On-orbit geometric calibration of satellite laser altimeters using infrared detectors and corner-cube retroreflectors

ABSTRACT After being launched into orbit, the geometric calibration of a satellite laser altimeter will reduce errors in laser pointing and ranging caused by satellite vibrations during launch, environmental changes, and thermal effects during long-term operation, which guarantees the accuracy of measurement data. In this study, a satellite laser geometric calibration method combining infrared detectors and corner-cube retroreflectors (CCRs) is proposed. First, a CCR-based laser ranging error calibration method was established, and then a laser pointing error calibration model was derived based on a single infrared detector array. Taking GaoFen-7 (GF-7) satellite laser beam 2 as the experimental object, laser geometric calibration was realized using an infrared detector and CCR-measured data. Then, the accuracy of the proposed method was compared with that of other calibration methods, the CMLID and the CMSPR. The results show that the accuracy of the proposed calibration method is equivalent to that of the CMLID and higher than that of the CMSPR. Among them, the accuracy of the laser pointing after calibration using the proposed method is better than 0.8 arcsec, and the elevation accuracy of the laser on flat, sloping, and mountainous terrains is better than 0.11 m, 0.30 m, and 1.80 m, respectively.


Introduction
In recent decades, with the rapid development of laser technology, lasers have been used on satellite platforms for Earth observation.In January 2003, NASA launched the world's first Earth observation satellite laser altimeter, the Geoscience Laser Altimeter System (GLAS) (Schutz et al. 2005), which was decommissioned in 2009.Subsequently, in the second half of 2018, NASA launched the Advanced Topographic Laser Altimeter System (ATLAS) (Neumann et al. 2019) and the Global Ecosystem Dynamics Investigation (GEDI) (Dubayah et al. 2022) laser altimeter systems.In addition, the Land Satellite Remote Sensing Application Center of the Ministry of Natural Resources of China launched the GaoFen-7 (GF-7) satellite laser altimeter and ZiYuan3-03 (ZY3-03) satellite laser altimeter in November 2019 and July 2020, respectively (Xie et al. 2020;Tang et al. 2019b).The National Forestry and Grassland Administration of China launched the Terrestrial Ecological Carbon Monitoring (TECM) Satellite laser altimeter in August 2022 (Sun, Wang, and Lv 2022).The resulting satellite laser data have been widely used in polar ice cover monitoring (Yang et al. 2010), land surface measurements (Abshire et al. 2005), vegetation height, forest carbon stock estimation (Simard et al. 2008), etc., achieving substantial progress in research and results.
A laser altimeter is affected by satellite jitter during the launch process, sudden changes in the orbital operating environment, and thermal effects during measurement, which lead to certain changes in satellite laser the pointing angle and ranging, resulting in a system error in the pointing angle and ranging.Assuming that the altitude of a satellite orbit is 500 km and the laser pointing angle error is 30 arcsec, there will be a laser footprint positioning error of 73 m, and when the footprint is on terrain with a slope of 1°, it will produce an elevation measurement error of 1.3 m (Luthcke et al. 2000).The satellite laser ranging error magnitude is usually small (decimeter level), which will directly contribute to errors in laser elevation measurements.On-orbit satellite laser geometric calibration is the main means of eliminating or reducing laser pointing angle and ranging errors and can improve satellite laser footprint positioning accuracy.In this way, high-precision laser data are provided for satellite lasers application in various fields, ensuring the accuracy and effectiveness of the application results.
Commonly used on-orbit satellite laser altimeter geometric calibration methods include the following.Magruder et al. proposed a calibration method based on laser infrared detectors (CMLID) (Magruder, Suleman, and Schutz 2003;Magruder et al. 2006;Magruder, Schutz, and Silverberg 2003), which captures two or more laser footprints by deploying multiple infrared detector arrays, extracts the centroid of the laser footprint, and uses the centroid as a ground control point to calibrate the satellite laser pointing angle and ranging error.This calibration method currently has the highest accuracy.Luthcke et al. achieved geometric calibration of the GLAS profile lasers (Luthcke et al. 2005) and ATLAS single-photon lasers (Luthcke et al. 2020) using a satellite attitude maneuver method.Magruder et al. successfully tagged ATLAS return photon clouds using corner-cube retroreflectors (CCRs) and verified a mean ICESat-2 geolocation measurement accuracy of 3.5 m ± 2.1 m (Magruder et al. 2021).Tang et al. (2019a) proposed a satellite laser pointing calibration method based on terrain matching.By traversing the laser pointing angles, the laser pointing angle with the smallest elevation residual between the laser and digital surface mode (DSM) was determined as the optimal pointing angle.Building on the research of Tang et al (2019a)., Liu et al. completed the calibration of GF-7 satellite lasers using a field-free calibration method based on the separation between satellite laser pointing and ranging (CMSPR) (Liu et al. 2022).This method compensates for the problem where a terrain matching method cannot be used to calibrate the ranging error, but there is still an error of 2 arcsec in the laser pointing following calibration.Xu et al. proposed a satellite laser and line array camera joint calibration method (Xu et al. 2022).With this method, the camera image is matched through the laser point cloud, thereby establishing a joint calibration model to achieve laser geometric calibration.The aforementioned studies have contributed to the field of satellite laser geometric calibration, and among them, high-precision calibration methods include the CMLID and satellite attitude maneuver calibration method.However, the latter methods are difficult to apply to platform stability satellite laser altimeters, such as the GEDI, GF-7, ZY3-03, TECM, and other satellite lasers.The CMLID method needs to deploy multiple infrared detector arrays, and field experiments are expensive.
In this paper, a satellite laser geometric calibration method that combines laser infrared detectors and CCRs (CMLIDC) is proposed, which can be used to complete the laser pointing angle and ranging error calibration by only deploying dense infrared detectors and sparse CCRs around the same laser footprint.In addition, this method changes existing calibration methods that calibrate the pointing angle and ranging error simultaneously or calibrate the pointing angle and then the ranging error.Rather, the laser ranging is calibrated first and then the laser pointing angle is calibrated.In this study, a CCR-based satellite laser ranging error calibration method was first established, and a laser pointing angle error calibration model based on a single infrared detector array was derived in detail.Then, taking laser beam 2 of the GF-7 satellite as an experimental object, infrared detectors and CCRs were deployed around the laser ground footprint at the same time, and the laser pointing angle and ranging errors were calibrated using the infrared detector and CCR measured data.Finally, the difference in accuracies between the CMLID, CMSPR, and the proposed calibration method was verified using the laser footprints captured by other infrared detectors.Additionally, the accuracy of satellite laser elevation measurements after calibration using these three methods is discussed.

Materials
This study conducted experiments from three aspects: satellite laser calibration, verification of laser positioning accuracy after calibration, and verification of laser elevation measurement accuracy after calibration.This section provides a detailed introduction to the experimental site, data, and equipment from four aspects, namely, the calibration and positioning accuracy verification site, the laser elevation measurement accuracy verification area and DSM data, the GF-7 satellite laser data, and the infrared laser detectors and CCRs.

Calibration and positioning accuracy verification site
For the experiment in this study, laser infrared detectors and CCRs must be deployed in the ground calibration field at the same time.The calibration field should be selected in a flat area, without tall vegetation on the ground surface and dry and rainless all year round.Therefore, the GF-7 satellite laser field calibration site in this study is located in the Sonid Right Banner, Inner Mongolia, China.The surface is mainly sand and stone, and the scope of the calibration site is shown in Figure 1.In June and September 2020, we used infrared detectors to capture 4 GF-7 satellite beam 2 laser footprints, denoted with the middle circle and the star shown in Figure 1, at the Sonic Right Banner calibration field in Inner Mongolia, China.On September 19, 2020, the infrared detector and CCR were simultaneously deployed at the laser footprint 212021051.00position, denoted with the pentagram in Figure 1, which serves as the calibration laser footprint in this study.Furthermore, the three red circles in Figure 1  each laser footprint was extracted from the infrared detector measured data, which serves as the actual centroid of the laser footprint for validating the laser positioning accuracy after calibration.

Laser elevation measurement accuracy verification area and DSM data
In this study, the eastern region of Kuqa city, Xinjiang, China was selected as the elevation accuracy verification experimental area, which is shown in Figure 2(a).This area includes three types of typical terrain: flat land, slopes, and mountains.The terrain undulations in the north-south direction in the experimental area are shown in the Figure 2(c-e) elevation profiles, which are the three-orbit satellite laser satellite point track elevation profiles shown in Figure 2(a).The surface of the experimental area is dominated by the Gobi, and its elevation is negligibly affected by time changes.The high-precision DSM is used as the elevation accuracy verification data.We used Leica Terrain Mapper light detection and ranging (LiDAR) to obtain the laser point clouds over the entire experimental area and generated a DSM with a resolution of 1 m and an elevation accuracy better than 0.17 m, as shown in Figure 2(b).After postfiltering and control point correction, the elevation accuracy of the DSM is better than 0.1 m.

GF-7 satellite laser data
The GF-7 satellite beam 2 employs a near-infrared laser with a wavelength of 1064 nm for Earth observation, and the measurement frequency is 3 Hz.The laser divergence angle is designed to be 35 urad, resulting in a footprint diameter of approximately 17.5 m on the surface.The GF-7 satellite laser also has the capability to record echo waveforms.For detailed hardware parameters and specifications, please refer to the literature by Xie et al. (2020).
In the field, we deployed infrared detectors and CCRs around the GF-7 satellite beam 2 laser footprint 212021051.00at the same time.Therefore, GF-7 satellite beam 2 was taken as the experimental object in this study.The GF-7 satellite laser data can be divided into three categories: calibration laser data, positioning accuracy verification laser data, and elevation measurement accuracy verification laser data.Each satellite GF-7 laser footprint has a unique time code (time label), and the time code is the accumulated seconds since January 1, 2014.The time accuracy recorded by the GF-7 satellite laser is at the microsecond level (10 −6 seconds).Considering that the GF-7 laser frequency is 3 Hz and to display the laser time code concisely, only two decimal places of the time code are retained in this study, and the GF-7 satellite laser time code was used as the only label for the laser footprint.The calibration laser data are laser footprint 212021051.00on September 19, 2020, as shown by the star in Figure 1.The laser data for positioning accuracy verification are the other laser footprints captured by the satellite GF-7 beam 2 infrared detector in the same year, and their distribution is shown in Figure 1.The basic calibration and positioning accuracy verification data information in this study is shown in Table 1.
As of September 30, 2022, three GF-7 satellite beam 2 laser orbits had passed over the experimental area in Figure 2(a).After removing the invalid data caused by the atmosphere, 77 laser footprints remained in the experimental area, including 45 laser footprints on flat terrain, 19 laser footprints on sloping terrain, and 13 laser footprints on mountainous terrain.Their distribution is shown in Figure 2(a).These 77 laser footprints were used as the experimental objects to verify the elevation measurement accuracy of the calibrated GF-7 satellite beam 2 in different terrain.The basic information of these laser footprints is shown in Table 2. Since the laser frequency of the GF-7 satellite is 3 Hz (Tang et al. 2019b), the laser time code is approximately fixed at *.00 seconds, *.33 seconds, and *.67 seconds.The time code intervals of the laser footprint are listed in Table 2.

Laser infrared detector and CCR
(1) Corner Cube Retroreflector Considering the uncertainty in the CCR placement and the fluctuations in atmospheric transmittance at the deployment site, along with the dynamic response range of the GF-7 satellite laser altimeter, a 12.5 mm aperture size was designed for the CCR.This size compensates for the satellite velocity aberration angle, ensuring that the laser detector receives the CCR echo signal from the central bright spot.During installation, there may be slight angular errors that affect the incident angle of the laser beam.However, if the angle error is controlled within a range of 2°, CCR tests have shown that although there may be slight fluctuations in the optical reflectivity of the CCR, its reflectivity remains above 80%.The CCR is precision-ground from K9 glass, and a physical photo is shown in Figure 3(a), while the internal structure is shown in Figure 3(b).
(2) Laser infrared detector The infrared laser detector for the GF-7 satellite employs a positive intrinsic-negative diode that covers the 1064 nm wavelength as its core photoelectric conversion device, supplemented by narrowband filters to detect the GF-7 satellite narrow laser pulses.According to laboratory tests, the infrared detector has a dynamic detection range of over 30 times from 3.46 nJ/cm 2 to 110.4 nJ/ cm 2 , and the equivalent surface reflectance ranges between 0.03 and 0.6.To ensure that the sensitivity change of the satellite laser within the ±5° incident angle range is less than 5%, a converging lens is installed at the front end of the photodiode inside the infrared detector.In addition, the infrared detector uses narrowband filters to prevent background scattered light in the air above the infrared detector from entering the photosensitive area of the photodiode.By reducing circuit noise through circuit noise reduction methods, the false trigger rate of the entire laser infrared detector is lowered, ultimately ensuring that the false trigger rate of the infrared detector is less than 0.1%.During the production of the infrared detector, electronic components with consistent performance, such as voltage regulators, divider resistors, and comparators, were chosen, and highprecision voltage and amplification circuits were used to compensate for the energy uniformity differences between infrared laser detectors, guaranteeing that the energy consistency difference for each GF-7 satellite infrared laser detector does not exceed 3%.A physical photo of the developed GF-7 satellite infrared laser detector is shown in Figure 3(c), while its internal structure is displayed in Figure 3(d).

Overall technical process
In this paper, a satellite laser geometric calibration method combining laser infrared detectors and CCRs is proposed, which simultaneously deploys laser infrared detectors and CCRs in the same laser footprint array.Using CCR-measured data for satellite laser ranging error calibration was prioritized, and then satellite laser pointing angle error calibration based on laser infrared detector-measured data was performed.Finally, the calibrated laser pointing angle and elevation measurement error were verified.The overall technical process is shown in Figure 4.
As shown in Figure 4, the method is divided into the following three steps.(1) Calibration of laser ranging error based on the CCRs Laser ranging error calibration refers to calibrating the systematic error of the laser ranging system.First, the actual laser ranging value is calculated using the CCRs.Then, based on the laser transit time and atmospheric parameters, the self-measured laser ranging value and the laser ranging refraction error are calculated.Finally, the laser ranging error is calculated based on the actual laser ranging value, the self-measured laser ranging value, and the laser ranging refraction error.
(2) Calibration of laser pointing angle based on the infrared laser detectors After the satellite passes over the deployed infrared detectors and the CCR calibration site, the triggered infrared detector coordinates and energy values are used to fit the laser footprint and extract its centroid.The laser footprint centroid is treated as a ground control point, and combined with satellite attitude, orbit data, and calibrated laser ranging values, the satellite laser pointing angle calibration model is used to calibrate the satellite laser pointing angle.
(3) Accuracy assessment after laser calibration After completing the calibration of the satellite laser pointing angle and ranging bias, the differences in laser pointing angles and ranging errors between the calibrated results are compared using the proposed method and those obtained from the CMLID and the CMSPR, thus directly verifying the accuracy of the proposed calibration method.Then, utilizing high-precision DSM data, the laser elevation measurement accuracy for three typical terrains calibrated using the proposed method is verified and evaluated.

Calibration of laser ranging error based on CCRs
As shown in Figure 5, the satellite laser altimeter can record the transit time between the laser and the CCR, and the laser ranging value r tof is calculated according to the speed of light c (Xie and Liu 2021).The atmospheric refraction error r atm caused by the influence of atmospheric refraction in laser transmissions can be calculated according to an atmospheric delay correction model (Thomas and Katherine 2012).In addition, some laser ranging errors r sys are caused by factors such as laser timing and circuit response in the laser ranging value, which can be calculated with the formula (1).
where r true refers to the actual ranging value from the laser to the CCR.If the laser light emitting point and CCR coordinates are known, the actual laser ranging value can be calculated according to the following formula.
where (x lp , y lp , z lp ) refers to the three-dimensional coordinates of the laser light emitting point, which is obtained by interpolating the postprocessed orbit determination data within the laser time; and (x ccr , y ccr , z ccr ) represents the CCR ground coordinates.The calculation of the CCR ranging value in formula (1) also requires the use of a triggered infrared detector.The specific implementation steps are as follows: (1) The CCR matches the echo waveform.The CCR inside the laser footprint is identified using the triggered infrared detectors, which is considered the effective CCR.The measured height of the effective CCR is used to match the waveform height of the CCR to determine the CCR corresponding to each echo peak.As shown in Figure 5, the last peak in the laser echo waveform is the ground echo signal, and the preceding peak is the CCR echo signal.The height of the CCR waveform is obtained by multiplying the time difference between each peak and the last peak in the waveform by the speed of light.(2) Accurate calculation of the CCR return signal time.The laser echo waveform can be considered the cumulative sum of multiple Gaussian components (Hofton, Minster, and Blair 2000).First, a Gaussian decomposition method is used to extract the CCR echo waveform.The Gaussian decomposition model is shown in formula (3).The single waveform peak time obtained via waveform decomposition is the precise return time of the CCR.For the detailed implementation process of waveform decomposition, please refer to the literature (Wang et al. 2013;Huang et al. 2018;Qin, Vu, and Ban 2012).
where f (t) is the approximated waveform; k is the number of CCR Gaussian components; A i is the amplitude of the i − th Gaussian component; m i is the time of the i − th Gaussian peak; and s i is the standard deviation of the i − th Gaussian component.
(3) Using the return time t ccr of the CCR signal calculated using the above steps, the laser light emitting time t 0 , and the speed of light c, the laser ranging value of the CCR is calculated according to formula (4).

Calibration of laser pointing angle based on the laser infrared detector
A satellite laser accurately measures the distance between the satellite and a ground target, considers the laser pointing angle and satellite orbit/attitude data, and corrects for atmospheric and tidal errors, thereby deriving a satellite laser geometric positioning model according to the geometric relationship between the satellite laser and the ground footprint, as shown in formula (5).denotes the offset of the laser relative to the center of mass of the satellite; r true is the actual ranging value of the laser after correcting the atmospheric delay and laser ranging errors; h tide is the tidal error; and a x , a y , and a z are the three components of the satellite laser pointing angle, that is, the angle between the optical axis and the X, Y, and Z axes of the satellite coordinate system, respectively.
The satellite laser pointing angle usually refers to the three-axis angle between the laser optical axis and the SBF.An infrared detector array can provide the three-dimensional coordinates of the laser footprint on the ground.According to formula (5), single laser footprints can only list three equations, and the pointing angle to be obtained also has three parameters.Since there are no redundant observations when calculating the pointing angle, errors occur in the pointing angle calibration.Therefore, in this study, the three-dimensional pointing angle of the satellite laser is converted into a two-dimensional pointing angle according to the projection relationship between the straight line and the surface, as shown in Figure 6.
According to Figure 6, the 8th item in formula (5) (i.e. the laser pointing angle matrix) can be converted into formula (6).
⎦ , and substitute formula(6) into formula (5) to establish the satellite laser pointing calibration condition equation, as shown in formula (7).
The condition equations in formula ( 7) are obviously nonlinear and are linearized using a firstorder Taylor expansion.
To simplify the derivation process, define- The linearized satellite laser pointing calibration error equation is derived as follows.
where F 0 X , F 0 Y , and F 0 Z are the calculation results from substituting the laser pointing angle before calibration as an input parameter into formula (7).Again, letting ⎦ , and X = da db   , the satellite laser pointing calibration error formula ( 8) is rewritten in matrix form.
According to the least squares principle, the correction amount of the two-dimensional pointing angle of the satellite laser can be solved, and the results are shown below.
where P is the weight matrix.For the same laser footprint, the satellite laser is under the observation condition of equal precision, that is, the matrix P is an identity matrix.
To improve the pointing calibration accuracy, multiple iterative solutions to the above equations must usually be performed.In the initial calibration experiment, the satellite laser pointing angle measured in laboratory is used as the input parameter.Then, the laser pointing after the last calibration is used as the initial parameter, and the satellite laser pointing angle is iteratively calculated until convergence.The experimental convergence condition is that da , 0.00001 o and db , 0.00001 o are satisfied simultaneously.Assuming that the input laser pointing angle for the ith calibration is a i and b i , and the pointing angle correction amount after calibration is da i and db i , the laser pointing angle after the ith calibration is as follows.

Laser elevation measurement accuracy assessment after calibration
The accuracy of satellite laser elevation measurements is an important index to measure the performance of satellite laser altimeters and to test the accuracy after satellite laser calibration.In this study, the GF-7 satellite laser point on three typical terrains of flat, sloping, and mountainous was taken as the experimental object, and high-precision DSM was used as the verification data.In the experiment, the coordinates of the GF-7 satellite laser point are used as the input data to determine the DSM grid where the laser point is located and obtain the elevations of the four corner points of the grid.According to these four corner point elevations, the DSM elevation of the laser point is calculated using the bilinear interpolation method, and the calculation formula is below.
where h(x, y) is the DSM elevation of the laser point; h 1 -h 4 refers to the elevation of the four corner points of the DSM grid; DSM is a regular grid, so the coordinates of h 1 -h 4 are (x 1 , y 1 ), (x 2 , y 1 ), (x 1 , y 2 ) and (x 2 , y 2 ).
Taking the DSM elevation of the laser point as the true elevation value, the differences between the elevation h ′ i of all the laser points on the three types of terrain and the true elevation are calculated.Finally, the mean value Dh and root mean square error (RMSE) dh of the laser point elevation difference are used as the evaluation index for laser height measurement accuracy, and the calculation formulas of Dh and the RMSE are shown below.

Detector deployment for field calibration
According to the estimated GF-7 satellite beam 2 laser ground position (the satellite laser footprint estimation method can be found in Xie et al. (2017)), the laser point 212021051.00on September 19, 2020, is selected as the laser footprint for the calibration experiment.Taking the estimated surface coordinates of the laser footprint as the center, the laser infrared detectors and CCRs are simultaneously deployed within a range of 175 m × 70 m (along the track × across the track), forming a detector array.To ensure that at least 9 laser infrared detectors and 1 CCR are triggered, the laser infrared detector deployment spacing is set to 5 m, and the CCR deployment spacing is set to 10 m.The joint deployment design scheme of the laser infrared detector and CCR is shown in Figure 7(a), and the site deployment layout is shown in Figure 7(b).In addition, the CCR aperture is 6 mm, and the CCR height is 1.5 m or 3 m.

Satellite laser ranging error calibration
The satellite laser footprint falls within the CCR array, and its echo waveform are marked as multiple peaks by the CCR inside the laser footprint and the ground.Figure 8 shows the original echo waveform of laser footprint 212021051.00with three peaks in the waveform, and the last peak is the ground surface echo waveform.The first two peaks are signals returned by CCRs at different heights.First, using the Gaussian decomposition method, the original waveform is decomposed into three independent waveforms defined as Waveforms #1, #2, and #3, as shown in Figure 8.Then, the time differences between the peak values of Waveforms #1 and #2 relative to that of Waveform #3 are calculated and converted to an elevation difference, and the results are shown in Table 3.
To match the decomposed waveform with the CCR, the shape of the laser footprint is fitted according to the energy value of the triggered laser infrared detector and its ground coordinates.The results are shown in Figure 9.Moreover, the nearest neighbor CCR is determined and named CCR #1 -#8.Subsequently, the surveyors used a tape measure to accurately measure the actual heights of the 8 CCRs, and the results are shown in Table 4. Comprehensively considering Table 3 and Table 4 indicates that Waveform #1 corresponds to CCRs #1 and #7, and Waveform 2 corresponds to CCRs #2 and #5. Figure 9 clearly shows that CCRs #5 and #7 are located outside the edge of the laser footprint and thus are invalid CCRs.Therefore, Waveform #1 corresponds to CCR #1, and Waveform #2 corresponds to CCR #2.
Finally, the precise satellite orbit data are interpolated using the calibration laser footprint time and converted to the coordinates of the laser light emitting point.The CCR coordinates are measured using RTK and are in China Geodetic Coordinate System 2000 (CGCS2000) three-degree zone projection coordinates.Additionally, the actual distance r true from a CCR to the laser light emitting point is calculated using formula (2).Based on the National Centers for Environmental Prediction (NCEP) atmospheric parameters, the atmospheric ranging correction value of the laser footprint position r atm is calculated.Based on the CCR echo time and the laser light emitting time, the CCR laser transit time r tof is calculated.Finally, the laser ranging errors r sys of CCRs #1 and #2 are calculated using formula (1).All the results of these calculations are shown in Table 5.These results show that the laser ranging errors calculated from different CCRs are not exactly the same, which is caused by random errors in field measurements.To balance these errors, the average of the laser ranging errors calculated using CCRs #1 and #2 is taken as the final laser ranging error.In summary, the laser ranging error of GF-7 satellite beam 2 after calibration using the CCRs is −0.43 m.

Calibration and verification of the satellite laser pointing angle
A laser infrared detector can respond to the energy of the laser footprint reaching the surface, and it is displayed as a DN value, as shown in Figure 7(b1).The larger the DN value is, the stronger the laser energy, which means that the laser infrared detector is closer to the center of the laser  footprint.The upper limit of the DN value is 4096.RTK is used to measure the surface coordinates of each triggered laser infrared detector.An elliptical Gaussian surface fitting method is used to fit the laser footprint and calculate the laser footprint centroid coordinates.The fitted laser footprint and its surrounding CCR distribution are shown in Figure 9.The fitted laser footprint centroid is used as the ground control point and substituted into the satellite laser pointing angle calibration model based on infrared detector array derived in this paper.After three iterations, the GF-7 satellite beam 2 pointing angle calibration is completed.Among them, the laser ranging value used in the calibration is obtained by subtracting the atmospheric ranging error and laser ranging error from the ranging value calculated using Waveform #3 in Figure 8.To verify the accuracy of the proposed calibration method, the differences between the calibrated laser pointing angles and ranging values using the proposed calibration method and those obtained using CMLID calibration are computed, with the latter serving as the ground truth.The results are shown in the second row of Table 6.In addition, we also computed the differences between the calibrated laser pointing angles and ranging values obtained using the proposed calibration method and those obtained using the CMSPR.The results are presented in the third row of Table 6.
The results in Table 6 show that approximately the same calibration accuracy is obtained using the proposed method and the CMLID.The difference in the pointing angle error of the two  methods is 0.37 arcsec, and the difference in the ranging error is 3 cm.However, there is a large pointing angle difference of 2.53 arcsec after calibration using the proposed method and the CMSPR.The reason is that the CMSPR uses DSM data for calibration, and its accuracy is affected by the accuracy of the DSM itself.Therefore, the calibration accuracy will be lower than that of the proposed calibration method.In addition, Liu et al. (Liu et al. 2022) verified that the calibration accuracy of the CMSPR is 2.2 arcsec, which agrees with the result in Table 6.
To further verify the difference in accuracy between the three calibration methods, the calibration accuracy is evaluated using the laser footprint positioning accuracy after calibration.First, the pointing angles after calibration using the CMLIDC, the CMLID and the CMSPR are used to calculate the surface positions of the four laser footprints in Figure 1.Then, the distances between the laser footprint centroid calculated using the different calibration methods and the actual laser footprint centroid are calculated, and the results are shown in Table 7.All four laser footprints in Figure 1 are successfully captured using laser infrared detectors, and the actual laser footprint centroids (ALFCs) are obtained by fitting the infrared laser detector data.The laser footprint centroid calculated using the three calibration methods and the fitted laser footprint (LF) shape are shown in Figure 10.Among them, the red pentagram is the actual laser footprint centroid; the orange '×', purple circle, and yellow triangle represent the laser footprint centroids after calibration using the CMLIDC (LFC-CMLIDC), the CMLID (LFC-CMLID) and the CMSPR (LFC-CMSPR), respectively.
The results in Figure 10 show that the position of the laser footprint centroid after calibration using the proposed method and the CMLID is closer to the actual laser footprint centroid, and both are located at the center of the laser footprint.However, the CMSPR calibration places the laser footprint centroid at the edge of the actual laser footprint.In addition, according to the quantification results in Table 7, the satellite laser positioning error from the proposed method (CMLIDC) is better than 2.0 m.The satellite laser positioning error is the smallest after the CMLID calibration, approximately 1.0 m.However, it is relatively large after the CMSPR calibration, approximately 6.0 m.Assuming that the satellite laser pointing angle error per arc second causes a satellite laser footprint positioning error of approximately 2.5 m, the laser pointing angle error of GF-7 beam 2 after CMLIDC, CMLID, and CMSPR calibration is better than 0.8 arcsec, 0.4 arcsec and 2.4 arcsec, respectively.Notably, in Figure 10 and Table 7, the centroid position of laser footprint 212021051.00calculated after CMLIDC calibration is identical to the actual footprint centroid because the actual centroid of laser footprint 212021051.00 is used as the ground control point during CMLIDC calibration.In summary, the laser pointing angle accuracy is higher after CMLID calibration than after calibration using the proposed method and the CMSPR.However, the calibration method proposed in the article still has very high precision for laser pointing angle calibration, with a calibrated laser pointing angle accuracy better than 0.8 arcsec.The difference between the laser ranging error of the proposed calibration method and the CMLID is only 3 cm.

Discussion
Satellite laser elevation measurement accuracy is an important indicator to measure the performance of satellite laser altimeters after calibration.This section discusses in detail the laser elevation measurement accuracy (including three types of terrain: flat, sloping and mountainous) of GF-7 satellite beam 2 after calibration using the CMLIDC, the CMLID and the CMSPR.All laser footprints of beam 2 from the GF-7 satellite shown in Figure 2 were selected as experimental objects.Then, the surface coordinates of these laser footprints are computed using the calibrated laser pointing angles and ranging errors from the CMLIDC, the CMLID, and the CMSPR as input parameters.Based on the longitude and latitude of the laser footprint and the DSM in Figure 2(b), the DSM elevation of each laser footprint using formula (12) is interpolated.The interpolated DSM elevation can be considered the actual elevation of the laser footprint.Finally, the elevation error between the footprint and the interpolated DSM elevation is calculated, and the results are shown in Figure 11.To compare the laser elevation measurement accuracy on different terrains, the laser footprints are classified into three categories based on the internal terrain of the laser footprint: laser footprints on flat terrain, laser footprints on sloping terrain, and laser footprints on mountainous terrain.The elevation errors of the laser footprints on the three types of terrain after calibration are shown in Figure 11(a-c).
From the results shown in Figure 11, it can be seen that the laser elevation error of GF-7 satellite beam 2 after calibration is smallest on flat terrain, followed by slopes and mountainous terrain.For any of the three terrain types, it is evident that the laser elevation error of GF-7 satellite beam 2 after CMSPR calibration is the largest, while the laser elevation error after calibration using the proposed method is comparable to that after calibration using the CMLID.To quantitatively analyze the laser elevation measurement accuracy after calibration using the three methods on different terrain types, the mean value and root-mean-square error (RMSE) of the elevation measurement accuracy of all laser footprints under each terrain were calculated, and the results are shown in Table 8.
According to the results in Table 8, for the three types of terrain, the laser elevation measurement accuracy of GF-7 satellite beam 2 after calibration using the proposed method is higher than that of the CMLID and the CMSPR.However, the difference in the laser elevation measurement accuracy between the proposed method and the CMLID method after calibration can be ignored.The results in Table 8 indicate that the laser elevation measurement accuracy on flat terrain is better than that on sloping terrain and mountainous terrain for all three calibration methods.When using the sum of the mean values and the RMSE shown in Table 8 as the accuracy indicator, the laser elevation measurement accuracy of GF-7 satellite beam 2 calibrated using the proposed method is better than 0.11m, 0.30 m, and 1.80 m on flat, sloping, and mountainous terrains, respectively.The laser elevation measurement accuracy after calibration using the CMLID is 0.12 m, 0.29 m, and 2.08 m on flat terrain, sloping terrain, and mountainous terrain, respectively.However, the laser elevation measurement accuracy calibrated using the CMSPR is poor, with values of 0.25 m, 0.39 m, and 3.56 m for flat, sloped, and mountainous terrain, respectively.Additionally, because the laser pointing angle accuracy after calibration using the proposed method is lower than that of the CMLID, and the laser elevation measurement accuracy after calibration using the proposed method is slightly higher than that of the CMLID, it can be concluded that the laser ranging accuracy after calibration using the proposed method is higher than that of the CMLID.

Conclusions
In this article, a satellite laser on-orbit geometric calibration method that combines laser infrared detectors and CCRs was proposed.Taking laser beam 2 of the GF-7 satellite as the experimental object, calibration of the laser pointing angle and ranging error was achieved using the measured data of laser infrared detectors and CCRs in the same array.The laser footprints captured using other infrared detectors were used to compare the accuracy of the proposed calibration method with that of the CMLID and the CMSPR.Finally, the laser elevation measurement accuracies calibrated using these three methods on three types of terrain were discussed.The conclusions drawn in this paper are summarized below.
(1) The proposed calibration method has a calibration accuracy similar to that of the CMLID, and both are higher than the calibration accuracy of the CMSPR.
(2) The proposed method has poorer pointing accuracy but better ranging calibration precision than the CMLID.After the proposed method, the CMLID, and the CMSPR were used to calibrate the GF-7 satellite beam 2 laser, and the pointing accuracies were 0.4, 0.8, and 2.4 arcsec, respectively.
(3) The elevation measurement accuracies of GF-7 satellite laser beam 2 on flat terrain after calibration using the proposed method, the CMLID, and the CMSPR are 0.11 m, 0.12 m and 0.25 m, respectively.(4) The elevation measurement accuracy of GF-7 satellite beam 2 after calibration is higher on flat terrain than on sloping or mountainous terrain.After using the proposed method for calibration, the laser elevation measurement accuracies on flat, sloping, and mountainous terrain are 0.11 m, 0.30 m, and 1.80 m, respectively.
In summary, the proposed calibration method only requires one set of detector array data to achieve the same accuracy as multiple sets of laser infrared detector array data calibration.This method can be used as a new solution for on-orbit geometric calibration of satellite lasers, thereby saving experimental costs.In the future, we will continue to carry out calibration experiments on GF-7 satellite beam 1 using the proposed method, and further verify the accuracy of the proposed method.However, the proposed method relies on the echo signal marked by a CCR, and the method cannot be applied to satellite lasers that cannot record waveforms, such as the ZY3-03 laser altimeter.
Figure 1.GF-7 satellite beam 2 laser footprint distribution map for calibration and accuracy verification.

Figure 2 .
Figure 2. Laser elevation accuracy verification area and data distribution map: (a) experimental area and GF-7 beam 2 laser footprints; (b) DSM data; (c-e) surface elevation profiles of the laser footprints for beam 2 of the GF-7 satellite from orbits 5278, 6160, and 8851, respectively.In addition, circles, squares, and triangles represent laser footprints on flat, sloping, and mountainous terrains, respectively.

Figure 3 .
Figure 3. Infrared laser detector and CCR for the GF-7 satellite laser: (a) physical photo of a CCR; (b) exploded view of the CCR; (c) physical photo of an infrared detector; (d) exploded view of the infrared detector.
5) where x S y S z S   T ITRF refers to the coordinates of the center of the laser footprint and x G y G z G   T ITRF represents the coordinates of the center of mass of the satellite, which is obtained by interpolating postprocessed orbit determination data.Its accuracy is approximately 5 cm (Zhao and Tang 2013), which can be ignored.x GB y GB z GB   T represents the offset from the center of the GNSS antenna to the center of mass of the satellite; R ITRF ICRF is the rotation matrix from the International Celestial Reference Frame (ICRF) to the International Terrestrial Reference Frame (ITRF); R ICRF SBF is the rotation matrix from the satellite body frame (SBF) to the ICRF; Dx Dy Dz   T

a
1 cos b cos a + a 2 cos b sin a + a 3 sin b b 1 cos b cos a + b 2 cos b sin a + b 3 sin b c 1 cos b cos a + c 2 cos b sin a + c 3 sin b x G + x GB − x S + a 1 Dx + a 2 Dy + a 3 Dz y G + y GB − y S + b 1 Dx + b 2 Dy + b 3 Dz a 1 cos b cos a + a 2 cos b sin a + a 3 sin b b 1 cos b cos a + b 2 cos b sin a + b 3 sin b c 1 cos b cos a + c 2 cos b sin a + c 3 sin b

Figure 7 .
Figure 7. Joint deployment scheme and field deployment photo of the infrared laser detectors and CCRs: (a) deployment scheme; (b) field deployment photo, where (b1) is a CCR.

Figure 8 .
Figure 8. Echo waveform of the laser footprint used for calibration and its waveform decomposition results.

Figure 9 .
Figure 9. Laser footprint and CCR distribution map.The contour lines are laser footprints fitted with laser infrared detectors.The yellow circles refer to CCRs.

Figure 11 .
Figure 11.Elevation errors between the laser footprints and DSM for GF-7 satellite Beam 2 after calibration on three types of terrain: (a) flat terrain; (b) slope terrain; (c) mountain terrain.

Table 1 .
Laser footprint information for GF-7 satellite beam 2 calibration and positioning accuracy verification.

Table 2 .
Laser footprint information for GF-7 satellite beam 2 elevation accuracy verification.

Table 3 .
CCR height calculated using echo waveforms.

Table 4 .
Actual CCR heights measured using RTK.

Table 5 .
CCR-based laser ranging error correction results.

Table 6 .
Differences between the calibration results from the proposed method and those calibrated by the other two calibration methods.

Table 7 .
The distance between the centroid of the laser footprint calculated using the three calibration methods and the actual footprint centroid.

Table 8 .
Laser elevation measurement accuracy of the calibrated GF-7 satellite beam 2 on different terrain types.