.0895	892.88	22.23	0.051	0.056	0.041	0.043	0.156	0.161
	PMS06	113.1770	38.0381	904.82	22.39	0.048	0.058	0.038	0.042	0.140	0.142
4	PMS01	−85.9760	33.9310	903.85	15.65	0.084	0.082	0.072	0.068	0.171	0.167
	PMS02	−85.6530	34.0943	916.65	16.59	0.078	0.081	0.069	0.067	0.186	0.191
	PMS03	−85.4410	33.8219	918.02	13.75	0.075	0.075	0.070	0.064	0.189	0.189
	PMS04	−85.0026	34.4065	905.62	12.07	0.082	0.082	0.070	0.067	0.171	0.164
	PMS05	−85.0134	33.2838	910.41	17.17	0.076	0.076	0.063	0.059	0.178	0.180
	PMS06	−84.3404	34.7174	908.57	21.21	0.080	0.081	0.066	0.062	0.164	0.154
5	PMS01	56.6032	36.7781	965.95	39.10	0.032	0.032	0.024	0.024	0.061	0.061
	PMS02	57.2162	37.9843	970.17	38.21	0.053	0.057	0.031	0.033	0.074	0.072
	PMS03	57.5006	37.9399	957.17	40.08	0.056	0.059	0.033	0.033	0.073	0.066
	PMS04	57.7809	37.8902	963.84	40.06	0.053	0.059	0.033	0.036	0.065	0.063
	PMS05	57.7198	36.6066	951.42	35.43	0.028	0.027	0.022	0.022	0.041	0.048
	PMS06	57.7097	35.5238	959.25	28.73	0.039	0.038	0.028	0.026	0.027	0.026
6	PMS01	115.9110	40.7808	923.60	5.39	0.057	0.062	0.039	0.042	0.082	0.083
	PMS02	115.9000	39.7075	919.35	9.77	0.051	0.055	0.035	0.039	0.076	0.074
	PMS03	116.5490	40.8880	932.99	10.34	0.061	0.062	0.040	0.040	0.106	0.108
	PMS04	116.3470	39.1887	950.56	9.27	0.069	0.077	0.048	0.055	0.072	0.067
	PMS05	116.7550	39.5424	925.59	5.20	0.075	0.083	0.049	0.054	0.078	0.080
	PMS06	117.4260	40.7341	988.08	8.67	0.055	0.056	0.037	0.037	0.114	0.114
7	PMS01	113.7650	29.7692	944.53	8.24	0.066	0.066	0.074	0.067	0.202	0.191
	PMS02	113.1380	25.9369	931.09	8.89	0.067	0.068	0.081	0.073	0.204	0.193
	PMS03	113.5780	26.7168	954.69	12.91	0.076	0.077	0.087	0.082	0.198	0.186
	PMS04	114.5390	29.6273	978.85	15.41	0.056	0.057	0.055	0.053	0.196	0.201
	PMS05	114.7990	29.5760	1005.45	12.15	0.057	0.057	0.057	0.051	0.146	0.132
	PMS06	115.0610	29.5277	991.39	11.14	0.064	0.064	0.066	0.059	0.168	0.149
8	PMS01	−49.9453	−8.6080	949.21	13.24	0.070	0.074	0.055	0.060	0.182	0.209
	PMS02	−50.2669	−11.3475	954.35	16.11	0.073	0.081	0.051	0.057	0.158	0.181
	PMS03	−49.4169	−8.2815	938.43	12.37	0.092	0.101	0.066	0.069	0.149	0.148
	PMS04	−49.4422	−9.5728	945.06	14.26	0.072	0.076	0.059	0.059	0.147	0.148
	PMS05	−48.9681	−8.3691	942.80	13.37	0.106	0.118	0.066	0.068	0.153	0.153
	PMS06	−48.7429	−8.4126	936.16	11.37	0.103	0.114	0.064	0.063	0.153	0.153
9	PMS01	106.4080	39.2100	959.16	7.55	0.046	0.042	0.038	0.032	0.152	0.145
	PMS02	106.5820	38.7373	949.88	7.64	0.065	0.069	0.056	0.057	0.192	0.186
	PMS03	107.7310	41.6466	937.45	8.09	0.034	0.033	0.026	0.025	0.081	0.073
	PMS04	108.0290	41.5941	946.14	8.58	0.040	0.039	0.029	0.028	0.106	0.101
	PMS05	107.6980	39.4598	939.56	3.88	0.037	0.038	0.031	0.031	0.109	0.118
	PMS06	108.3000	40.4499	941.41	7.56	0.033	0.033	0.023	0.024	0.072	0.078
10	PMS01	113.0890	31.1257	942.86	10.47	0.063	0.065	0.059	0.059	0.195	0.206
	PMS02	114.2620	34.6828	947.98	4.68	0.089	0.099	0.077	0.084	0.180	0.187
	PMS03	113.6700	31.2467	947.46	12.30	0.074	0.073	0.072	0.067	0.187	0.173
	PMS04	114.6980	34.1625	936.93	5.92	0.068	0.071	0.054	0.055	0.175	0.155
	PMS05	115.3740	35.5943	928.93	10.70	0.066	0.070	0.053	0.055	0.150	0.137
	PMS06	115.2430	34.0617	923.47	8.54	0.073	0.081	0.056	0.058	0.176	0.158
11	PMS01	118.9490	34.4785	909.25	15.04	0.048	0.054	0.041	0.046	0.111	0.106
	PMS02	119.4830	35.4579	928.73	10.08	0.052	0.063	0.039	0.049	0.108	0.108
	PMS03	118.7450	31.5122	915.27	8.03	0.056	0.057	0.054	0.054	0.182	0.190
	PMS04	119.7440	34.3367	918.89	10.32	0.051	0.050	0.045	0.041	0.177	0.166
	PMS05	120.0090	34.2875	913.93	12.41	0.050	0.049	0.048	0.041	0.179	0.146
	PMS06	120.1070	33.6233	912.32	9.87	0.057	0.057	0.046	0.044	0.147	0.145
12	PMS01	117.3390	33.3438	937.33	7.01	0.053	0.054	0.057	0.055	0.157	0.141
	PMS02	117.6590	33.5040	919.27	8.63	0.049	0.053	0.054	0.055	0.159	0.145
	PMS03	117.9310	33.4745	926.65	7.94	0.047	0.052	0.049	0.050	0.148	0.136
	PMS04	118.1960	33.4255	907.80	10.29	0.050	0.051	0.047	0.046	0.152	0.142
	PMS05	118.4640	33.3930	920.41	11.95	0.034	0.037	0.038	0.040	0.110	0.112
	PMS06	118.7260	33.3436	925.97	9.78	0.025	0.025	0.024	0.024	0.069	0.071
13	PMS01	118.1850	34.8946	863.36	6.33	0.081	0.086	0.059	0.064	0.169	0.157
	PMS02	118.4500	34.8483	922.37	8.65	0.078	0.083	0.056	0.058	0.191	0.167
	PMS03	118.7160	34.7966	928.66	11.13	0.074	0.079	0.051	0.055	0.202	0.184
	PMS04	118.9830	34.7488	936.67	10.68	0.076	0.090	0.058	0.071	0.196	0.190
	PMS05	119.2510	34.6943	944.59	10.57	0.070	0.080	0.055	0.062	0.149	0.148
	PMS06	119.5210	34.6449	968.92	13.16	0.051	0.052	0.042	0.043	0.139	0.148
14	PMS01	118.2360	33.4411	915.86	13.18	0.068	0.071	0.075	0.071	0.219	0.210
	PMS02	118.6570	34.0131	947.96	10.95	0.087	0.095	0.076	0.078	0.218	0.212
	PMS03	118.9230	33.9701	871.01	10.32	0.079	0.084	0.074	0.073	0.227	0.219
	PMS04	119.1340	33.7160	920.24	11.47	0.089	0.092	0.076	0.075	0.241	0.235
	PMS05	119.3990	33.6744	911.57	13.08	0.073	0.074	0.072	0.070	0.212	0.211
	PMS06	119.6060	33.4203	928.29	13.72	0.068	0.072	0.071	0.071	0.217	0.219
15	PMS01	118.4560	37.2542	865.14	16.86	0.066	0.077	0.058	0.064	0.189	0.175
	PMS02	118.7330	37.2055	921.19	16.78	0.057	0.065	0.050	0.055	0.173	0.162
	PMS03	119.0150	37.1816	914.49	11.91	0.051	0.055	0.040	0.045	0.179	0.191
	PMS04	119.2880	37.1324	913.75	14.63	0.039	0.042	0.032	0.034	0.150	0.153
	PMS05	119.5670	37.1044	923.79	14.36	0.045	0.048	0.039	0.041	0.168	0.177
	PMS06	119.8360	37.0546	918.19	14.73	0.072	0.087	0.060	0.071	0.186	0.171
16	PMS01	117.3120	35.8074	889.44	10.53	0.080	0.083	0.064	0.066	0.173	0.165
	PMS02	118.9450	40.6145	902.49	7.64	0.086	0.095	0.059	0.063	0.105	0.108
	PMS03	119.2260	40.5571	903.80	8.96	0.082	0.090	0.056	0.060	0.135	0.140
	PMS04	119.5130	40.5128	888.48	8.35	0.084	0.092	0.057	0.060	0.132	0.135
	PMS05	119.8010	40.4609	893.51	9.57	0.096	0.107	0.064	0.068	0.165	0.177
	PMS06	120.0880	40.4098	907.35	11.64	0.089	0.097	0.064	0.067	0.171	0.166
17	PMS01	118.6950	36.7780	958.36	7.05	0.122	0.129	0.082	0.085	0.152	0.172
	PMS02	118.2070	33.7786	956.61	7.27	0.072	0.077	0.053	0.056	0.159	0.149
	PMS03	118.4770	33.7334	910.51	6.35	0.061	0.066	0.049	0.051	0.196	0.200
	PMS04	118.7460	33.6844	923.00	6.42	0.069	0.073	0.055	0.053	0.153	0.141
	PMS05	119.0180	33.6436	932.99	4.29	0.077	0.081	0.056	0.057	0.186	0.188
	PMS06	119.2860	33.5937	935.50	7.39	0.061	0.063	0.048	0.050	0.170	0.168
18	PMS01	118.1720	36.9991	874.04	7.05	0.091	0.112	0.059	0.077	0.123	0.119
	PMS02	117.3240	32.7071	905.22	10.18	0.082	0.091	0.068	0.073	0.202	0.193
	PMS03	117.5730	32.6353	886.60	9.14	0.069	0.075	0.056	0.060	0.193	0.206
	PMS04	117.8300	32.5891	903.52	10.10	0.072	0.078	0.060	0.063	0.160	0.155
	PMS05	119.0780	36.1847	901.45	9.87	0.072	0.081	0.051	0.059	0.111	0.101
	PMS06	118.3410	32.4639	897.44	8.00	0.079	0.085	0.059	0.062	0.189	0.189
19	PMS01	117.1290	34.5434	912.84	19.90	0.062	0.067	0.055	0.058	0.193	0.189
	PMS02	117.5050	34.9116	898.89	14.00	0.077	0.084	0.063	0.067	0.203	0.197
	PMS03	117.7740	34.8587	910.04	12.49	0.075	0.080	0.060	0.062	0.196	0.184
	PMS04	118.0450	34.8105	889.97	12.46	0.085	0.093	0.068	0.073	0.210	0.199
	PMS05	118.2020	34.3356	899.41	14.76	0.072	0.079	0.063	0.063	0.214	0.201
	PMS06	118.5890	34.7046	905.39	17.39	0.074	0.075	0.059	0.057	0.221	0.195
20	PMS01	118.0990	34.6793	894.49	6.59	0.083	0.088	0.067	0.070	0.200	0.182
	PMS02	118.3580	34.6340	895.90	4.45	0.068	0.069	0.052	0.052	0.159	0.138
	PMS03	118.6150	34.5860	893.38	7.00	0.064	0.074	0.045	0.049	0.189	0.184
	PMS04	118.8740	34.5396	929.68	5.36	0.066	0.076	0.054	0.056	0.190	0.189
	PMS05	119.1330	34.4890	902.90	5.71	0.068	0.083	0.054	0.060	0.142	0.138
	PMS06	119.3930	34.4414	918.47	6.49	0.065	0.078	0.056	0.057	0.184	0.196

. The geometric correction with GCPs experiment was conducted using 10 scenes data in Lanzhou (36.03°N, 103.73°E), Zhangye (38.93°N, 100.45°E), Jiuquan (39.74°N, 98.51°E), Jiayuguan (39.77°N, 98.27°E), Wuwei (37.93°N, 102.64°E), and Baiyin (36.54°N, 104.18°E) in Gansu Province. The multi-temporal data geometric positioning consistency experiment used 34 scenes of data distributed across Gansu (103.55°E–104.17°E, 35.73°N–36.35°N), Linyi (118.32°E–118.96°E, 35.63°N–35.07°N), Qingdao (120.08°E–120.64°E, 36.06°N–36.62°N), and Xinjiang (87.14°E–87.86°E, 44.24°N–43.65°N).

3.2. Calibration Results of Jilin1-KF01B WF Camera

Based on the TDI CCD grouping rules described earlier, the on-orbit geometric calibration was performed for the data of each band within the six groups using the GCGC method. The geometric positioning residuals after calibration are shown in

Table 5. The statistics of on-orbit calibration residual.

Group ID	Band	Geometric Positioning Residuals After Calibration (pixel)
		Column Direction			Row Direction
		Max	Min	RMSE	Max	Min	RMSE
PMS01	Pan	−0.366257	−0.000008	0.145391	−0.371304	0.000003	0.144718
	Blue	−0.564352	0.000002	0.126919	0.557150	−0.000001	0.129533
	Green	−0.530157	0.000001	0.102103	−0.533769	−0.000001	0.103027
	Red	0.264572	0.000000	0.100068	0.262534	−0.000003	0.098821
	NIR	0.534556	0.000001	0.086426	0.518925	0.000000	0.087164
PMS02	Pan	−0.801025	−0.000006	0.159698	0.441153	−0.000005	0.146380
	Blue	−0.528487	0.000002	0.100682	−0.607542	0.000001	0.096686
	Green	0.504580	0.000001	0.073226	−0.550179	0.000001	0.073398
	Red	−0.315732	−0.000006	0.132981	−0.278999	−0.000009	0.116255
	NIR	−0.547901	−0.000000	0.102177	−0.556096	0.000001	0.099198
PMS03	Pan	0.454667	0.000003	0.147296	0.496185	−0.000001	0.142020
	Blue	0.512209	0.000000	0.078841	0.546872	−0.000001	0.077803
	Green	−0.509249	0.000000	0.056408	−0.496459	−0.000000	0.058658
	Red	0.390876	−0.000012	0.141304	0.279552	0.000012	0.139158
	NIR	0.480456	−0.000000	0.063747	−0.519422	−0.000000	0.063199
PMS04	Pan	0.485560	−0.000000	0.145864	0.425948	−0.000000	0.143750
	Blue	−0.477041	−0.000000	0.054909	0.541585	−0.000001	0.049627
	Green	0.462559	−0.000000	0.037815	0.475999	−0.000000	0.035560
	Red	0.360066	0.000002	0.138781	0.336034	0.000004	0.139590
	NIR	0.490291	0.000003	0.058698	0.470296	0.000001	0.053294
PMS05	Pan	−0.486315	−0.000005	0.147013	−0.420472	−0.000005	0.139291
	Blue	0.538902	−0.000001	0.078787	0.550656	0.000001	0.075120
	Green	−0.538176	−0.000000	0.064479	−0.524089	0.000000	0.062171
	Red	0.370930	0.000005	0.139559	0.312009	−0.000013	0.137802
	NIR	0.533537	0.000000	0.064541	−0.531552	0.000001	0.060180
PMS06	Pan	−0.718997	−0.000004	0.151824	0.647674	−0.000000	0.151632
	Blue	−0.538045	−0.000001	0.091801	0.604879	0.000001	0.093526
	Green	−0.514802	−0.000001	0.068905	0.501747	0.000000	0.067017
	Red	0.357659	−0.000006	0.140000	0.383580	0.000005	0.139645
	NIR	−0.541270	−0.000001	0.090876	0.531835	−0.000001	0.094335

, the band-to-band registration accuracy for the multispectral bands is presented in

Table 6. The statistics of the band-to-band registration accuracy of calibration images.

Group ID	Band-to-Band Registration Accuracy (pixel)
	Reference Band	Test Band	Cross-Track Direction	Along-Track Direction
PMS01	Red	Blue	0.047369	0.042034
		Green	0.037828	0.033754
		NIR	0.099154	0.091168
PMS02	Red	Blue	0.042712	0.040210
		Green	0.031912	0.029589
		NIR	0.104993	0.092015
PMS03	Red	Blue	0.040804	0.035482
		Green	0.029230	0.025333
		NIR	0.093654	0.082061
PMS04	Red	Blue	0.040669	0.036317
		Green	0.027555	0.024206
		NIR	0.094967	0.080833
PMS05	Red	Blue	0.037810	0.034158
		Green	0.027813	0.023955
		NIR	0.065563	0.055794
PMS06	Red	Blue	0.049055	0.047041
		Green	0.034215	0.030350
		NIR	0.082013	0.069267

, and the image stitching between adjacent CCDs before and after calibration is shown in Figure 9. The calibration results indicate that due to the presence of non-linear geometric distortions in the original camera parameters, there was a significant misalignment between images captured by adjacent CCDs, which could not meet the high-precision geometric stitching requirements for multi-CCD images. After the on-orbit calibration of the exterior orientation elements, the installation errors in the geometric calibration model were corrected. The on-orbit calibration of the interior orientation elements with additional inter-chip geometric positioning consistency constraints effectively eliminated the internal non-linear distortions of the camera, achieving consistent geometric positioning accuracy across all CCD images and seamless stitching between adjacent CCDs. The RMSE (Root Mean Square Error) of the calibration residuals in both row and column directions for the entire spectrum was better than 0.16 pixels. Additionally, the relative calibration between multispectral bands eliminated geometric positioning errors between the bands, with a registration accuracy better than 0.1 pixels. The geometric positioning consistency between panchromatic and multispectral data are shown in

Table 7. The statistics of geometric positioning consistency accuracy between panchromatic and multispectral data for calibration data.

Group ID	Geometric Positioning Consistency (Multispectral Pixel)
	Column Direction RMSE	Row Direction RMSE	Plane RMSE
PMS01	0.105	0.133	0.169
PMS02	0.178	0.147	0.231
PMS03	0.147	0.144	0.206
PMS04	0.163	0.124	0.205
PMS05	0.143	0.138	0.199
PMS06	0.189	0.203	0.277

, where the RMSE for geometric positioning consistency in the column direction was better than 0.19 multispectral pixels, in the row direction better than 0.21 multispectral pixels, and the overall planar consistency better than 0.28 multispectral pixels. The calibration residuals for the red band based on panchromatic data and simulated red band data are shown in

Table 8. Statistical results of calibration residuals comparison for the red band based on panchromatic data and simulated red band data.

Group ID	Red Band Calibration Residual RMSE (Pixels)
	Panchromatic Data as Control Data				Simulated Red Data as Control Data
	xMax	xRMSE	yMAX	yRMSE	xMax	xRMSE	yMAX	yRMSE
PMS01	0.597	0.138	0.735	0.130	0.286	0.111	0.272	0.105
PMS02	0.816	0.214	0.564	0.145	0.316	0.133	0.279	0.116
PMS03	1.170	0.268	1.749	0.265	0.391	0.141	0.280	0.139
PMS04	0.869	0.248	1.281	0.289	0.360	0.139	0.336	0.140
PMS05	1.694	0.248	1.202	0.266	0.371	0.140	0.312	0.138
PMS06	0.827	0.259	1.357	0.280	0.358	0.140	0.384	0.140

. This demonstrates that using the simulated red band data, based on the calibrated panchromatic band, as control data for the multispectral reference band achieved high-precision geometric positioning consistency between the panchromatic and multispectral data.

3.3. Basic Products Geometric Accuracy Evaluation

The geometric accuracy of the calibration data used for solving on-orbit geometric calibration parameters was analyzed earlier. To ensure the reliability and applicability of the calibration parameter results, the geometric accuracy of basic product data from the Jilin1-KF01B satellite during its on-orbit operation was verified. The on-orbit calibration parameters obtained were used as input for the production of basic data products. For this study, 20 orbits of data were randomly selected from August 2021 to June 2024 to test the seamless stitching accuracy, RPC fitting accuracy, positioning accuracy without GCPs, band-to-band registration accuracy, and geometric positioning consistency between panchromatic and multispectral data for the sub-meter WF camera. The experimental data covered different regions of the world, with acquisition times and angles as shown in the “ Basic Products Geometric Accuracy “ section of Table 4. Additionally, 10 scenes covering PMS01-06 in Gansu Province were selected for geometric correction with GCPs experiments to evaluate the absolute positioning accuracy after control points correction. The acquisition times and angles are shown in

Table 9. The statistics of positioning accuracy with GCPs.

ID	Imaging Time	Image Angle (°)			Group ID	GCPs	Plane RMS (meter)
		Roll	Pitch	Yaw
LanZhou 1	2022-04-03	3.00	−0.16	3.06	PMS05	167	1.90
ZhangYe 1	2022-04-24	3.00	−0.16	3.07	PMS03	36	1.98
ZhangYe 2	2022-06-17	−3.00	0.16	2.99	PMS01	94	1.51
ZhangYe 3	2022-06-17	−3.00	0.16	2.98	PMS02	72	1.81
JiuQuan 1	2022-07-07	0.40	−0.02	3.00	PMS04	92	2.00
ZhangYe 4	2022-07-07	0.40	−0.02	3.00	PMS04	130	2.08
JiuQuan 2	2022-07-07	0.40	−0.02	3.00	PMS05	125	1.58
JiaYuguan 1	2023-04-08	−2.29	0.12	3.10	PMS05	133	1.63
WuWei 1	2023-09-12	−3.00	0.17	3.21	PMS06	88	2.09
BaiYin 1	2023-10-30	3.00	−0.17	3.17	PMS06	110	1.82

. Finally, multi-temporal data from Gansu, Linyi, Qingdao, and Xinjiang regions were used to verify the geometric positioning consistency accuracy across multiple periods using bundle adjustment. The acquisition times and angles are shown in the

Table 10. The statistics of geometric positioning consistency accuracy for multi-temporal data.

Experimental Area	Feature Type	Imaging Time	Image Angle (°)			ID	Group ID	OGPCA (meters)	NGPCA (meters)
			Roll	Pitch	Yaw
GanSu	Mountain	2024-02-11	3.00	−0.16	3.10	1	PMS03	5.521	1.350
		2024-04-10	−1.91	0.10	3.12	2	PMS03	5.408	1.435
		2021-12-22	1.06	−0.06	3.15	3	PMS03	5.078	1.295
		2023-03-04	2.99	−0.18	3.37	4	PMS06	4.937	1.490
		2023-03-04	2.99	−0.18	3.37	5	PMS05	5.490	1.227
		2023-08-27	3.00	−0.17	3.19	6	PMS05	5.815	0.847
		2023-08-27	3.00	−0.17	3.19	7	PMS05	5.981	1.565
		2023-10-30	3.00	−0.17	3.17	8	PMS01	7.479	1.565
		2023-07-16	3.00	−0.17	3.25	9	PMS02	10.546	1.932
LinYi	Plain	2023-03-29	3.00	−0.17	3.21	1	PMS02	3.957	1.293
		2023-03-29	3.00	−0.17	3.21	2	PMS02	3.279	1.377
		2024-01-28	−0.50	0.03	3.20	3	PMS03	7.096	1.878
		2024-01-28	−0.50	0.03	3.20	4	PMS02	3.522	0.933
		2023-08-16	−3.00	0.17	3.17	5	PMS02	3.980	1.153
		2024-02-17	3.00	−0.18	3.36	6	PMS04	4.321	1.237
		2024-02-17	3.00	−0.18	3.36	7	PMS04	3.501	1.263
		2024-05-13	2.99	−0.18	3.41	8	PMS06	7.597	1.499
QingDao	Plain	2023-05-23	0.00	0.00	3.27	1	PMS03	5.473	1.687
		2024-05-14	0.50	−0.03	3.39	2	PMS01	9.768	1.555
		2021-12-30	1.18	−0.06	3.08	3	PMS06	4.192	1.165
		2023-11-12	0.38	−0.02	3.24	4	PMS06	4.069	1.056
		2023-11-12	0.38	−0.02	3.24	5	PMS05	3.981	0.847
		2023-12-06	2.00	−0.11	3.09	6	PMS05	3.509	1.003
		2023-12-06	2.00	−0.11	3.09	7	PMS05	3.352	0.918
		2023-07-05	−1.00	0.05	3.12	8	PMS05	3.452	0.942
		2023-07-05	−1.00	0.05	3.12	9	PMS02	9.323	1.440
XinJiang	Mountain	2023-05-10	0.98	−0.05	3.00	1	PMS02	8.191	1.975
		2023-05-10	0.98	−0.05	3.00	2	PMS03	5.609	1.289
		2023-07-27	−1.00	0.05	2.75	3	PMS04	3.450	1.197
		2021-11-20	0.00	0.00	2.78	4	PMS01	6.392	0.968
		2022-04-06	0.00	0.00	3.07	5	PMS01	6.211	1.001
		2022-06-21	0.73	−0.04	2.75	6	PMS01	4.688	1.066
		2023-08-04	−0.31	0.02	2.89	7	PMS04	5.438	1.026
		2023-08-04	−0.31	0.02	2.89	8	PMS04	5.700	1.021

.

3.3.1. In-Scene Stitching Accuracy Evaluation of Basic Products

For PMS01-06, rigorous geometric models were constructed using the calibration coefficients for each detector, and matching points were obtained through image matching in the overlapping regions of adjacent detectors. The reprojection error RMSE of matching points for the same ground object between adjacent detectors was used to evaluate the stitching accuracy of the basic products. The acquisition times and angles of the experimental data are shown in the “ Basic Products Geometric Accuracy “ section of Table 4, and the statistical results are presented in Figure 10. The circular points in the figure represent the distribution of the RMSE of stitching accuracy for each scene of the grouped cameras. The outer contour lines in the figure illustrate the distribution of data RMSE, with wider contour line intervals indicating regions where the RMSE is concentrated along the y-axis, while narrower contour lines indicate areas with fewer data points along the y-axis. The figure shows that the rigorous geometric model constructed using the calibration coefficients achieved stitching accuracy better than 1 pixel in both row and column directions for all logical cameras, with an average RMSE better than 0.6 pixels, achieving sub-pixel seamless stitching between adjacent CCD images.

3.3.2. RPC Fitting Accuracy Evaluation of Basic Products

During basic product production, the rigorous geometric model was used to generate corresponding RPC coefficients, which serve as the geometric positioning model for basic products. The accuracy of these coefficients directly affects geometric positioning accuracy without and with GCPs of basic products. Satellites like IKONOS have achieved fitting accuracy RMSE better than 0.01 pixels [1]. In this study, the rigorous geometric model was used to calculate object-space virtual control points, which were projected into the image space using the RPC coefficients, and the errors in the X and Y directions were evaluated. The acquisition times and angles of the experimental data are shown in the “Basic Products Geometric Accuracy” section of Table 4. The distribution of RPC coefficient fitting accuracy to the rigorous geometric model data are shown in Figure 11, where the mean fitting accuracy in the X direction was better than 0.000025 pixels, and RMSE better than 0.000031 pixels, while in the Y direction, the mean fitting accuracy was better than 0.0002 pixels, and RMSE better than 0.00023 pixels. This fitting accuracy, better than 0.0003 pixels, indicates that the RPC coefficients accurately fit the rigorous geometric model based on the calibration coefficients, allowing the RPC geometric positioning accuracy of basic products to accurately reflect the original satellite positioning accuracy.

3.3.3. Positioning Accuracy Without GCPs and Band-to-Band Registration Accuracy Evaluation of Basic Products

Basic data achieves geometric positioning through RPC coefficients. By performing automatic dense matching between the experimental data and reference images, uniformly distributed match points were obtained, and the positioning results of these match points based on RPC coefficients were used to evaluate positioning accuracy without GCPs. Additionally, the band-to-band registration accuracy for multispectral images in the experimental data was assessed by automatically performing dense matching between the red band (used as the reference) and the blue, green, and near-infrared (NIR) bands, with the deviations in the along-track and cross-track directions of the match points being statistically analyzed. The acquisition times and angles of the experimental data are shown in the “Basic Products Geometric Accuracy “ section of Table 4, and the statistical results are presented in Table A1. Table A1 shows that after on-orbit geometric calibration, the positioning accuracy without GCPs of basic product data improved from around 1 km to within 20 m in most cases, and remained stable over time, with band-to-band registration accuracy better than 0.3 pixels.

3.3.4. Geometric Positioning Consistency Accuracy Evaluation Between Panchromatic and Multispectral Data of Basic Products

This paper proposes using simulated multispectral data based on the calibrated panchromatic image as control data for calibrating the multispectral reference band to improve geometric positioning consistency between panchromatic and multispectral data. High-precision dense matching between panchromatic and multispectral data was performed to obtain matching point data, and the projection error of matching points combined with RPC coefficients was used to evaluate the geometric positioning consistency between panchromatic and corresponding multispectral data. The acquisition times and angles of the experimental data are shown in the “Basic Products Geometric Accuracy” section of Table 4, and the statistical results are presented in Figure 12. The figure shows that the geometric positioning consistency RMSE for all data in all logical cameras ranges from 0.13 to 0.32 multispectral pixels, with a mean RMSE better than 0.25 multispectral pixels, achieving high-precision geometric positioning consistency between panchromatic and multispectral data.

3.3.5. Positioning Accuracy with GCPs Evaluation of Basic Products

When using remote sensing image data for high-precision mapping, control points are needed to further eliminate geometric positioning errors in the data, making positioning accuracy with GCPs a crucial factor in determining whether remote sensing image data can be used for high-precision mapping. A geometric correction with GCPs experiment was conducted using 10 scenes data in Lanzhou, Zhangye, Jiuquan, Jiayuguan, Wuwei, and Baiyin in Gansu Province, with high-precision control data. The acquisition times and angles are shown in Table 9, and the distribution of control points is shown in Figure 13. The number of control points and the statistical results of controlled geometric accuracy are shown in Table 9. The experiment used the RPC image affine transformation model as the geometric correction model (as shown in Equation (8)). The results indicate that the controlled accuracy of the basic products produced by the Jilin1-KF01B satellite, using the on-orbit geometric calibration coefficients, was better than 2.1 m in urban, agricultural, and mountainous areas, meeting the requirements for 1:5000 scale mapping.

$$\left\{\begin{array}{c}x+a_0+a_{1}x+a_{2}y=RP{C}_x(lat,lon,h)\\ y+b_0+b_{1}x+b_{2}y=RP{C}_y(lat,lon,h)\end{array}\right.$$

(8)

3.3.6. Multi-Temporal Data Geometric Positioning Consistency Accuracy Evaluation of Basic Products

The wide-swath characteristics of the Jilin1-KF01B satellite enable it to acquire data with high frequency, multiple time phases, and multiple viewing angles. Therefore, the geometric positioning consistency accuracy of multi-temporal standard image products is a key metric for evaluating applications such as target extraction and change detection across multiple periods. The geometric positioning consistency accuracy of multi-temporal data was evaluated using data from four regions: Gansu, Linyi, Qingdao, and Xinjiang, covering both plain and mountainous terrains. The geographic distribution of experimental data are shown in Figure 14, and the acquisition times and angles of the 34 scenes of experimental data are shown in Table 10. In each experiment, based on the overlap relationships between the multi-temporal data, corresponding range DOM (Digital Orthophoto Map) and DEM (Digital Elevation Model) data were obtained as control data to extract control points. Additionally, tie points between overlapping images were extracted using a matching algorithm, and these control and tie points were used in a bundle adjustment based on the RPC image affine transformation model to correct the RPC model coefficients for each image. The geometric positioning consistency accuracy of each image was then calculated using the tie points.

The geometric positioning consistency accuracy of multi-temporal basic products requires determining the tie points and their observation dimensions for each image. Suppose image A has “n” connection points, each with “m” observation dimensions; the geometric positioning consistency error for each tie point can be calculated using Equation (9), where $X_p,Y_p$ represents the object-space coordinates of tie point “p” in image A (in UTM coordinate system), and $X_p^i,Y_p^i$ represents the object-space coordinates of tie point “p” in image B (in UTM coordinate system). After obtaining the geometric positioning consistency errors for all tie points, the geometric positioning consistency accuracy of the image can be calculated using Equation (10). The geographic distribution of experimental data are shown in Figure 14, and the experimental results are shown in Table 10, the effect of geometric positioning consistency before and after correction art shown in Figure 15, Figure 16, Figure 17 and Figure 18, indicating that the original geometric positioning consistency accuracy (OGPCA) is about from 3 m to 11 m, and the new geometric positioning consistency accuracy (NGPCA)after bundle adjustment in all experimental groups was better than 2 m, achieving high-precision geometric positioning consistency across multiple periods.

$${\Delta }_p=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^{m-1}\left[{(X_p-X_p^i)}^2+{(Y_p-Y_p^i)}^2\right]}}{m-1}}}$$

(9)

$$\xi =\sqrt{{\displaystyle \frac{{\displaystyle \sum _{p=1}^n{\Delta }_p^2}}{n}}}$$

(10)

4. Conclusions

On-orbit geometric calibration is a crucial prerequisite to ensure the practical application of the Jilin1-KF01B satellite. This paper, based on the sub-meter wide-field camera’s characteristics—large aperture, wide field of view, independent control of TDI CCD timing by multiple IPS, and the use of ultra-multi-chip TDI CCD mechanical staggered stitching—presents a detailed introduction to the GCGC method. The method employs a stepwise iterative strategy to solve for exterior and interior calibration elements separately, with a particular focus on the independent time control among multiple IPS. In response to this situation, the wide-field camera is logically grouped, with exterior orientation elements calibrated independently for each group, enhancing data positioning accuracy through independent exterior orientation correction. Additionally, an inter-chip geometric positioning consistency constraint is applied to improve the geometric positioning consistency in the overlapping areas between adjacent CCDs, enabling seamless stitching of adjacent CCD images. For the absolute calibration of the multispectral reference band data, the GCGC process involves simulating multispectral reference band data as control data, which is generated using a band correlation-based image simulation algorithm from the calibrated panchromatic data. This approach reduces the difficulty of control point extraction for the multispectral reference band data in traditional calibration algorithms and improves geometric positioning consistency between panchromatic and multispectral band data. The grouping calibration strategy in the proposed GCGC method is suitable for wide-swath satellites. The geometric positioning consistency constraint between adjacent detectors is not limited to such satellites, it is also applicable to optical satellites with mechanically spliced detectors. Furthermore, the method based on simulated spectral bands can be widely applied to all optical satellites.

A series of comprehensive geometric accuracy validation experiments were conducted on the long-term basic products of the Jilin1-KF01B satellite. The results show that after on-orbit geometric calibration, the average seamless stitching accuracy is better than 0.6 pixels, achieving sub-pixel level stitching, RPC fitting accuracy was better than 0.0003 pixels, positioning accuracy without GCPs was better than 20 m (CE90), band-to-band registration accuracy was better than 0.3 pixels, the geometric positioning consistency accuracy between panchromatic and multispectral data was better than 0.25 multispectral pixels, geometric correction accuracy with GCPs was better than 2.1 m, and the geometric positioning consistency accuracy across multiple periods was better than 2 m. These experimental results demonstrate that the GCGC method proposed in this paper is reasonable and effective, significantly improving the geometric positioning accuracy of the Jilin1-KF01B WF camera, proving that the basic products can meet the requirements for high-precision mapping and data application needs.