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Abstract
Satellite laser altimeter data are used for polar ice sheet elevation mapping, vegetation mapping, etc. Data quality mainly depends on complex relationships among several factors in the path of laser transmission and on illuminated surfaces, including clouds, atmospheric aerosol, satellite pointing, laser energy, topography, footprint size, shape and orientation. The precise pointing of the transmitted laser pulse is critical for improving the horizontal accuracy of the footprint on the ground. Thus, we extracted the centroid of the laser profile array (LPA) image of ICESat/GLAS by 1/e2 maximum energy distribution method. The results show that the accuracy of extraction of the LPA’s centroid improved by 0.3 pixel, and the relative positioning accuracy improved by 0.11 pixel. The fast Fourier transform and Fourier series fitting of the LPA centroid has been implemented to detect the periodic change and analyze the model regularity. The results show that the centroid of the LPA undergoes four periodic changes: 1.83 × 10−4, 3.36 × 10−4, 5.19 × 10−4, and 6.71 × 10−4Hz. The correlation of fit is a good indicator (R2=0.86) and accurate up to 0.4 arcsec (approximately 0.13 pixel). Finally, we extract and estimate the LPA characteristic parameters (eccentricity, orientation, total intensity, and major axis) in different campaigns. We observe that the results obtained by the 1/e2maximum energy distribution are only approximate.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The Geoscience Laser Altimeter System (GLAS) payload is a high-precision satellite laser lidar onboard NASA’s Ice, Cloud, and Land Elevation Satellite (ICESat). The ICESat was launched on 13 January 2003 [1] with the principle objective of measuring long-term polar ice-sheet elevation changes, atmospheric profiles of cloud and aerosol properties [2], and all surface types globally, including land topography, vegetation canopies and sea ice [3]. Achieving these objectives requires high-precision measurement of the surface elevation at each laser footprint. GLAS combines its own geocentric position vector—a range vector formed by a 1,064-nm laser, inferred from the round-trip travel time of the laser pulse—and a laser pointing angle determination system, ascertained by Precision attitude determination, to obtain the precision location of each footprint [4,5].
The satellite orbit has an average altitude of 600km, and an inclination of 94°. GLAS carries three identical lasers (designated laser 1, 2, and 3), with only one laser operating at a time to transmit a laser pulse of 10-ns duration, and consecutively record the return pulse as reflected from the approximately 70-m-diameter footprint on the ground. Thus, it records every 175 m along each track [6,7]. It uses a primary wavelength of 1064nm for the altimetry measurements of surface topography, and a secondary wavelength of 532nm for a more sensitive determination of the vertical distribution of clouds and aerosols [8]. A major advantage of the ICESat/GLAS is that it utilizes the all waveform data that are available from nearly around the earth. For the flat surfaces, the vertical accuracy of ICESat/GLAS is better than 10cm, with a vertical precision of 2–3cm for non-vegetated surfaces. Its horizontal accuracy, however, is relatively poor (approximately 5m), and can be improved further [9,10]. The laser pointing data are obtained onboard with the Stellar Reference System, comprising two main components, namely the attitude determination system (ADS) and laser reference camera (LRC), that include shot-to-shot laser pointing direction to determine the orientation of the optical bench, which is crucial for the laser beam [11]. The LRC contains a laser reference sensor (LRS), which is used to record 10-Hz laser images, and laser profile array (LPA), which is used to record 40-Hz laser images [12].
Fig. 1. (a) ICESat/GLAS LPA images of the laser far-field patterns measured during different campaigns [14]. (b) Sketch of the LPA image shape, major axis and orientation angle. (c) Three-dimensional intensity distribution within an LPA images, of which the boundary is determined by 1/e2.
ICESat/GLAS transmits a laser pulse in the shape of an ellipse. The pulse has a maximum intensity at the center of the LPA image, and plummets at the LPA image boundary by about 1/e2 (Fig. 1(c)) [13]. The LPA image is a far-field projection of the laser spot measured onboard, and it is used to approximately represent the footprint size and shape on the ground. A few typical ICESat/GLAS LPA images are shown in Fig. 1(a).
The shape of the LPA image is determined by four main parameters, i.e., LPA azimuth angle (i_tpazimuth_avg), LPA major axis (i_tpmajoraxis_avg), LPA eccentricity (i_tpeccentricity_avg), and transmit pulse total intensity (i_tpintensity_avg), which are provided in the GLAS products in GLAH05. It also has two secondary parameters, i.e., maximum intensity and number of pixels used. As shown in Fig. 1(b), the LPA includes two azimuth angles. These angles are computed as the angle between the major axis and LPA x-axis as well as the angle between the major axis and ground laser orientation from the North, which is in turn obtained from the geometrical relationships between the GLAS frame, LPA frame, and ICESat orbit at the measurement time. The LPA major axis represents the spot size on the ground, in meters (m). The LPA eccentricity is computed according to e2 = 1 – b2/ a2, where a is the major axis and b is the minor axis. The total intensity, maximum intensity and number of pixels used are identified to apply the 1/e2 cut-off criterion, where e is the natural logarithm [15].
There are several methods to extract the centroid of image data formed by a point light source, such as laser spot and star images. Van Waerbeke et al. [16] proposed the SOURCE EXTRACTOR algorithm to obtain the ellipsoid parameters. The algorithm obtained that the major (and minor) axis of the LPA image represents twice the maximum (and minimum) spatial RMS (one sigma) of the laser image along all each direction. Yuan et al. [17] determined the centroid coordinate parameters of the LPA image, using the gray center extraction algorithm, and found that the variation range of the laser pointing angle was 9 arcsec. Quine et al. [18] developed a new sub-pixel interpolation technique to process image centroids. When applied to a conventional CCD camera and active-pixel sensor, the technique achieves a centroid performance of 11.6 times and 12.8 times the raw pixel resolution, respectively. Bae et al. [4] extracted the centroid of an LPA image using the gray centroid method. The precision laser pointing information was obtained via converting the LPA centroid from the LPA frame to the International Terrestrial Reference Frame. Sirota et al. [19], through statistical changes of the laser centroid on the LRS for several orbits, found that the peak-to-peak per-axis orbital variation was 2 arcsec and presented two peaks per orbit that coincided with the passes through the terminator for the spacecraft.
2. Data and methods
2.1 LPA image data of ICESat/GALS
The ICESat was operated in a near-repeat ground track orbit to provide repeatable measurements throughout the mission, using approximately 33days of a 91-day repeat orbit used for each campaign [1]. The major axis, LPA orientation angle, LPA eccentricity and total intensity are significant during the span of each campaign, over the course of one orbit, and even shot-to-shot. Therefore, the LPA parameters reported in the data products were examined and each mission campaign was named according to the laser number and a sequential letter. The examination data included L2a (first campaign from laser 2, day of years (DOY):283–321 in 2003, 38 days in total); L3b (second campaign from laser 3, DOY:47–83 in 2005, 36 days in total); L3d (fourth campaign from laser 3,DOY:293–328 in 2005, 35 days in total); L3f (sixth campaign from laser 3, DOY:143–177 in 2006, 34 days in total); and L3i (ninth campaign from laser 3, DOY:274–309 in 2007, 35 days in total).
The LPA measures the spatial energy distribution of each transmitted laser pulse using an 80×80 pixels array image, with a 20×20 pixels data provided on the GLAS products in GLAH04 [20]. Each pixel in the LPA image represents an instantaneous Field-of-view (FOV) of 3.388 arcsec, a ground equivalent of approximately 10.46m, which is defined in Eq. (1) [21].
where r is the spatial resolution of each pixel on the ground in the LPA image; $\varphi $ is the size of the FOV of LPA, which is equivalent to 0.08°; R is the altitude of the ICESat (approximately 600km, represented in meters for calculations); n is the total number of rows or columns of the LPA image (80×80 pixels).The centroid coordinates of LPA are determined from 20×20 pixels (united in pixels) in the paper, however they are provided in GLAH05, a data product of GLAS, and determined from 80×80 pixels (united in arcsec). For comparison, the pixel is used as the unit of the centroid coordinates. Thus, a transformation relationship between the two coordinate frames is established by Eq. (2).
2.2 LPA image centroid positioning and parameters extraction
Most digital images are based on point light sources imaging. In an ideal situation, the energy intensity distribution in the spot is approximately Gaussian. Considering the effect of the attenuation of the transmitted laser energy in the atmosphere, the size of the image boundary of the spot can be determined by the 1/e2 maximum energy after eliminating the influence of background noise in the LPA image. A method to find the centroid (xLPA, yLPA) of the LPA image is expressed in Eq. (3).
When the LPA laser spot is approximately elliptic, the coordinates can be extracted according to the edge detection of the LPA spot image, and then these parameters, comprising the LPA orientation angle (computed as the angle between major axis and LPA x-axis in this paper), LPA major axis, and LPA eccentricity, which determine the shape of the LPA spot image, can be obtained via the least-square ellipse fitting method. The expression of the elliptic curve is shown in Eq. (4).
where A, B, C, D, E, and F are parameters of the ellipse, whose optimal solution can be obtained via the least-square ellipse fitting. The centroid of the elliptic spot can be defined by Eq. (3) after the parameters have been determined. However, this equation may cause errors in positioning the spot centroid and determining the parameters. The erroneous parameters may deviate from the actual ground distance when the shape of the spot is asymmetric, or when the edge is unclear. Despite this, the equation is still highly efficient and simple to use [17]. Centroid (x0, y0), LPA semi-major axis a and LPA orientation angle θ are determined by Eqs. (5), (6) and (7), respectively.2.3 Data fitting
For each mission campaign, the raw laser image centroids obtained from LPA were smoothed to minimize the infulence of noise, and handle the observed data gaps. Consequently, the nonlinear Levenberg–Marquardt fitting algorithm was adopted, using the Fourier series as shown in Eq. (8).
The experimental procedure is illustrated in Fig. 2. We first extracted the LPA image and obtain the maximum intensity value, which was used to determine the size of the LPA image. Then, we determined the centroid and parameters using the method described in Section 2.2. Subsequently, we compared the official parameters provided in GLAH05 with our results. We applied the Fast Fourier transform (FFT) to the time series of centroid coordinates to determine the order of higher-order Fourier series fitting. The FFT is a mathematical method that associates the spatiotemporally sampled signal with its frequency-sampled counterpart. It can detect significant parameters of the time series signal, such as the frequency component, period, initial phase and amplitude, which are regarded as references for the initial value of higher-order Fourier series fitting. The corresponding value of a0 is the direct current (DC) component, and its actual value is half that of the amplitude.
Fig. 2. Flowchart of the method to determine the laser pointing and characterization parameters based on laser profile array of ICESat/GLAS.
3. Results
3.1 LPA image centroid positioning and data fitting
The results of LPA centroid coordinates $({g\_CentX,g\_CentY} )$ on 283/2003 (DOY/y) determined by Eq. (2) and the provided official centroid $({CentX,CentY} )$, which were converted by Eq. (3) are shown in Table 1 (each value is displayed in intervals of 5Hz in 1s; total 40Hz). As in Table 1, $({CentX,CentY} )$ is converted to the frame of 20×20 pixels, with values ranging from to 13.7–14.1 and 7.3–7.5pixels, respectively. The variation ranges of $({g\_CentX,g\_CentY} )$ are extremely close to $({CentX,CentY} )$. The smallest difference in the X and Y directions were 0.02 and 0.04 pixel, respectively. We found that the LPA centroid extraction accuracy was better than 0.3 pixel, and the relative positioning accuracy was better than 0.11 pixel, according to the statistical results of the centroid determined by Eq. (2) in the L2a campaigns.
Table 1. Centroid extraction results in intervals of 5Hz in 1s.
The changes in the X and Y centroids of the LPA for several orbits after approximately 12h on 283/2003 are shown in Fig. 3. Finally, centroids with an average value of 40 Hz required to dilute the original coordinate series and 45,113 LPA images of centroid coordinates with a frequency of 1Hz were obtained. The peak-to-peak per-axis centroid variation between the sunlit and shadowed portions of the orbit (approximately 1.5 h cycle) was approximately 1.2 pixel (approximately 4 arcsec). There were two peaks and two valleys in each cycle. The complex motions of the LPA illustrate the importance of the pointing monitoring devices.
As shown in Fig. 4 and Table 2, we analyzed the centroid series in Fig. 3 using the FFT, to obtain initial parameters described in Section 2.3. The results are shown in Fig. 4 and Table 2. There are four notable different frequencies: ${1.83} \times {1}{{0}^{{ - 4}}}$,${3.36} \times {1}{{0}^{{ - 4}}}$,${5.19} \times {1}{{0}^{{ - 4}}}$, ${6.71} \times {1}{{0}^{{ - 4}}}$ Hz, which are approximately equal to the cycle of 91.07, 49.6, 32.11, and 24.84 min, respectively. The value with a 0-Hz frequency is the DC component. Therefore, the actual amplitudes (1/2 of the DC component) of X and Y are 13.81 and 7.405pixels, respectively.
Table 2. Amplitude result of the FFT detection frequency
Based on the fast Fourier change detection results and satellite orbit cycle variation, the frequency of the LPA centroid change can be divided into four frequencies. They comprise ultrahigh frequency, high frequency, intermediate frequency, and low frequency, which are caused by the difference between the environment of the satellite in orbit and ground environment, changes in gravitational field, temperature gradient, internal stress variation, and disturbance from solar wind and particle flow on solar panels [24].
- (1) Ultrahigh frequency: caused by a complex mechanical environment that leads to the micro-displacement of internal structural changes, which appears as cyclic changes (approximately 24.84 min) of a small amplitude and high frequency in the LPA centroid.
- (2) High frequency: cause by the temperature change of the internal assemblies and the couplings and forces between various loads in the satellite, which leads to changes in the LPA centroid with a high frequency cycle (approximately 32.11min).
- (3) Intermediate frequency: When passing through over different latitudes, the satellite is affected by the change in external surroundings such as solar radiation and solar wind, which causes temperature fluctuations and jitter of the satellite platform and makes the frequency and amplitude of the LPA centroid appear stable and as a cycle (approximately 49.6min).
- (4) Low frequency: the satellite passed over different latitudes and through the sunlit and shadowed portions of the orbit during its operation, resulting in changes of temperature and surroundings, and the LPA centroid has a long cycle (approximately 91.07min), which corresponds to the pass through the terminator is manifested on the secondary peaks.
Table 3. Fourier coefficients of X and Y LPA centroid
The angular frequency (ω) of the LPA centroid in Table 3 is 0.001081 rad/s, which is approximately equal to 96.87min. The remaining angular frequencies are 2, 3, and 4 times ω, according to Eq. (8). Therefore, these remaining frequencies are 0.002162 rad/s (approximately 48.45min), 0.003243 rad/s (approximately 32.29min), and 0.004324rad/s (approximately 24.21min), which are similar to the FFT. The correlation of fit in the LPA x-axis and y-axis is above 0.86, and the coordinate fitting accuracy is approximately 0.4”, which is better than 0.1 pixel. The correlation of fit served as a good indicator and the fitting accuracy was up to 0.12 pixel (approximately 0.37 arcsec), which was computed from the RMSE of the x-axis and y-axis. Moreover, according to Kepler’s third law, the satellite cycle (T) can be defined by Eq. (9).
where a is the semi-major axis, and the geocentric gravitational constant GM is equal to $\mu = 3.9861 \times {10^5}k{m^3}/{s^2}$. The official semi-major axis of the ICESat orbit is 6,879km. Therefore, a cycle of 94.63 min is obtained by Eq. (9), which is almost the same as the results of the Fourier fitting. Then, we discretized the fitting result, and calculated the residual between the LPA centroid and discrete values that is shown in Fig. 6. The maximum and minimum residual errors of the x-axis and y-axis are 0.2429, −0 .2817, 0.1865, −0 .2657pixels, respectively.We extracted a parts of the satellite orbit and the Instrument Star Tracker (IST) CCD temperature data that corresponded to the LPA centroid in the above experiment (Fig. 7). We found noticeable changes in the LPA centroid with the ICESat orbit and IST CCD temperature. (1) For the x-axis, in 1 cycle, when the ICESat moved from North to South (descending orbit), the centroid of the LPA decreased from the highest peak in the North Pole to the lowest peak in 30S within 60S, and then, began to rise. Further, a secondary highest peak appeared near the South Pole. It dropped again when the ICESat moved from the South Pole to 30N (ascending orbit), where a second lowest peak appeared. Subsequently, the highest peak appeared again when the satellite reached the North Pole, where a new cycle began. (2) For the y-axis, the LPA centroid rose from the secondary lowest peak above the North Pole to the secondary highest peak near the equator, and then began to decrease the lowest peak near 60S within the South Pole during the descending orbit. Next, in the ascending orbit, the centroid rose from the lowest peak until the highest peak near 30N. Subsequently, it dropped to the secondary lowest peak above the North Pole. (3) For the IST CCD temperature, it changes more frequently in the interval of 30N to 30S, owing to the jitter of ICESat and the amount of sunlight it receives as the orbit descends. The centroid fluctuates slightly near 60N and 60S for both the descending and ascending orbits.
3.2 Parameters extraction and statistic
A high-spatial-resolution camera in the LPA images the spatial distribution of laser energy. The images for L2a, L3b, L3d, L3f, and L3i of LPA are shown in Fig. 8, and the results of the parameter extraction are shown in Table 4. For most GLAS campaigns, the lasers produced an elliptical shape with a central maximum and radial decrease in energy. This simple geometry permits the use of the centroid, major axis, eccentricity, orientation, and azimuth angle as close representations of the GLAS footprint geolocation; further, the total intensity of the transmitted laser can accurately reflect the conditions of the lasers in different campaigns. Therefore, it is essential to extract these parameters accurately.
Fig. 8. LPA images observed during different GLAS campaigns. These images have been smoothed; however, the calculations are performed on unsmoothed data.
Subsequently, we calculated the standard deviation (σ) and relative error (Δσ) of L2a. We found that the LPA centroid extraction accuracy is better than 0.3pixel (approximately 0.9″/40 Hz), as determined by the standard deviation, and the relative positioning accuracy was better than 0.11 pixel (approximately 0.37″/40Hz), as determined by dx and dy in Table 4. As shown in Table 4, the relative error (Δσ) of each campaign was less than 2%. The minimum and maximum differences in the x-axis were 0.001 pixel (L3d) and 0.065 pixel (L2a), respectively, and those in the y-axis were 0 pixel (L3i) and 0.108 pixel (L3f), respectively. The orientation, eccentricity, total intensity (the sum of gray counts of all pixels), and major axis are close to the value provided in the GLAS data product. The total intensity parameter has a maximum value of 111 (L2a) and minimum value of 7 (L3f). The differences of the major axis and orientation of the majority campaigns are equal to approximately 3m and 2° (the maximum value approached to approximately 3° in L2a), respectively. Therefore, we conclude that the results of orientation, eccentricity, total intensity, and major axis were only approximations using ${{1} / {{e^2}}}$ maximum energy distribution.
All of the five campaign-averaged LPA parameters (±standard deviation), which are computed as the average of the daily means and plotted in Figs. 9–13, inclusive of any anomalies in the time series (see Table 7), are listed in Tables 5 and 6.
Table 5. Mean LPA characteristic ± standard deviation averaged over each campaign: LPA-X, LPA-Y, and orientation
Table 6. Mean LPA characteristic ± standard deviation averaged over each campaign: Eccentricity, total intensity, and major axis
Table 7. Anomalies observed in the time series plots of LPA characteristics
As shown in Figs. 9–13. For each campaign, one figure (with six plots) was generated that represented the evolution of the six LPA parameters containing the plots of LPA-x, LPA-y, orientation, eccentricity, total intensity, and major axis. Each plot point is a daily average, and the error bars represent the standard deviation.
The orientation of L2a had a maximum average (125°) in the five campaigns, and the change in standard deviation exhibited an increasing trend after 302 days, as shown in Fig. 9. In Fig. 10, all the parameters of L3b showed the same trend, with a slight anomalous jump and higher RMS on day 54, and a much smaller jump in the subsequent days. The standard deviations of orientation and eccentricity of L3d, L3f, and L3i are unstable and large, as shown in Figs. 11–13, except for the LPA centroid, in which all the campaigns were highly stable and experienced a receding trend in the L3d and L3i campaigns. The changes in the total intensity and major axis were also stable, but still experienced a few jumps and increased the standard deviation. The maximum major axis is 105 m (L2a), and the corresponding laser energy is 70.7 mJ. After the L3b campaign, the major axis stabilized between 50 and 60 m, and the corresponding laser energy reduced to 30 ± 10 mJ, which affected the size of the laser footprint.
Meanwhile, the average and standard deviation of the parameters showed obvious anomalies on 285/2003 (Oct. 13). To verify this abnormal phenomenon, we plotted the change curve of the LPA centroid on that day, and found a slight anomalous jump in Figs. 14(a) and 14(b). Subsequently, we found that the LPA temperature (the parameter name in the GLAS data product in GLAH04 is d_LPAC13_t1) change curve had the same slight jump at the same time node with the LPA centroid obtained by extracting and plotting the LPA temperature on day 285 as shown in Fig. 14(c). Compared with the trend after the slight jump node, the temperature change ranges before the node was equal to approximately 24.6–26.2°C (the difference is approximately 1.6°C), which is massive and unstable, and after the node was equal to approximately 24.5–25.6°C (the difference is approximately 1.1°C), which is more concentrated and stable. The difference of nearly 0.5°C caused a deviation of approximately 6 pixels of the x-axis and approximately 0.5 pixel on the y-axis of the LPA centroid. Therefore, we also confirmed that the temperature change of the satellite onboard would displace the centroid of the LPA.
Fig. 14. Change of LPA center and temperature on day 285; (a)LPA-x centroid time series; (b) LPA-y centroid time series; (c) The temperature of LPA, which is calculated with the average and standard deviation every half hour.
4. Discussion and conclusion
Spaceborne laser altimeter measurement data for ICESat/GLAS are widely used in various field. Although the redundancy and cross checks of the ADS and LRC system provided reliability of laser pointing when some of the components perform sub-optimally, and the high-spatial-resolution images recorded by the LPA and LRS of the laser beam to determine laser pointing, as well as engineering analysis and diagnosis of laser issues. There are still several uncertain factors affecting laser pointing. Therefore, to meet this challenge, we extracted the centroid, and other significant parameters of the LPA image within the 1/e2 criterion. The results of the experiment led to following conclusions.
- (1) The peak-to-peak per-axis centroid time-series variation of the LPA determined by Eq. (3) is approximately 1.2 pixel (approximately 4 arcsec) between the sunlit and shadowed portions of the orbit (approximately 1.5 h cycle). Each cycle contains two peaks and two valleys, which correspond to the line passing through the terminator, are reflected in the secondary peaks. Consequently, the centroid extraction accuracy is better than 0.3pixel (approximately 0.9″/40Hz) and the relative positioning accuracy is better than 0.11 pixel (approximately 0.37″/40Hz).
- (2) The cycle of the LPA centroid time-series was detected using the FFT, and four different frequencies were found: ${1.83} \times {1}{{0}^{{ - 4}}}$,${3.36} \times {1}{{0}^{{ - 4}}}$,${5.19} \times {1}{{0}^{{ - 4}}}$, and ${6.71} \times {1}{{0}^{{ - 4}}}$ Hz. Meanwhile, the cycle (similar to FFT) was also obtained using fourth-order least-square Fourier fitting (correlation R2=0.86) and Kepler’s third law. The parameters orientation, major axis, eccentricity, and total intensity, obtained using the maximum energy distribution in different campaigns, were only approximations.
- (3) (3) For the current results of laser pointing, certain information, including LRS, ADS, and other transform parameter data, could not be obtained. Therefore, the centroid in the LPA frame could not be converted to the LRS frame, or even the celestial reference frame. Consequently, researchers should further study methods for determining the precise laser pointing for satellite laser altimeter in the future.
Funding
Common key technology projects for major special applications of High-resolution Earth Observation Systems (11-Y20A11-9001-17/18, 11-Y20A13-9001-17/18); National Natural Science Foundation of China (41871382, 41971425).
Acknowledgments
We thank the NASA National Snow and Ice Data Center (NSIDC) for providing us with the ICESat/GLAS data.
Disclosures
The authors declare that no conflicts of interest.
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