3.1. External Calibration
Among external errors, the orbital measurement error can achieve cm-level positioning accuracy through post-processed precise orbit determination [
27]. When post-processed precise attitude data are used, the attitude measurement error fluctuates with time but is systematic in a short time [
28,
29]. Systematic errors occur because of the vibration during launch and the difference in the environmental conditions before and after entering the orbit. External calibration is mainly used to eliminate the systematic errors that occur during instrument installation.
A rotation matrix is introduced into the rigorous geometric imaging model Equation (1) of the LFC to compensate for the systematic error caused by instrument installation:
where
is the compensation matrix of the system error, which is composed of three rotation angles, i.e.,
,
, and
, as follows,
a linearized error equation was constructed to solve these three rotation angles. Regarding the orbit, attitude, and other data as true values, Equation (2) can be written as the following equation:
where
,
.
On the left-hand side of Equation (4),
is the direction of the real incident light determined by the ground point coordinates in the image space coordinate system; on the right-hand side of the equation,
is the direction of the incident light determined by the focal sensor of the LFC in the image space coordinate system. According to the collinear condition, these two directions should be parallel in theory; however, because of various errors, there is a small systematic deviation between them, which causes the final geometric positioning error. The following formula was obtained by removing the unknown parameter
from Equation (4):
where
.
For a ground-control point (GCP), the error equation can be constructed by linearizing Equation (5) as follows:
where
indicates the residuals for x and y coordinates;
is the correction of the rotation angles;
is the constant of the error equation, calculated according to the initial value;
is the linearized coefficient matrix; and
is the index of the GCP. By combining the error equations for all GCPs, we obtain the following overall error equation:
where
,
,
, in which
is the total number of GCPs. The three corrections are solved via least square adjustment:
After calculating the compensation matrix and eliminating the external error, the geometric positioning of the LFC can be obtained using Equation (2). During external calibration, two equations can be obtained from one GCP. Therefore, in theory, the compensation matrix can be calculated using the two GCPs. However, more GCPs are recommended to ensure the robustness of the solution.
3.2. Internal Calibration
After external calibration, the global positioning error of the footprint image was eliminated, leaving the local positioning error in the image range. The internal calibration was based on the pointing angle model. For image point coordinates
on the footprint image, the pointing angle in the image space coordinates of the LFCs can be expressed as follows:
where
and
are the pixels corresponding to the pointing angle;
and
are the coordinate values of the principal point of the image;
is the camera constant.
Owing to internal errors, there is a certain deviation between the real and theoretical pointing angles corresponding to an image point. The theoretical pointing angle was used to build a rigorous geometry model, and the image was captured according to the real pointing angle. Internal calibration was used to make the theoretical pointing angle as close to the real pointing angle as possible. For this purpose, the general cubic polynomial, which has high orthogonality and low correlation, was used to fit the real pointing angle of the image point [
19]:
where
denote the internal calibration coefficients. The process of internal calibration involves solving 20 calibration parameters in Equation (10). The error equations of the internal calibration can be constructed using Equations (3), (9) and (10). These error equations constitute a linear system of equations that can be solved directly via least-squares adjustment.
3.3. On-Orbit Geometric Calibration Process
The on-orbit calibration of GF-7 LFC adopts an iterative method of external and internal calibration. The overall calibration process is illustrated in
Figure 2.
In the internal calibration process, it is necessary to obtain a large number of uniformly distributed GCPs to more accurately calculate the internal calibration parameters. Many factors, such as small image size, weak image texture, and large differences in rotation and resolution, make it difficult to match GCPs from high-accuracy reference images (Digital Orthophoto Map, DOM). It is difficult to obtain a sufficient number of GCPs using only single-scene images. To solve this problem, we adopted the joint calibration of multi-scene footprint images in the calibration area. According to the exposure time of each laser footprint image, combined with the post-precision orbital and attitude data and various initial parameters, a rigorous geometric positioning model was used to simulate the footprint images [
30]. Each pixel of the simulated footprint image was ray-traced onto the reference digital elevation model. The pixel value was then obtained via Gaussian filtering around the corresponding location in the reference DOM. Because the resolution is relatively low, the geometric accuracy loss during the simulation can be ignored. After the simulation, the GCPs were matched automatically combined with Harris features [
31] and least-squares matching between the real laser footprint images and corresponding simulated footprint images. Least-squares matching can achieve subpixel precision and is robust for local distortion and radiative transformation because it makes full use of the local information for adjustment calculations [
25,
32].
When only a single measurement area is used for external calibration, random errors caused by satellite-specific conditions, such as instantaneous jitter of the satellite platform or temperature changes of the instrument are also systematic. As a result, the geometric positioning accuracy of the calibration results in the calibration area is very high, but the accuracy in other areas may be significantly degraded, resulting in unreliable calibration results. After the internal calibration was completed, we used multi-track ground reference data in different areas to perform external calibration for reducing the influence of random errors on calibration results.