A Separate Calibration Method of Laser Pointing and Ranging for the GF-7 Satellite Laser That Does Not Require Field Detectors

Abstract

Satellite laser altimeters have been widely used in the surveying, mapping, forestry, and polar regions and by other industries due to their excellent elevation measurement accuracy. Satellite laser on-orbit geometry calibration is a necessary means to ensure elevation accuracy. This study proposes an iterative geometry calibration method for satellite laser altimeter pointing and ranging separation that does not require the use of field detectors. The DSM data were first used to complete the laser pointing calibration, and then the laser footprint elevation was measured accurately to complete the laser ranging calibration. The iterative calibration experiment was repeated until the convergence condition (i.e., the laser point difference was less than 1 × 10^-5 degrees and the laser ranging difference was less than 0.01 m) was met, with the calibrated laser pointing angle and ranging separation used as the input parameters. In this work, the GaoFen-7 (GF-7) satellite laser was used as the test object and the actual laser pointing and ranging values derived from ground detector calibrations. The results verified that the pointing accuracy of the GF-7 beam 1 was 2 arcsec and that the ranging accuracy was 2 cm after applying the calibration method presented in this paper. The pointing accuracy of the GF-7 beam 2 was 2.2 arcsec, and the ranging accuracy was approximately 1 cm. This analysis demonstrated that the GF-7 laser mission exceeded its pointing angle requirement of 3 arcsec after laser pointing and ranging separation iterative calibrations were applied. Finally, ground control points were used to verify the calibrated elevation accuracy of the GF-7 satellite laser, and its accuracy on flat terrain was 0.18 m. In summary, it was proven that the satellite laser geometry calibration method proposed in the article is effective.

1. Introduction

The GaoFen-7 (GF-7) satellite was launched on 3 November 2019 and is equipped with two full-waveform profile laser beams for synchronous Earth observation [1]. The operating modes of the two GF-7 laser beams are identical. For example, the wavelength is 1064 nm, the measurement frequency is 3 Hz, and the transmitted and received waveforms are recorded at a sampling rate of 2 GHz. The two laser beams are positioned on either side of the across track at the nadir of the satellite, and each beam is at an angle of 0.7 degrees from the nadir; that is, the distance between the two laser footprints is approximately 12.25 km across track. For more details, please refer to reference [2]. The GF-7 satellite has been in orbit for 3 years and has acquired a large amount of data from the surface of the globe. These data products have been released on the Natural Resources Satellite Remote Sensing Cloud Service Platform (http://sasclouds.com/chinese/normal/, accessed on 20 November 2022) in the *.SLA format. The detailed application process for the GF-7 satellite laser data is in Appendix A. These data have been widely studied by domestic and foreign researchers and engineers. Therefore, ensuring the accuracy of GF-7 satellite laser data is one of the most vital tasks during satellite operation. Thus, the periodic on-orbit geometry calibration of the lasers is an important means to ensure data accuracy.

At present, there are two primary high-precision calibration methods for existing profile satellite lasers: (1) The satellite attitude maneuver calibration method [3,4,5,6], which causes the satellite laser to rotate within a cone over the middle of the ocean. Then, according to the principle of minimum residual error of satellite laser ranging, the geometric calibration of the satellite laser is completed. (2) The calibration method based on laser ground detectors (LGDs) [7,8,9,10]. This method actively captures laser footprints by placing multiple infrared detectors on the surface, and satellite laser calibration is accomplished by triggering the detector. Other satellite laser calibration methods have not been used for high-precision satellite laser calibration due to low accuracy, a low success rate, or difficulty in implementation and so on. For example, the calibration method based on the synchronous imaging measurements through an infrared camera [11] requires the aircraft to fly synchronously with the satellite at night. This method is difficult to implement and has a low success rate. The simulation waveform matching calibration method [12] has some uncertainties with high requirements for the local terrain type, and the reliability is low. The traditional terrain matching calibration method [13] cannot calibrate laser range errors, and the calibration accuracy is low. Additionally, there is an iterative pointing angle calibration method based on small-range terrain matching [14]. This method establishes a calibration error model, and the calibration is carried out by solving the partial derivatives of the pointing angle and ranging. However, this method requires a large number of laser spots and thus cannot be used for profile satellite lasers with low pulse frequencies. However, high-precision calibrations can be accomplished by maneuvering the satellite at attitude, though this method requires the high attitude maneuverability of the satellite platform. For satellites with platform-stabilized lasers, such as the GF-7 satellite, rapid cone measurements cannot be performed. Therefore, only the calibration method based on LGDs is widely used on platforms such as the GF-7 and ZY3-02/03 satellites. The disadvantage of this method is that it requires that staff place a large number of LGDs in the calibration field, and the workload is substantial.

Multibeam satellite lasers are the current trend in satellite laser development, such as the GF-7 dual-beam lasers and the five-beam lasers of terrestrial ecosystem carbon monitoring satellites. The cost of the LGD calibration method is large, and the cost doubles with the increase in beams. Therefore, it is critical to develop a low-cost satellite laser geometry calibration method that does not require ground detectors. Due to this urgent demand, this work proposes an iterative geometry calibration method for satellite laser altimeter pointing and ranging separation that does not require field detectors. First, the residual sum of the elevation of all the GF-7 laser footprints and the reference digital surface model (DSM) were obtained by terrain matching. The satellite laser was then calibrated according to the sum of the minimum elevation residuals. Then, a flat and hard surface area was used as the test area for laser range calibration, and the elevation difference between the laser-derived elevations and the actual elevation was defined as the laser ranging error. The actual elevation of the test location was obtained by accurately measuring the flat surface ground with a precise level. Next, taking the calibrated pointing angle and range of the satellite laser as the initial value, the satellite laser pointing and ranging calibration experiment was repeated until the laser point difference was less than 1 × 10⁻⁵ degrees and the laser ranging difference was less than 0.01 m, and the calibrated satellite laser pointing and ranging error values were the output. In this work, the GF-7 satellite laser was used as the test object, and the calibrated pointing and ranging error of the GF-7 satellite laser based on LGDs were used as the true values, which were used to verify the laser pointing and ranging accuracy of the GF-7 satellite after the calibrations based on the method presented in this paper were applied. Additionally, the ground control points (GCPs) were used to verify the laser elevation accuracy of the GF-7 satellite after the calibration.

2. Materials and Methods

2.1. Study Area

2.1.1. Calibration Area

(1)

Point Calibration Area

Considering the minimal time phase difference between the laser data and the reference DSM data, there was a little elevation error between the systems caused by the natural growth of surface vegetation. Even so, this study preferentially selected the bare natural surface as the calibration area. A region in the east Gobi desert near the city of Kuqa, Xinjiang, China, was selected as the GF-7 satellite laser pointing calibration area. The area was 83.30–83.86°E, 41.19–42.03°N and consisted of a total area of approximately 2000 square kilometers. The topography was very complex in this area, the landforms undulated considerably, and there was little tall vegetation on the surface. The elevation changes within the calibration area have been negligible over recent decades; thus, it is an ideal experimental area for satellite laser pointing calibration. This experimental area is shown in Figure 1a.

(2)

Ranging Calibration Area

According to the calibration strategy proposed in this paper, a hard and flat surface was selected for the laser ranging calibration area. An airport runway near Beijing city, as shown in Figure 1b, was selected as the GF-7 laser beam 1 ranging calibration area. The ranging calibration laser footprint time code of the GF-7 beam 1 was 203685561.00 (the time code is the accumulated seconds since 1 January 2014), and there was no elevation difference over approximately 200 m, and thus this location can be regarded as falling within the same elevation plane. The ranging calibration area of GF-7 laser beam 2 was selected along a flat highway in the suburbs of Beijing city, as shown in Figure 1c. The ranging calibration laser footprint time code of GF-7 beam 2 was 203685553.00, and the surface elevation was the same within approximately 100 m of this feature. The satellite laser pointing accuracy is normally approximately 30 m [13] after the first terrain matching calibration. Within this error range, the actual elevation of the ground for the selected laser footprints would not change, and the selected ranging calibration area was reasonable. Finally, the DS05 level was used to accurately measure the surface elevation of the ranging calibration area, which was used as the true elevation for the calibration of the laser ranging error.

2.1.2. Elevation Accuracy Validation Area

The accuracy of satellite-derived laser elevation measurements is a topic that has garnered extensive attention. Surface GCPs are measured by real-time kinematic GPS (RTK) to accurately verify satellite-derived elevation data. The areas used for verification should be flat terrain. Sonid Youqi, Inner Mongolia, China, was selected as the area to verify the GF-7 laser calibrations discussed in this work, as shown in Figure 2. The terrain was relatively flat in this location and tall vegetation was minimal, consisting primarily of xerophytes and small grasses.

2.2. Data

2.2.1. GF-7 Satellite Laser Data

In this article, the GF-7 satellite laser data are divided into three categories: laser pointing calibration data, laser ranging calibration data, and elevation accuracy verification data, as shown in

Table 1. The basic information of all test laser data of the GF-7 satellite in this study.

Type	File Name	Number of Laser Points
		Beam 1	Beam 2
Pointing calibration	KRN-GF7-20201014-005278-0000005703	37	39
	KRN-GF7-20201212-006160-0000006634	9	10
	KSC-GF7-20210607-008851-0000010008	37	19
Ranging calibration	KSC-GF7-20200615-003419-0000003495	1	1
Elevation accuracy verification	SYC-GF7-20200609-003326-0000003289	7	5
	SYC-GF7-20200614-003402-0000003397	5	3
	SYC-GF7-20200619-003478-0000003529	4	6
	KSC-GF7-20200624-003555-0000003615	8	10

. The GF-7 pointing calibration data came from the 5278th track on 14 October 2020, the 6160th track on 12 December 2020, and the 8851th track on 7 June 2020, of which beam 1 had a total of 83 laser points and beam 2 had a total of 68 laser points. The laser spatial distribution is shown in Figure 1a. The GF-7 ranging calibration data came from the 3419th track on 15 June 2020. The laser point timecode of beam 1 was 203685561.00 and 203685553.00 for beam 2, and their distributions are shown in Figure 1b,c, respectively. The GF-7 elevation verification data were from the 3326th track on 9 June 2020, the 3402th track on 14 June, the 3478th track on 19 June, and the 3555th track on June 24. Among these, beam 1 and beam 2 both contained 24 laser points used for elevation accuracy verification, as shown in Figure 2.

2.2.2. Reference Terrain Data

(1)

AW3D30 DSM

ALOS World 3D-30 m (AW3D30) is a global DSM dataset with a 30 m grid resolution released by the Japan Aerospace Exploration Agency (JAXA) Earth Observation Research Center in 2016 [15]. Users can download the global dataset after registration. The data download link of the AW3D30 dataset is https://www.eorc.jaxa.jp/ALOS/en/aw3d30/data/index.htm (accessed on 20 November 2022). The global elevation accuracy of the AW3D30 is better than 5 m [16]. We adopted the national control point image database to verify the elevation accuracy of the AW3D30 DSM, and the elevation accuracy of the AW3D30 DSM in the laser pointing calibration area was approximately 2 m [17]. The AW3D30 DSM of the pointing calibration area is shown in Figure 3a.

(2)

LiDAR Data

We used a fixed-wing piloted aircraft to carry the Leica TerrainMapper Light Detection and Ranging (LiDAR) to obtain surface terrain data of the pointing calibration area in September 2021. During the surface terrain data measurement, the aircraft maintained a flight altitude of 5200 m, and the laser emission frequency was 436 kHz. The average point density on the surface was guaranteed to be 1.6 points per square meter when the aircraft was at the maximum flight speed of 309 km/h. After completing field measurements, the filtered LiDAR point cloud data were used to generate a DSM image with a resolution of 1 m and an elevation accuracy of 0.17 m, which is shown in Figure 3. In this article we refer to this DSM image as LiDAR data.

2.2.3. GCPs Dataset

The GCP data in the paper refer to the RTK-derived surface elevation control points set around the GF-7 laser footprint. An example of a GCP measurement is shown in Figure 4b. First, the 81 GCPs were measured around the laser footprint after calibration, such that an array of GCPs was formed around the laser footprint. The GCP array is shown in Figure 4a, where the black dots represent the GCPs and the green dot is the laser point 203168336.67 from GF-7 laser beam 1. The distance between the GCPs in the GCP array was approximately 3 m, and the elevation accuracy of each GCP was better than 5 cm. Subsequently, the GCP arrays surrounding all the GF-7 laser points shown in Figure 2 were combined to form a GCP dataset.

2.3. Methodology

This study proposes an iterative geometry calibration method for satellite laser altimeter pointing and ranging separation that does not require the use of field detectors. The workflow of our proposed method can be summarized as follows (Figure 5):

Step 1:

Experimental data preparation: initial pointing parameters, initial ranging error (0 by default), GF-7 laser data, reference DSM data, and the GCPs around the laser footprint.

Step 2:

Laser pointing calibration: First, a pointing angle grid is established with the initial laser pointing angle as the center. Then, the residual sum of the laser and the reference DSM elevation for each grid is calculated. The laser pointing angle with the smallest elevation residual sum between the laser and the DSM is used as the laser pointing angle after calibration.

Step 3:

Laser ranging calibration: taking the laser pointing angle after calibration as the input parameter, the coordinates of the laser footprint on the flat hard surface are calculated, and the elevation difference between the laser and the surface is used as the ranging error after calibration.

Step 4:

Iterative of laser pointing and ranging calibration: It must be assessed whether the laser point difference (LPD) $\overrightarrow{\Delta p}$ between the calibrated pointing and the input pointing in step 2 is less than 1 × 10⁻⁵ degrees and whether the laser ranging difference (LRD) $\Delta \rho$ between the calibrated ranging error and the input ranging error in step 3 is less than 0.01 m. This is called the iterative convergence condition. When the convergence condition is satisfied, the output pointing and ranging error are the final iterative calibration results. Otherwise, redo steps 2–4 with the calibration result of this iteration as an input parameter. Typically, it takes 2–3 iterations to converge.

Step 5:

Accuracy verification after calibration: the laser pointing and ranging error value after calibration by LGDs is first used as the true value to verify the laser pointing and ranging accuracy following the calibration method outlined in this paper. Then, the GCPs are used to verify the elevation accuracy of GF-7 satellite laser after calibration.

2.3.1. Satellite Pointing Error Calibration

The primary mathematical model for obtaining the satellite calibration data of the laser pointing, ranging error, and laser elevation accuracy verification presented in this paper is the satellite laser geometric positioning model. The geometric relationship between the satellite laser and the surface laser footprint was established, as shown in Figure 6.

The geometric relationship between the laser emission point, the satellite centroid, the GNSS antenna, the center of the Earth, and the laser footprint on the surface is shown in Figure 6. The laser footprint position vector $\overrightarrow{r_S}$ is the sum of the GNSS antenna center position vector $\overrightarrow{r_G}$; the vector $\overrightarrow{r_{GB}}$ from the GNSS antenna center to the satellite centroid; the vector $\overrightarrow{r_{GL}}$ from the satellite centroid to the laser emission point; and the laser ranging vector ρ ● $\overrightarrow{p}$, where $\rho$ is the laser ranging value and $\overrightarrow{p}$ is the laser pointing unit vector. Converting it into matrix form, the geometric positioning model of the satellite laser is established, as shown in the following formula [18]. The formula can be used to calculate the three-dimensional coordinates of laser footprints based on satellite orbit, attitude data, and laser ranging value.

$${\left(\begin{array}{c}X\\ Y\\ Z\end{array}\right)}_{ITRF}={\left(\begin{array}{c}{X}_G\\ Y_G\\ Z_G\end{array}\right)}_{ITRF}+{\left(\begin{array}{c}{X}_{GB}\\ Y_{GB}\\ Z_{GB}\end{array}\right)}_{ITRF}+R_{ICRF}^{ITRF}{R}_{SBF}^{ICRF}\left[\left(\begin{array}{c}{X}_{BL}\\ Y_{BL}\\ Z_{BL}\end{array}\right)+\rho \cdot \left(\begin{array}{c}\cos ({\alpha }_x)\\ \cos ({\alpha }_y)\\ \cos ({\alpha }_z)\end{array}\right)\right]$$

(1)

where ${\left(\begin{array}{ccc}X& Y& Z\end{array}\right)}_{ITRF}^T$ are the surface coordinates of the laser footprint. ${\left(\begin{array}{ccc}{X}_G& Y_G& Z_G\end{array}\right)}_{ITRF}^T$ are the satellite centroid coordinates in the International Terrestrial Reference Frame (ITRF). $R_{ICRF}^{ITRF}$ is the rotation matrix from the International Celestial Reference Frame (ICRF) to ITRF. $R_{SBF}^{ICRF}$ is the rotation matrix from the satellite body frame (SBF) to ICRF. ${\left(\begin{array}{ccc}{X}_{GB}& Y_{GB}& Z_{GB}\end{array}\right)}_{ITRF}^T$ are the offset values between the GNSS antenna center and the satellite centroid. ${\left(\begin{array}{ccc}{X}_{BL}& Y_{BL}& Z_{BL}\end{array}\right)}^T$ are the offset values between the satellite centroid and the laser emission point. $\rho$ is the laser range value, which includes the ranging errors caused by tides and the atmosphere, as well as the ranging errors of the laser itself, where the latter ranging errors need to be corrected by geometric calibration. ${\left[\begin{array}{ccc}\cos ({\alpha }_x)& \cos ({\alpha }_y)& \cos ({\alpha }_z)\end{array}\right]}^T=\overrightarrow{p}$ is the laser pointing unit vector; ${\alpha }_x$ is the intersection angle between the laser optical axis and X axis in the SBF; ${\alpha }_y$ is the intersection angle between the laser optical axis and Y axis in the SBF; and ${\alpha }_z$ is the intersection angle between the laser optical axis and Z axis in the SBF. Among them, ${\alpha }_z$ can be directly calculated from the pointing angles ${\alpha }_x$ and ${\alpha }_y$, and the calculation formula is shown in Formula (2). Due to the satellite vibration during launch and environmental changes after the satellite reaches orbit, there are pointing differences between the actual alignment ${\alpha }_x$ and ${\alpha }_y$ and the laboratory measurement results that also need to be calibrated.

$${\alpha }_z=a\cos \left(\sqrt{1-\cos ({\alpha }_x)\cdot \cos ({\alpha }_x)-\cos ({\alpha }_y)\cdot \cos ({\alpha }_y)}\right)$$

(2)

The satellite laser pointing error calibration method adopts the terrain matching method. First, the initial laser pointing angle $({\alpha }_x,{\alpha }_y)$ is used as the center to establish a rectangular pointing grid with the pointing angles ${\alpha }_x$ and ${\alpha }_y$. It is assumed that the pointing grid range is $L_x$ and $L_y$, and grid spacing is $\Delta L_x$ and $\Delta L_y$, respectively. Then, each pointing angle in pointing grid is taken out as the input pointing angle, and the surface coordinates of all the laser footprints are calculated according to the satellite laser geometric positioning model. According to the longitude and latitude of each laser footprint, the elevation of the reference DSM is interpolated to calculate the elevation residual value of all laser footprint elevations and the DSM elevation. All laser footprint elevation residual sums are counted (abbreviated as laser elevation residual sums here). Finally, the previous step is repeated until all value in the pointing grids have been processed. The laser pointing angle with the smallest laser elevation residual sum is the optimal pointing angle, that is, the laser pointing angle after calibration. The detailed implementation process of the Algorithm 1. When the range of the pointing grid is large or the grid spacing is small, the experimental processing speed is slow, and the reference [13] can be used to improve the pointing calibration processing speed by using a pyramid method.

Algorithm 1: Satellite Laser Pointing Calibration

1: Input: The Initial laser pointing $\left({\alpha }_x,{\alpha }_y\right)$. The pointing grid range $\left(L_x,L_y\right)$. The pointing grid interval $\left(\Delta L_x,\Delta L_y\right)$. The original laser data set $PT=\left\{P{t}_1,P{t}_2,\dots P{t}_n\right\}$.  2: Output: The pointing after calibration $\left({\alpha }_{x-opt},{\alpha }_{y-opt}\right)$.  3: Generate a laser pointing grid with a length of $2L_x$ and a width of $2L_y$. So, the laser pointing set is $LPS=\left\{\left({\alpha }_1,{\alpha }_1\right),\dots ,\left({\alpha }_{2L_x/\Delta L_x},{\alpha }_1\right),\dots ,\left({\alpha }_{2L_x/\Delta L_x},{\alpha }_{2L_y/\Delta L_y}\right)\right\}$. There is total $m=\left(\frac{2L_x}{\Delta L_x}\times \frac{2L_y}{\Delta L_y}\right)$ laser pointing  4: For $i$ = 1 to $m$ do  5:  For $j$ = 1 to $n$ do  6:   Compute the surface coordinates of the $j-th$ laser footprint ${\left(\begin{array}{ccc}{X}_j& Y_j& Z_j\end{array}\right)}_{ITRF}^T$ using laser pointing $\left({\alpha }_x^i, {\alpha }_y^i\right)$.  7:   Interpolate the elevation $h_j$ of the reference DSM using ${\left(\begin{array}{cc}{X}_j& Y_j\end{array}\right)}_{ITRF}$.  8:   Calculate the elevation difference $\Delta h_j$ between $h_j$ and $Z_j$, $\Delta h_j=abs\left(Z_j-h_j\right)$.  9:  end for  10:  Count the mean of all laser elevation differences $\Delta H_i=\left(\Delta h_1+\Delta h_2+\dots +\Delta h_n\right)/n$.  11: end for  12: Find the smallest in the elevation difference from dataset $\left\{\Delta H_1,\dots ,\Delta H_m\right\}$, and assume that its corresponding laser pointing is $\left({\alpha }_{x-min},{\alpha }_{y-min}\right)$. Thus, $\left({\alpha }_{x-opt},{\alpha }_{y-opt}\right)=\left({\alpha }_{x-min},{\alpha }_{y-min}\right)$.

2.3.2. Satellite Ranging Error Calibration

Satellite lasers are usually measured in the vertical nadir direction [2,19,20]; that is, the intersection angle between the laser optical axis and the $Z$ axis in the SBF can be ignored. However, for the multibeam lasers of the GF-7 satellite, the intersection angle between the laser optical axis and the $Z$ axis in the SBF is relatively large. That is, ${\alpha }_z$ is approximately 0.7 degrees, so $\cos {\alpha }_z\approx 1.0$. Therefore, according to Formula (1), it is easy to deduce that the laser footprint elevation error will be completely caused by the ranging error. Based on this theory, we propose a method to directly correct the laser ranging error by using the laser footprint elevation error. The key issue of this method is to accurately measure the surface elevation of the laser footprints. We will address these issues in two aspects. The first approach pertains to laser footprint selection, and the other involves accurately measuring the surface elevation of the laser footprints.

First, laser pointing after calibration is used to calculate the surface coordinates of the laser footprints that may fall on flat and hard ground, such as cement courts and ice. Additional laser footprints within 50 m of the initial laser footprint that fall within the same elevation plane were selected. Then, the DS05 precision level is used to precisely measure the leveling points to establish a fourth order leveling network. Then, the laser footprints are placed on the level network path to measure their elevation values. The elevation accuracies of the laser footprints is guaranteed to be better than 3 cm. Assuming that the elevation of the laser footprint after the pointing calibration is $h_{laser}$, and assuming that the actual surface elevation of the laser footprint measured by the DS05 level is $h_{surf}$, then the satellite laser ranging error $\Delta \rho$ can be directly calculated according to the following formula:

$$\Delta \rho =h_{laser}-h_{surf}$$

(3)

2.3.3. Elevation Accuracy Verification of Satellite Laser

After iterative calibration, the satellite laser pointing and ranging values are brought into Formula (1) to calculate the initial coordinate ${\left(\begin{array}{ccc}X& Y& Z\end{array}\right)}_{ITRF}^T$ of the laser footprint, which is converted to geodetic coordinates using the WGS84 ellipsoid according to Formula (4). According to the coordinates of the laser footprint and laser emission time, the National Centers for Environmental Prediction (NCEP) data are used to correct the laser ranging delay error caused by atmospheric interference [21,22], and the FES2014 ocean tide model is used to correct the laser elevation tide error [23]. The coordinates of the laser footprint are recalculated using the corrected laser range as the final coordinates of the laser footprint.

$$\left\{\begin{array}{l}L=\arctan \left(\frac{Y}{X}\right)\\ B=\arctan \left(\frac{Z\cdot (N+H)}{\sqrt{(X^2+Y^2)[N(1-e^2)+H]}}\right)\\ H=\frac{\sqrt{X^2+Y^2}}{\cos B}-N\end{array}\right.$$

(4)

where $N$ is the radius of the curvature of the prime vertical for the WGS 84 ellipsoid; $e$ is the first eccentricity of the WGS84 ellipsoid; and $H$ is the ellipsoid height of the laser footprint.

Taking the continuously operating reference stations (CORS) as the RTK base station, RTK GPS is used to measure the geodetic elevation value of the laser footprint, $H^{\prime }$. $H^{\prime }$ is considered to be the true value of the laser footprint elevation in order to evaluate the elevation accuracy of the GF-7 satellite laser $H$ after the calibration. Assuming that there are laser footprints, the elevation residual values $dH$ of n laser footprints are calculated by Formula (5). Finally, the mean value $\Delta \overline{H}$ and root mean square error (RMSE) $\sigma H$ of the laser footprint elevation residuals are counted as the evaluation index of the laser elevation accuracy, and the calculation formula of the mean value and RMSE are as follows:

$$dH=H-H^{\prime }$$

(5)

$$\Delta \overline{H}=\frac{{\displaystyle \sum _{i=1}^{i=n}d{H}_i}}{n}$$

(6)

$$\sigma H=\sqrt{\frac{{\displaystyle \sum _{i=1}^{i=n}d{H}_i^2}}{n}}$$

(7)

3. Results

3.1. Calibration of Laser Pointing and Ranging

In this work, all laser footprints of GF-7 beams 1 and 2 from the pointing calibration area in Figure 1a were used as the experimental objects, and AW3D30 DSM was used as the reference terrain data to conduct terrain matching experiments according to the method shown in Figure 7. For each beam of the GF-7 laser, the initial laser center pointing angle was measured in the laboratory before the satellite launch. A pointing grid range with a laser pointing angle ${\alpha }_x$ and ${\alpha }_y$ range of ±0.5 degrees and grid spacing of 0.003 degrees was created. This pointing grid had a total of 110,889 pointing angles. Assuming that the satellite orbits at an altitude of 500 km, the laser footprint position error was approximately 2.5 m per arcsec. The resolution of the AW3D30 DSM was 30 m, which is equal to approximately 0.003 degrees of the laser pointing angle. For each pointing angle in the pointing grid, the absolute value of the elevation residuals between the 83 laser footprint elevation of beam 1 and the AW3D30 DSM elevation were calculated. Then, the sum of the absolute values of all the laser footprint elevation residuals of beam 1 was calculated, and the result is referred to as the sum of elevation residuals. Finally, after traversing all the pointing angles in the pointing grid, a set of elevation residuals was generated for the grid pointing angles and the sum of the elevation residuals. The calibration experiment was consistent for beams 2 and 1, and the three-dimensional surface plots of the grid pointing and the sum of the elevation residuals of beam 1 and beam 2 are shown in Figure 7a,b. In the figure, $\Delta H$ refers to the elevation residuals between the laser and reference terrain at different pointing angles, and ${\alpha }_x$ and ${\alpha }_y$ are two pointing angles of laser boresight in Formula (1).

From the results of Figure 7a,b, there is an obvious minimum value on the surface between the laser pointing angle and the sum of the elevation residuals. It is easy to understand that as the laser pointing accuracy increases, the sum of the laser elevation residuals decreases. Therefore, the laser pointing angle that corresponds to the minimum sum of the elevation residuals in Figure 7a,b is the optimal pointing angle of GF-7 laser beams 1 and 2 after the first iterative calibration.

After the first calibration was completed, the laser pointing angle was used to calculate the elevations of the laser footprint in Figure 1b,c for the ranging calibration. The elevation difference between the laser elevation and surface elevation was measured by RTK, and this elevation difference value was taken as the laser ranging error after the first calibration and is shown in

Table 2. The ranging error of the GF-7 laser based on three iterations of AW3D30 calibration.

Experiment	Beam Type	Surface Coordinates of Laser FOOTPRINTS			Surface Actual Elevation (m)	Ranging Error (m)
		Latitude (°)	Longitude (°)	Elevation (m)
First iteration	Beam 1	39.24695	115.82774	12.19	12.95	−0.76
	Beam 2	39.72343	116.10928	35.78	36.14	−0.36
Second iteration	Beam 1	39.24692	115.82793	12.30	12.95	−0.65
	Beam 2	39.72347	116.10908	35.86	36.14	−0.28
Third iteration	Beam 1	39.24692	115.82793	12.30	12.95	−0.65
	Beam 2	39.72347	116.10908	35.86	36.14	−0.28

. Then, the GF-7 laser beam 1 and beam 2 pointing angle after the first iterative calibration were taken as the center pointing angle, and a new pointing grid with the pointing range of ± 0.03 degrees and the grid spacing of 0.003 degrees was created. The first iterative calibration was repeated using the new pointing grid, and the GF-7 laser pointing after the second iterative calibration was calculated. The sum of the elevation residual surfaces after the second iterative calibration is shown in Figure 7c,d. After the second pointing iterative calibration, the new laser pointing angle was used to recalculate the elevations of the two laser footprints in Figure 1b,c. The elevation difference between the laser elevation and the true surface elevation was used as the laser ranging error to complete the second iterative pointing and ranging calibration. Finally, the third iterative calibration takes the laser pointing angle and ranging error after the second iterative calibration as the input parameters and completely repeats the second iterative calibration experiment. The sum of the residual surface elevation after the third iteration calibration is shown in Figure 7e,f, and the laser ranging error after the calibration is shown in Table 2.

Table 2 shows the coordinates of the laser footprints of GF-7 beam 1 (laser time code is 203685561.00) and beam 2 (laser time code is 203685553.00) after each calibration iteration and true surface elevation value. Note: the true surface elevations of the laser footprints calculated after each iterative calibration were the same because the selected surface was very flat, as shown in Figure 1b,c. Although the position of the footprint changed after iterative calibration, the surface elevation remained the same across the plane. As shown in Table 2, the calculated coordinates of the same laser footprint were completely consistent after the second and third iterations, which indicates that the GF-7 laser pointing and ranging calibration had converged after two iterations using AW3D30 DSM as the reference terrain. Figure 7 also illustrates the same result; for example, the surface plot of the sum of laser elevation residuals was completely consistent after the second and third iterations of calibration.

From the results of the first and second iterative calibrations shown in Figure 7 and Table 2, it can be seen that the laser pointing angle could still be changed after the pointing calibration was repeated by correcting the laser ranging error. However, whether the laser pointing accuracy was improved after the change needs further verification. We used the GF-7 laser pointing after calibration based on LGDs as the true value [10] to verify the GF-7 laser pointing accuracy before and after each iterative calibration.

Table 3. Laser pointing and ranging accuracy before and after three iterative calibrations based on AW3D30.

GF-7 Laser Pointing Type	Beam 1			Beam 2
	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{x}}(°)$	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{y}}(°)$	$\mathbf{\Delta }\mathit{\rho }\left(\mathbf{m}\right)$	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{x}}(°)$	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{y}}(°)$	$\mathbf{\Delta }\mathit{\rho }\left(\mathbf{m}\right)$
Initial pointing	0.031	−0.038	−1.01	0.107	−0.044	−1.26
Pointing after first calibration	−0.00043	0.00316	−0.08	0.00056	0.0033	0.14
Pointing after second calibration	−0.00043	0.00116	0.03	0.00056	0.0013	−0.06
Pointing after third calibration	−0.00043	0.00116	0.03	0.00056	0.0013	−0.06

shows the difference between the initial pointing of the GF-7 laser, the laser pointing and ranging after iterative calibration based on the AW3D30 DSM, and the real laser pointing and ranging. Among them, $\Delta {\alpha }_x$ is the difference between the laser pointing angle ${\alpha }_x$ after the calibration and the real pointing angle. $\Delta {\alpha }_y$ is the pointing difference of the pointing angle ${\alpha }_y$. $\Delta \rho$ is the difference between the laser ranging error after the calibration and the real ranging error.

The results in Table 3 show that the calibration method proposed in the paper had a fast convergence speed, and the results converged directly after two iterations. The main reason for this is that the input ranging error of beam 1 changed from 0 to −0.76 m, and that of beam 2 changed from 0 to −0.36 m (as shown in Table 2) during the second iteration. When performing the third iterative calibration, the ranging error of beam 1 changed by 0.11 cm, and the ranging error of beam 2 only changed by 0.08 cm. The change in the ranging error in the third iteration was much smaller than the change in the ranging error from the first to the second calibration. According to the error of the ranging error in Table 3, the ranging error was very close to the true value after the second iteration. Therefore, when there was a small change in the laser ranging, the result of the laser pointing calibration would not change again.

After the first iterative calibration, the laser pointing angle quickly approached the true value, in which the pointing angle accuracy of beam 1 was improved by more than 12 times, from the initial −0.038 degrees to 0.00316 degrees. The beam 2 pointing angle accuracy was improved by an order of 13 from the initial −0.044 degrees to 0.0033 degrees. After iterative calibration and convergence using AW3D30 DSM, the pointing accuracy of GF-7 beam 1 reached 0.00116 degrees (4.2 arcsec), the beam 2 pointing accuracy reached 0.0013 degrees (4.7 arcsec), and the ranging accuracies became 0.03 m and −0.06 m, respectively.

3.2. Elevation Accuracy Validation after Calibration

The accuracy of satellite-derived laser elevation measurements is a topic that has gained extensive interest in various industries. In this study, high-precision GCPs were used as elevation data to verify the elevation accuracy of GF-7 lasers after iterative calibration based on the AW3D30 DSM. First, taking the laser pointing and ranging parameters after the first and second calibration iteration as the input parameters, the coordinates of the 24 laser footprints of beams 1 and beam 2 were calculated, as shown in Figure 2. The four nearest GCPs were chosen to calculate the actual laser footprint elevation using bilinear interpolation. Then, the elevation differences between all laser elevations of beam 1 and beam 2 (as shown in Figure 2) and the actual elevation were calculated to verify the GF-7 laser elevation accuracy after calibration, and the results are shown in

Table 4. The elevation accuracy of GF-7 laser beam 1 after iterative calibration based on AW3D30.

Point Number	First Iteration					Second Iteration
	Latitude (°)	Longitude (°)	Laser Elevation	Latitude (°)	Longitude (°)	Laser Elevation	Latitude (°)	Longitude (°)
1	43.11004	111.78070	993.81	993.35	0.46	993.51	993.18	0.33
2	43.06836	111.76772	999.31	998.22	1.09	999.01	998.42	0.59
3	43.04752	111.76124	1046.36	1045.91	0.45	1046.06	1045.87	0.19
4	43.02668	111.75476	1043.75	1043.37	0.38	1043.45	1043.23	0.22
5	43.00584	111.74828	1029.45	1028.87	0.58	1029.15	1028.88	0.27
6	42.99202	112.13091	1074.51	1074.32	0.19	1074.21	1074.21	0
7	42.97117	112.12443	1084.89	1084.43	0.46	1084.59	1084.45	0.14
8	42.95032	112.11797	1092.55	1092.27	0.28	1092.25	1092.17	0.08
9	42.92947	112.11149	1092.13	1091.92	0.21	1091.83	1091.79	0.04
10	42.88124	112.46241	1039.26	1039.81	−0.55	1038.96	1038.76	0.2
11	42.86038	112.45595	1041.01	1040.58	0.43	1040.71	1040.51	0.2
12	42.83952	112.44949	1041.04	1040.45	0.59	1040.74	1040.45	0.29
13	42.81866	112.44304	1051.06	1050.69	0.37	1050.76	1050.68	0.08
14	42.77694	112.43014	1050.33	1050.10	0.23	1050.03	1049.95	0.08
15	42.75607	112.42369	1051.81	1051.45	0.36	1051.51	1051.26	0.25
16	42.73521	112.41726	1067.35	1066.96	0.39	1067.05	1066.87	0.18
17	42.71435	112.41082	1075.33	1074.93	0.4	1075.03	1074.85	0.18
18	42.87520	112.82920	1076.66	1076.42	0.24	1076.36	1076.25	0.11
19	42.72912	112.78404	1086.30	1085.93	0.37	1086.00	1086.71	−0.71
20	42.68738	112.77117	1090.22	1089.78	0.44	1089.92	1089.85	0.07
21	42.66651	112.76474	1089.20	1088.89	0.31	1088.91	1088.94	−0.03
22	42.56215	112.73264	1119.02	1118.76	0.26	1118.72	1118.74	−0.02
23	42.54128	112.72623	1133.86	1133.69	0.17	1133.56	1133.53	0.03
24	42.52040	112.71982	1136.73	1136.40	0.33	1136.43	1136.34	0.09
Mean value (m)					0.35	-	-	0.12
RMSE (m)					0.27	-	-	0.22

and

Table 5. The elevation accuracy of GF-7 laser beam 2 after iterative calibration based on AW3D30.

Point Number	First Iteration					Second Iteration
	Latitude (°)	Longitude (°)	Laser Elevation(m)	GCP Elevation(m)	Elevation Difference(m)	Laser Elevation(m)	GCP Elevation(m)	Elevation Difference(m)
1	43.04254	111.91306	1053.10	1053.50	−0.41	1053.50	1053.65	−0.15
2	43.02171	111.90653	1056.06	1056.84	−0.78	1056.47	1057.08	−0.61
3	43.00087	111.89999	1065.71	1066.36	−0.65	1066.12	1066.58	−0.46
4	43.00798	112.28877	1044.74	1045.48	−0.74	1045.15	1045.39	−0.24
5	42.98714	112.28224	1071.27	1071.65	−0.38	1071.68	1071.77	−0.10
6	42.96629	112.27571	1078.84	1079.30	−0.46	1079.24	1079.38	−0.14
7	42.94545	112.26918	1079.56	1080.14	−0.57	1079.97	1080.25	−0.28
8	42.92460	112.26266	1081.05	1081.82	−0.77	1081.46	1082.01	−0.55
9	42.90376	112.25614	1087.56	1088.05	−0.49	1087.97	1088.16	−0.19
10	42.87648	112.61299	1075.63	1074.01	1.62	1076.04	1075.71	0.33
11	42.85562	112.60648	1073.92	1074.32	−0.40	1074.33	1074.38	−0.05
12	42.83477	112.59997	1077.86	1078.32	−0.45	1078.27	1078.38	−0.11
13	42.81391	112.59347	1084.12	1084.59	−0.47	1084.53	1084.66	−0.14
14	42.79305	112.58697	1083.87	1084.27	−0.40	1084.27	1084.33	−0.06
15	42.75134	112.57397	1093.29	1093.78	−0.49	1093.69	1093.84	−0.15
16	42.73048	112.56748	1092.46	1092.87	−0.42	1092.86	1092.97	−0.11
17	42.70962	112.56099	1101.00	1101.40	−0.40	1101.40	1101.67	−0.26
18	42.68877	112.55450	1109.16	1109.65	−0.49	1109.56	1109.78	−0.22
19	42.66791	112.54801	1117.98	1117.92	0.06	1118.38	1118.03	0.36
20	42.70360	112.92741	1093.63	1094.06	−0.43	1094.03	1093.72	0.31
21	42.57840	112.88855	1120.68	1121.04	−0.36	1121.08	1121.12	−0.04
22	42.55754	112.88208	1125.42	1125.80	−0.39	1125.82	1125.59	0.23
23	42.51580	112.86916	1128.16	1128.44	−0.28	1128.56	1128.59	−0.03
24	42.49494	112.86270	1124.02	1124.25	−0.23	1124.42	1124.37	0.05
Mean value (m)					0.37	-	-	−0.11
RMSE (m)					0.46	-	-	0.25

. Because of the large initial pointing error of the GF-7 satellite laser, the GCP array measured by RTK was far from the initial laser footprint position; thus, it was difficult to verify the elevation accuracy of the GF-7 laser prior to calibration. The results in Table 3 show that the pointing and ranging error after the third calibration iteration were consistent with the results after the second iteration. Therefore, the elevation accuracy of the GF-7 laser after the third iteration was no longer verified here.

Note that the GCP elevations in Table 4 and Table 5 were different in the first and second iterations. The reason is that the laser pointing angle changed after the first and second iterations during calibration. This means that the positions of the laser footprints also changed. Thus, the GCPs around the laser footprints changed accordingly, resulting in different GCP elevations. To show the changes more clearly in the elevation accuracy of the GF-7 laser beam 1 and beam 2 after the iterative calibration, Table 4 and Table 5 are presented graphically in Figure 8.

The above figure and table results show that the iterative geometry calibration method for laser altimeter pointing and ranging separation performed without field detectors, as proposed in this paper, obviously improved the elevation accuracy of the GF-7 laser. Among these, the elevation accuracy of beam 1 was improved from 0.35 ± 0.27 m to 0.12 + 0.22 m after the first iteration, and the elevation accuracy of beam 2 was improved from 0.37 ± 0.46 m to −0.11 ± 0.25 m. It is worth noting that the GCP elevation in Table 4 and Table 5 used to verify the laser footprint elevation did change after the first and second iterations of calibration. The reason is that the laser pointing of the GF-7 laser changed after the second iterative calibration, resulting in a change in the surface position of the laser footprint. Therefore, the GCPs used to verify the elevation of the laser footprint were also different.

After iterative calibration using the AW3D30 DSM, the maximum elevation error of GF-7 laser beam 1 was approximately 0.34 m, and the maximum elevation error of beam 2 was approximately −0.36 m. In addition, beam 1 still had a systematic error of 0.12 m, and beam 2 also had a systematic error of −0.11 m. The main reason is that beam 1 and beam 2 of the GF-7 lasers still had pointing and ranging errors after calibration. The maximum pointing error of beam 1 was 4.2 arcsec after calibration, and that of beam 2 was 4.7 arcsec. If a satellite laser pointing error of 1 arcsec will lead to a 2.5 m positioning error in the laser footprint [24], there remains a positioning error of 10.5 m in the laser footprint of beam 1 and a positioning error of 11.8 m in beam 2. Because the grassland in Sonid Youqi, Inner Mongolia, is not as flat as the cement road, the changes to the laser-derived elevation accuracy resulting from a laser footprint positioning error of approximately 11 m cannot be estimated. Therefore, it is reasonable that an elevation error of 0.34 m persists after calibrating the GF-7 satellite.

4. Discussion

When the AW3D30 DSM was used as the reference terrain, the GF-7 satellite laser footprint still had a positioning error of 11 m after the iterative calibration of the pointing and ranging separation. We consider that the error was mainly caused by the accuracy of the reference terrain itself. The resolution of the AW3D30 DSM is 30 m, and the elevation accuracy is approximately 2 m [17]. The 30 m resolution of the reference terrain data resulted in the spacing of the laser pointing grids being only divided to 0.003°. The low elevation accuracy of the DSM also limited the laser pointing accuracy after iterative calibration. For all the above reasons, higher-precision reference terrain data will be used to discuss the calibration accuracy of this method.

The DSM image of Kuqa City in Xinjiang generated by the airborne LiDAR data was used as the reference terrain and had a horizontal 1 m resolution. The methods discussed in this paper were used to repeat the pointing and ranging calibration of the GF-7 beams 1 and 2. Here, the first iterative calibration based on LiDAR took the GF-7 laser pointing angle after the second iterative calibration based on AW3D30 (as presented in Table 3) as the initial pointing, and the grid range was set to three times the grid spacing of the second iterative calibration (which was based on the AW3D30 DSM). The grid spacing was 0.0001 degrees (corresponding to the 1 m resolution of the LiDAR data). The surface plots of the sum of the laser elevation residuals during the first iterative calibration based on the LiDAR data are shown in Figure 9a,b. Similar to the ranging calibration method based on the AW3D30 DSM, the two laser footprint elevations in Figure 1b,c were calculated using the pointing angles derived following after the LiDAR calibration, and the elevation difference between the laser elevation and the true surface elevation was used as the laser ranging error. The ranging error of GF-7 laser beam 1 was −0.65 m, and beam 2 was −0.28 m. Subsequently, the second iterative calibration experiment used the first LiDAR calibration result as the input to completely repeat the previous calibration experiment, including the grid pointing range and spacing settings. The surface plots of the sum of the laser elevation residual were calculated as shown in Figure 9c,d.

In Figure 9, there is still an obvious minimum value in the surface plot of the sum of the elevation residuals, but the grid spacing was 30 times smaller than the grid spacing set during the AW3D30 DSM calibration. That is, with the improvement in the accuracy of the reference terrain, the laser pointing optimal solution still existed. According to Figure 9, however, the surface plots of the sum of the elevation residuals after the second calibration were basically the same as the surface plots of the sum of the elevation residuals after the first calibration. This indicates that the GF-7 laser converged after the first calibration when using LiDAR data for reference. Similarly, the GF-7 laser pointing angle and ranging errors derived from the LGD calibration were used as the true values to verify the pointing and ranging accuracies derived from the LiDAR data calibration. The results are shown in

Table 6. Laser pointing angle and ranging accuracy of GF-7 laser calibrations based on AW3D30 and LiDAR.

GF-7 Laser Pointing Type	Beam 1			Beam 2
	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{x}}(°)$	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{y}}(°)$	$\mathbf{\Delta }\mathit{\rho }\left(\mathbf{m}\right)$	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{x}}(°)$	$\mathbf{\Delta }{\mathit{\alpha }}_{\mathit{y}}(°)$	$\mathbf{\Delta }\mathit{\rho }\left(\mathbf{m}\right)$
Initial pointing	0.031	−0.038	−1.01	0.107	−0.044	−1.26
Pointing after second iteration based on AW3D30	−0.00043	0.00116	0.03	0.00056	0.0013	−0.06
Pointing after first iteration based on LiDAR	0.00033	−0.00056	−0.02	0.00036	0.0006	−0.01
Pointing after second iteration based on LiDAR	0.00033	−0.00056	−0.02	0.00036	0.0006	−0.01

.

The calibration results in Table 6 also show that the calibration based on LiDAR data converged after one iteration. The reason is that the laser pointing and ranging accuracy were already very high after the first iterate when using LiDAR. Compared with the calibration results based on the AWD3D30 DSM, the method discussed in this paper uses LiDAR data as the reference terrain to further improve the laser pointing and ranging accuracy of the GF-7 satellite. Using this method and LiDAR data nearly doubled the accuracy. Among these, the pointing accuracy of GF-7 beam 1 was −0.00056 degrees (−2 arcsec) and the ranging accuracy was −2 cm after calibration, while the pointing accuracy of GF-7 laser beam 2 was 0.0006 degrees (2.2 arcsec) with a ranging error of −1 cm after calibration. If the pointing error was roughly converted into the GF-7 satellite laser footprint positioning error, the positioning error of beam 1 after calibration was 5 m, and that of beam 2 was 5.5 m.

The above results verified that the laser pointing and ranging accuracy of the GF-7 satellite was improved by using the LiDAR data as the reference terrain. However, whether the GF-7 satellite laser pointing and ranging accuracy can improve the laser elevation accuracy needs further verification. Based on the LiDAR data, the results of the second iterative calibration were consistent with the results of the first iterative calibration. We directly used the GF-7 satellite laser pointing and ranging parameters after the first iterative calibration to calculate the surface coordinates of the same GF-7 laser footprints in Table 4 and Table 5. Finally, the calculated laser elevation was verified with GCPs, and the results are shown in

Table 7. The elevation accuracy of the GF-7 laser beams 1 and 2 after iterative calibration based on LiDAR.

Beam 1				Beam 2
Point Number	Laser Elevation	GCP Elevation	Elevation Difference	Point Number	Laser Elevation	GCP Elevation	Elevation Difference
1	993.51	993.48	0.03	1	1053.51	1053.58	−0.07
2	999.01	998.63	0.37	2	1056.48	1056.79	−0.31
3	1046.06	1045.91	0.15	3	1066.13	1066.41	−0.29
4	1043.45	1043.28	0.17	4	1045.16	1045.34	−0.18
5	1029.15	1028.99	0.15	5	1071.68	1071.70	−0.02
6	1074.21	1074.23	−0.02	6	1079.25	1079.31	−0.06
7	1084.59	1084.61	−0.02	7	1079.98	1080.16	−0.18
8	1092.25	1092.26	−0.02	8	1081.47	1081.72	−0.25
9	1091.83	1091.82	0.00	9	1087.97	1088.04	−0.07
10	1038.96	1038.84	0.12	10	1076.04	1076.04	0.00
11	1040.71	1040.59	0.12	11	1074.34	1074.30	0.04
12	1040.74	1040.68	0.07	12	1078.28	1078.27	0.01
13	1050.76	1050.74	0.02	13	1084.54	1084.55	−0.01
14	1050.03	1049.87	0.16	14	1084.28	1084.27	0.01
15	1051.51	1051.36	0.15	15	1093.70	1093.80	−0.10
16	1067.05	1066.93	0.12	16	1092.87	1092.91	−0.04
17	1075.03	1074.92	0.10	17	1101.41	1101.34	0.07
18	1076.36	1076.30	0.06	18	1109.57	1109.72	−0.14
19	1086.00	1086.02	−0.03	19	1118.39	1118.16	0.23
20	1089.92	1089.95	−0.04	20	1094.04	1094.02	0.02
21	1088.90	1089.06	−0.15	21	1121.09	1121.04	0.05
22	1118.72	1118.80	−0.08	22	1125.83	1125.81	0.02
23	1133.56	1133.58	−0.02	23	1128.57	1128.55	0.03
24	1136.43	1136.46	−0.03	24	1124.43	1124.37	0.06
Mean value (m)		0.06		Mean value (m)		−0.05
RMSE (m)		0.11		RMSE (m)		0.13

.

The line graphs shown in Figure 10 are used to visually display the GF-7 satellite laser elevation accuracies after the first and second calibrations using the AW3D30 DSM and the first iteration using the LiDAR reference data.

The results in Figure 10 show that when higher-precision reference terrain data are used for calibration, the elevation accuracy of the GF-7 satellite laser can be further improved. Compared with the elevation error of the GF-7 satellite laser by the AW3D30 DSM, the systematic laser elevation error after calibration using LiDAR data was suppressed and basically fluctuated around the 0 value. Additionally, the trend of the laser elevation accuracy was completely different after calibration with ALOS, as shown in Figure 10a,b. This is because there was still approximately a 10 m positioning error in the GF-7 laser footprints after calibration. In addition, the larger elevation errors for the individual laser footprints that remained after calibration using the AW3D30 DSM became relatively small after the LiDAR-based calibration, such as the 19th point of GF-7 laser beam 1. After calibration based on the LiDAR data, the elevation accuracy of GF-7 beam 1 was improved from 0.12 ± 0.22 m to 0.06 ± 0.11 m and beam 2 was improved from −0.11 ± 0.25 m to −0.05 ± 0.13 m. Based on the calibration method proposed in this paper, the GF-7 satellite laser elevation accuracy was approximately 0.18 m after being calibrated with the high-precision LiDAR data.

In summary, when using higher-precision LiDAR data for terrain matching, the calibration method proposed in the article can double the pointing and ranging accuracy of the satellite laser. The elevation accuracy of the GF-7 satellite was 0.18 m after using the LiDAR data for calibration. The elevation accuracy of the GF-7 satellite laser after being calibrated by LGDs was 0.1 m [10]. The calibration accuracy of the method presented here was lower than that based on LGDs. The main reason is that there was a positioning error of approximately 5 m in the laser pointing angle after applying the calibration method discussed in this paper.

5. Conclusions

This work presents an iterative geometry calibration method for satellite laser altimeter pointing and ranging separation that is not dependent on field detectors. The satellite laser pointing calibration is first completed based on the terrain matching method, and then the satellite laser ranging error calibration is achieved by accurately measuring the ground elevation of the laser footprint. The calibrated laser pointing and ranging errors are used to iteratively calibrate the satellite laser pointing and ranging errors. Then, this work also discussed the calibration accuracy of our calibration method when using the AW3D30 DSM and high-precision LiDAR data. Finally, after the laser calibration of the GF-7 satellite using the method outlined in this paper is complete, the pointing and ranging error accuracy of the GF-7 satellite laser was verified by using the calibration results from LGDs, and the GCPs were used to verify the elevation accuracy of the GF-7 satellite laser. According to the experimental results in this article, the following four main conclusions are drawn.

(1)

The iterative calibration method for the separation of laser pointing and ranging proposed in this paper solves the ranging calibration problem that cannot be solved using traditional terrain matching laser calibration methods. The method proposed here can further improve the calibration accuracy of satellite lasers.

(2)

The convergence speed of this calibration method is fast. Typically, the satellite laser pointing angle converges after 2–3 iterations. Taking the AW3D30 DSM as the reference terrain data, the pointing angle accuracy of GF-7 beam 1 was 4.2 arcsec, and the ranging accuracy was 3 cm after the calibration. The pointing accuracy of the beam 2 was 4.7 arcsec, and the ranging accuracy was 6 cm.

(3)

High-precision DSM data can improve the accuracy of the calibration method. Based on the calibration results from the AW3D30 DSM as the reference terrain, when LiDAR data was used as the reference terrain data, the laser pointing and ranging accuracy of the GF-7 satellite was improved. The improved laser pointing accuracy of GF-7 satellite beam 1 was 2 arcsec, and the ranging accuracy was 2 cm. The GF-7 laser beam 2 pointing accuracy was 2.2 arcsec, and the ranging accuracy was 1 cm.

(4)

Using high-precision LiDAR data as the reference terrain data, the elevation accuracy of the GF-7 satellite laser was 0.18 m in flat terrain after the calibration, which was slightly lower than that of the calibration method based on the ground detector. However, it was still slightly lower than that of the calibration method based on LGDs.

In summary, the application and implementation of the calibration method presented in this paper can provide a new high-precision geometric calibration technique for satellite lasers when ground detector calibration cannot be performed. However, at present, we are only conducting calibration experiments on barren, sloped terrain in northern China. To further examine the applicability of this method, subsequent studies will specifically consider more varied terrain types with different slopes.