Assessment of ICESat-2’s Horizontal Accuracy Using an Iterative Matching Method Based on High-Accuracy Terrains

Abstract

The Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2), launched in September 2018, has been widely used in forestry and surveying. A high-accuracy digital elevation model (DEM)/digital surface model (DSM) for terrain matching can effectively evaluate the ICESat-2 design requirements and provide essential data support for further study. The conventional terrain-matching methods regard the laser ground track as a whole, ignoring the individual differences caused by the interaction of photons during flight. Therefore, a novel terrain-matching method using a two-dimensional affine transformation model was proposed to describe the deformation of laser tracks. The least-square optimizes the model parameters with the high-accuracy terrain data to obtain the best matching result. The results in McMurdo Dry Valley (MDV), Antarctica, and Zhengzhou (ZZ), China, demonstrate that the proposed method can verify geolocation accuracy and indicate that the average horizontal accuracy of ICESat-2 V5 data is about 3.86 m in MDV and 4.67 m in ZZ. It shows that ICESat-2 has good positioning accuracy, even in mountainous areas with complex terrain. Additionally, the random forest (RF) model was calculated to analyze the influence of four factors on geographic location accuracy. The slope and signal-to-noise ratio (SNR) are considered the crucial factors affecting the accuracy of ICESat-2 data.

1. Introduction

The Geoscience Laser Altimeter System (GLAS) on NASA’s Ice, Cloud, and Land Elevation Satellite (ICESat) is the first satellite-based laser altimetry in the world, and has provided high-accuracy data on a global scale [1,2]. However, in 2009, ICESat stopped working because of the failure of its laser sensor. Fortunately, the Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2), equipped with an advanced topographic laser altimeter system (ATLAS), was launched from California in 2018. As the successor of ICESat, the primary goal of ICESat-2 is to monitor the changes in sea ice freeboard at centimeter accuracy [3]. Additionally, it provides repeated altimetry measurements for land, vegetation, ocean, and inland water [4].

Accurate calibration and verification of measurements are crucial for all satellite missions, especially those involving novel instruments or technologies such as the photon-counting laser altimeter on ICESat-2. Over the past few years, the positioning accuracy of ICESat-2 products has been verified under different conditions. Wang et al. [5] compared the ground elevations retrieved from the ICESat-2 data and the airborne LiDAR-derived ground elevation. The calculated mean difference and root mean square error (RMSE) of the ICESat-2 product are −0.61 m and 1.96 m, respectively. However, only the vertical accuracy of ICESat-2 data is verified, while the horizontal accuracy is ignored. The alignment errors and time-varying pointing biases of the on-orbit laser will significantly affect the horizontal positioning accuracy [6]. The stringent laser surface bounce point horizontal (on the surface) geolocation knowledge requirement is 6.5 m, and the best estimate before launch is 4.9 m [7].

When the ground is uneven, the deviation of horizontal photon coordinates can lead to an additional elevation error, as shown in equation ${\sigma }_v={\sigma }_h\tan \alpha$, where ${\sigma }_h$ is the horizontal positioning error of the photon, $\alpha$ is the slope angle and ${\sigma }_v$ is the quadratic elevation error of the photon. Assuming that the terrain slope is 30°, the horizontal positioning accuracy is 6.5 m (as required), which could lead to a potential elevation error of 5.94 cm. In mountainous areas, the elevation error caused by the quadratic effect may become larger with the increase in slope. For a massive area of ice, a small change in surface height can considerably impact sea level estimation. For the area above 2000 m in Greenland, ice sheet change rates correspond to a volume change rate of $21{ \mathrm{km}}^3/\mathrm{yr}$ [8]. It is necessary to evaluate the horizontal positioning accuracy of ICESat-2. However, it is a great challenge to verify the horizontal accuracy of ICESat-2. Firstly, the slice data obtained by ICESat-2 lacks an appropriate accuracy index. Secondly, the proposed accuracy verification methods regard the laser track as a whole, without considering the individual differences of photons.

At present, the verification methods for the horizontal accuracy of satellite-based lidar include GPS/GNSS differential technology [9], an analysis method based on echo waveform [10,11], and matching with the high-accuracy digital elevation model (DEM)\digital surface model (DSM) [12,13]. By comparing the GNSS and ATL03 photon-based heights, Brunt et al. [14] proved that the current accuracy of ATL03 is better than 5 cm, and the surface measurement accuracy is better than 13 cm. The conventional geodetic survey method relies on high-accuracy ground control points (GCPs), which require massive human and material resources. The analysis method based on echo waveform can only be used for laser altimeters in a full waveform system, which is unsuitable for ICESat-2 ATLAS [10]. The novel photon-counting laser altimeter provides high-density laser photons on the ground surface that have never been heard before. It can gather along-track surface profiles even when surveying topographically complicated terrains. Therefore, it is feasible to use high-accuracy DSM to check the accuracy of the ICESat-2 data. Tang et al. [13] proposed a terrain-matching method based on a hierarchical pyramid searching for satellite-based laser altimeters. Using Tang’s method, the best matching position can be quickly determined.

Nevertheless, the parameters must be provided in advance based on experience. Nan et al. [15] achieved the fast iterative calibration of the ICESat-2 pointing angle using the small-range terrain generated by airborne lidar. A terrain-matching method [16] is used to iteratively estimate the systematic biases between laser pointing and laser ranging of satellite-based photon-counting laser altimeters. Schenk et al. [17] used mathematical expressions to approximate the terrain near ICESat-2 ground tracks (GTs). The optimal matching position is determined by calculating the minimum distance between the laser photons and all the planar patches. The proposed terrain-matching methods can correct the coordinates of the laser ground track. The forward scattering effect of photons caused by clouds, aerosols, and other particles in the atmosphere will distort the laser track, affecting the matching accuracy [18].

A new terrain-matching method is proposed to represent the changes in the ICESat-2 laser photon tracks accurately. The state of all photons in the tracks is described using a two-dimensional affine transformation model. The model parameters are refined by a least square iteration to minimize the elevation differences between the photons and reference terrain. Due to the use of high-accuracy DSM, the correction of the laser photon track can be considered the ICESat-2’s horizontal accuracy. To verify our method, the McMurdo Dry Valleys (MDV) and Zhengzhou (ZZ) are selected as the study sites, while the results of the proposed methods are compared. Through the results of 673 laser photon sets, the ICESat-2’s positioning performance is verified. Additionally, a random forest (RF) model is constructed to analyze the influence of different factors (slope, signal-to-noise ratio (SNR), latitude, and photon density) on photon localization accuracy.

2. Materials and Methods

2.1. Study Site

The horizontal accuracy of ICESat-2 data was investigated in two different sites as shown in Figure 1, MDV in East Antarctica and ZZ in China. MDV is located in Victoria Land in the western part of the McMurdo Sound Mountains in Antarctica. The East Antarctic ice sheet does not cover these valleys due to the blockage to flow by the Transantarctic Mountains and the severe rain shadow caused by these mountains [19]. The unique geographical environment in this region leads to a cold and dry climate all year round. MDV consists of permafrost and exposed rocks, with almost no snow. The cold temperature and lack of strong hydrological cycling suggest that MDV is a relatively stable landscape [20]. Zhengzhou is located in north-central China, with complex and diverse landforms. With the uneven terrain, ZZ is higher in the southwest and lower in the northeast. The elevation difference in ZZ is 1440 m. The obvious topographic features and stable geological structure of the study sites ensure the verification of the matching method.

The horizontal accuracy of the DSM in the study sites should have better accuracy than the laser altimetry to reduce the possibility of mismatch. The airborne lidar data of the MDV were collected [21] with an Optech Titan multispectral sensor during the summer of 2014–2015. The vertical and horizontal accuracy is estimated at 0.07 and 0.03 m, respectively. The high-accuracy lidar data were made available via the OpenTopography platform. The airborne lidar was used to obtain the surface data of Zhengzhou in 2018. The data resolution is 1 m, and the coverage is ~5000 ${\mathrm{km}}^2$. The specific study sites’ parameters are illustrated in

Table 1. Introduction of research sites’ parameters.

Site	Range	Acquisition Time	Access	Resolution (m)	Elevation (m)	Coordinate System
MDV	∼77.5°S, ∼162.5°E	2014–2015	Airborne lidar	1	−52–2070	WGS84
ZZ	34°0′N–35°05′N, 112°30′E–114°0′E	2018	Airborne lidar	1	74–1494	CGCS2000

.

2.2. ICESat-2 Data

ICESat-2 can obtain dense photons with 0.7 m spacing in the along-track direction from a 496 km orbit [22]. The satellite obtains laser data on a 91-day repetition period, with an orbital inclination of 94°. The observation coverage range is 88°S–88°N, which achieves global coverage. The onboard ATLAS can send a single laser pulse to the ground at a repetition frequency of 10 kHz [23]. Figure 2 shows the beam pattern on the ground. The pattern has a total of six laser beams organized in a 3 × 2 array. Each beam pair consists of a strong and a weak beam with an energy ratio of 4:1. The generated laser tracks are 3.3 km apart and 90 m wide between the strong and weak beams. The beam intensity depends on the orbital direction of the satellite. In the along-track direction, the strong beam maps to the right orbit (R), the weak beam maps to the left orbit (L), and vice versa.

ICESat-2/ATLAS data products are divided into five levels, which contain 21 standard data products from ATL00 to ATL21. The publicly available products are obtained from the NASA Distributed Active Archive Center (DAAC) at the National Snow and Ice Data Center (NSIDC). The ATL03 product provides each photon’s longitude and latitude, elevation, and measurement time, as well as relevant parameters such as photon confidence labels and elevation correction information [7]. Additionally, it also includes a set of parameters that are useful for higher-level user-oriented products, such as the rough distinction between signal and noise, surface classification of land, ocean, land ice, sea ice, and inland waters, several geophysical corrections for photon heights to account for tidal and atmospheric effects, measurements of the ATLAS system impulse response function, and so on [24].

Here, the fifth version (labeled as V5) of the ATL03 product was adopted to train and validate results. Twelve tracks are selected, covering December 2018 to July 2022. The specific track information chosen is shown in

Table 2. ICESat-2 data in this study. The data of two laser ground tracks passing through the study site at different times are selected, respectively. RGT0390 corresponds to MDV, and RGT0560 corresponds to ZZ.

RGT	Date	Cycle	Spacecraft Orientation	Granule
0390	21 January 2020	06	Forward	ATL03_20200121071113_03900612_005_01
0390	21 April 2020	07	Forward	ATL03_20200421025102_03900712_005_01
0390	20 July 2020	08	Backward	ATL03_20200720223049_03900812_005_01
0390	18 January 2021	10	Forward	ATL03_20210118135026_03901012_005_01
0390	19 April 2021	11	Forward	ATL03_20210419093021_03901112_005_01
0390	19 July 2021	12	Forward	ATL03_20210719051013_03901212_005_01
0390	18 October 2021	13	Backward	ATL03_20211018005010_03901312_005_01
0390	17 April 2022	15	Backward	ATL03_20220417160941_03901512_005_02
0390	17 July 2022	16	Forward	ATL03_20220717114952_03901612_005_01
0560	1 February 2020	06	Forward	ATL03_20200201091359_05600602_005_01
0560	2 May 2020	07	Forward	ATL03_20200502045345_05600702_005_01
0560	1 August 2020	08	Backward	ATL03_20200801003333_05600802_005_01
0560	30 October 2020	09	Backward	ATL03_20201030201321_05600902_005_01
0560	29 January 2021	10	Forward	ATL03_20210129155316_05601002_005_01
0560	30 April 2021	11	Forward	ATL03_20210430113305_05601102_005_01
0560	30 July 2021	12	Forward	ATL03_20210730071259_05601202_005_01
0560	29 October 2021	13	Backward	ATL03_20211029025257_05601302_005_01
0560	28 July 2022	16	Forward	ATL03_20220728135237_05601602_005_01

. Due to time-dependent distance deviation, the V5 data update some product parameters by adding the ATL03 photon height correction reference value. The relevant data on the website (https://search.earthdata.nasa.gov, accessed on 22 November 2022) are free to access.

3. Method

Because of the oscillation of photons in the air, laser trajectory would be deformed when it reaches the ground, and simple translation parameters are not enough to describe it accurately. The two-dimensional affine transformation model can effectively describe the deformation of the GTs caused by various factors, such as alignment errors and laser on-orbit time-varying pointing deviation. Therefore, a two-step terrain matching method is adopted. Firstly, a rough matching is performed using a pyramid iterative search algorithm which quickly obtains the approximate position of the GTs through layered search and provides an initial value for precise matching. Subsequently, an iterative matching method based on high-accuracy terrains is proposed, which refines the parameters of the affine transformation model of the GTs using the least square method, obtaining the position with the minimum elevation difference between laser photons and the high-accuracy DSM.

The flowchart of the proposed method is shown in Figure 3, including signal photon extraction, rough matching, and precise matching. The official confidence label and the quadtree isolation (QI) noise removal method is used to preprocess the ATL03 data to obtain high-quality signal photons. The two-step terrain matching method is then employed to correct the photon coordinates. The proposed algorithm considers the deformation of GTs in rotation and scaling scales, thus having the potential to achieve more accurate matching positions between laser photons and the reference terrain.

3.1. Signal Photon Extraction

ATLAS uses a high-sensitivity photon-counting system. It is easily affected by solar background noise, atmospheric scattering noise, and instrument noise [25]. Noise photons are widely distributed, so extracting signal photons from raw data is challenging. The result of signal photon extraction directly affects the follow-up steps [26]. Most proposed noise removal methods are based on spatial distribution characteristics, which are not friendly enough for users [27,28,29,30]. Photons’ local density distributions differ considerably by observing the raw data, signal, and noise. Signal photons are densely distributed, while noise photons are more sparsely scattered.

To avoid the influence of noise photons on the matching results, we use the signal label in ATL03 data and quadtree isolation to participate in the signal photon extraction. The parameter $signal\_conf\_ph$ is provided to users in ATL03 data to judge the confidence of photons. The value of $signal\_conf\_ph$ −2, −1, 0 or 1 indicates that the photon is an independent, noise, or buffer photon. The value of $signal\_conf\_ph$ 2, 3 or 4 indicates that the photon is a low-confidence, medium-confidence, or high-confidence signal. Therefore, the signal label in ATL03 data can be used to remove the obvious noise photons away from signals.

Zhang et al. [31] proposed a quadtree isolation noise removal method that employs a quadtree to partition each photon and determine each photon’s tree depth. The outliers are identified as photons with smaller tree depths. This method outperforms conventional noise removal methods in accuracy. It does not rely on external parameter input and has yielded promising results in various terrain experiments. Therefore, quadtree isolation can be used to accurately remove the remaining noise photons. The photon denoising result is shown in Figure 4.

3.2. Rough Matching

The conventional pyramid iterative search method roughly matches the laser photon tracks. It provides an initial value for the following least square iteration. Figure 5 shows the diagram of the rough matching method.

Two levels of pyramid search are set up, with a search step of 1 and 0.2 m, respectively. When searching each layer, a bilinear interpolation method is used to obtain DSM elevation values corresponding to ground photons. The best matching position is determined by calculating the ground laser and DSM correlation coefficient. Then, the next layer is searched. Once the step size is determined, we can obtain the corresponding matching coordinates $x_i$ and $y_i$, along with the ground elevation value $h_i$:

$$\{\begin{array}{c}{x}_i=x_0+s\ast l\\ y_i=y_0+s\ast l\\ h_i=f\left(x_i,y_i\right)\end{array},$$

(1)

where $l$ is the number of times the laser track moves within the search window with the value of $\left\{0,\pm 1,\pm 2\right\}$. The coordinate with the maximum correlation coefficient is selected as the result of rough matching:

$$r\left(x_i,y_i\right)=\frac{{{\displaystyle \sum }}_{i=1}^n\left(h_i-\overline{h}\right)\left(H_i-\overline{H}\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^n{\left(h_i-\overline{h}\right)}^2·{{\displaystyle \sum }}_{i=1}^n{\left(H_i-\overline{H}\right)}^2}},$$

(2)

$$x_{rough},y_{rough}=x_j,y_j \left(where r\left(x_i,y_i\right)=\stackrel{i\in \left(1,25\right)}{max}\right),$$

(3)

where $r\left(x_j,y_j\right)$ is the matching correlation coefficient between the laser photon track and topographic profile, $n$ is the number of photons in the laser track, and $H_i$ is the elevation values from ATL03 data. The search area is traversed to determine $x_{rough}$ and $y_{rough}$ with the highest correlation coefficient. The result of rough matching provides initial values for photon coordinates in precise matching. This not only reduces the time required for terrain matching, but also avoids getting trapped in local optima during the iterative process.

3.3. Precise Matching

In Section 3.2, the approximate location of matching between ground track and terrain profiles is determined, but the accuracy is not high enough due to the former step’s limitation. Photons oscillate in the air, leading to the laser track’s deformation. Simple translation parameters are not sufficient to describe these complexities. Therefore, a two-dimensional affine transformation model is developed, which contains rotation and scaling factors to describe the photon track’s deformation accurately. Then, the model parameters are optimized by the least square iteration to obtain more accurate terrain-matching results. The schematic representation of the iterative matching method based on high-accuracy terrains is shown in Figure 6.

The affine transformation model is constructed using photon coordinates obtained by rough matching:

$$x_{after}=a_0+a_1{x}_{before}+a_2{y}_{before},$$

(4)

$$y_{after}=b_0+b_1{x}_{before}+b_2{y}_{before},$$

(5)

where $a_{\mathrm{i}}$ and $b_{\mathrm{i}}$ ($i=0, 1, 2$) are the parameters of the affine transformation model, and $\left(x_{before},y_{before}\right)$ and $\left(x_{after},y_{after}\right)$ are the photons positions obtained after the rough match and the precise match, respectively.

Considering the deformation of elevation value and geometric distortion of the topographic profile, the elevation of the precise matched ground track can be written as:

$$H_{before }\left(x_{before},y_{before}\right)=r_0+r_1{H}_{after}\left(x_{after},y_{after}\right),$$

(6)

where $H_{before}$ and $H_{after}$ are the elevation values of photons before and after matching, respectively, $r_0$ is the translation of the elevation, and $r_1$ is the scaling factor.

The amount of difference before and after matching can be expressed as follows:

$$F\left(r_0,r_1,a_i,b_i\right)=r_0+r_1{H}_{after}\left(x_{after},y_{after}\right)-H_{before}\left(x_{before},y_{before}\right),$$

(7)

We perform Taylor processing on the expression and keep the first term as follows:

$$F\left(r_0,r_1,a_i,b_i\right)=F^0+\frac{\partial F}{\partial r_0}d{r}_0+\frac{\partial F}{\partial r_1}d{r}_1+{\displaystyle \sum }_{i=0}^2\frac{\partial F}{\partial a_i}d{a}_i+{\displaystyle \sum }_{i=0}^2\frac{\partial F}{\partial b_i}d{b}_i,$$

(8)

Therefore, the error equation of the $ith$ photon’s least square track matching can be listed as:

$$v_i=F\left(r_0,r_1,a_i,b_i\right)-\mathrm{\Delta }H,$$

(9)

The unknown parts are the correction values of various undetermined parameter values, and the initial value is given as follows: $r_0^0=0,r_1^0=1,a_0^0=0,a_1^0=1,a_2^0=0,b_0^0=0,b_1^0=0,b_2^0=1$.

The constant term is:

$$\mathrm{\Delta }H=r_0^0+r_1^0{H}_{after}\left(x_{after}^0,y_{after}^0\right)-H_{before}\left(x_{before}^0,y_{before}^0\right),$$

(10)

The signal photon extraction method cannot completely remove noise photons. The remaining noise photons can affect the results of least square adjustment, making it challenging to achieve the best matching between laser tracks and reference terrain. The robust estimation is introduced, and an appropriate equivalent weight matrix is chosen to replace the prior weight matrix. We ultimately choose the Tukey function by comparing the ability of different equivalent weight functions to suppress noise photons [32]. The specific experiments are described in Section 5.3.

The Tukey function is:

$${\overline{p}}_i=\{\begin{array}{cc}{p}_i{\left[1-{\left(\frac{v_i}{4.685{\sigma }_0}\right)}^2\right]}^2& \left|v_i\right|⩽4.685{\sigma }_0\\ 0& \left|v_i\right|>4.685{\sigma }_0\end{array},$$

(11)

where ${\overline{p}}_i$ is the weight of the error equation of the $ith$ photon calculated by the Tukey function, and ${\sigma }_0$ is the standard error of the mean.

If the number of photons involved is $n$, the equivalent weight matrix is obtained as follows:

$$\overline{P}=diag\left({\overline{p}}_1,{\overline{p}}_2,\dots ,{\overline{p}}_n\right),$$

(12)

Therefore, the solution of the equation can be written as

$$\widehat{X}={\left(C^T\overline{P}C\right)}^{-1}{C}^T\overline{P}L,$$

(13)

where $L$ is the constant vector of each photon calculated by Equation (10), $C={\left[\begin{array}{ccc}\frac{\partial F}{\partial r_0}& \frac{\partial F}{\partial r_1}& \begin{array}{ccc}\frac{\partial F}{\partial a_0}& \frac{\partial F}{\partial a_1}& \begin{array}{ccc}\frac{\partial F}{\partial a_2}& \frac{\partial F}{\partial b_0}& \begin{array}{cc}\frac{\partial F}{\partial b_1}& \frac{\partial F}{\partial b_2}\end{array}\end{array}\end{array}\end{array}\right]}^T$, $\widehat{X}={\left[\begin{array}{ccc}d{r}_0& d{r}_1& \begin{array}{ccc}d{a}_0& d{a}_1& \begin{array}{ccc}d{a}_2& d{b}_0& \begin{array}{cc}d{b}_1& d{b}_2\end{array}\end{array}\end{array}\end{array}\right]}^T$.

The adjustment results are used to correct the parameters of the affine transformation model, and Equations (4)–(13) are repeated until the difference between the two iterations is less than a threshold value.

4. Results

4.1. Terrain Matching

As shown in Figure 7, the elevation difference between most photons and the reference terrain is more than 2 m. Our method significantly improves the consistency between the corrected laser ground track and the reference DSM, indicating that most horizontal errors have been eliminated.

In Figure 8, most of the ATL03 photons are below the terrain profile before matching. The absolute value of the height difference between more than 80% photons and the reference terrain can be reduced to less than 1.5 m after coordinate correction. For RGT0390, the mean absolute value (MAV) before and after matching is 1.04 m and 0.64 m, respectively. For RGT0560, the MAV before and after matching is 2.64 m and 2.15 m. The matching of laser tracks and reference terrain in MDV shows better consistency.

The detailed values of the geolocation errors of laser ground tracks are listed in

Table 3. Detailed values of the geolocation errors of ICESat-2.

RGT	Beam	MAE (m)		STD (m)		RMSE (m)		${\mathit{R}}^2\times {10}^2$
		Before	After	Before	After	Before	After	Before	After
0390 (MDV)	gt1l	1.431	0.964	1.442	1.215	1.769	1.272	99.986	99.993
	gt1r	1.409	0.927	1.456	1.211	1.762	1.243	99.985	99.993
	gt2l	0.798	0.752	1.009	0.960	1.019	0.971	99.989	99.991
	gt2r	1.366	0.9556	1.496	1.267	1.697	1.282	99.975	99.985
	gt3l	0.608	0.530	0.756	0.684	0.781	0.685	99.990	99.992
	gt3r	0.9683	0.677	1.173	0.897	1.240	0.906	99.975	99.986
0560 (ZZ)	gt1l	2.954	1.981	3.053	2.608	3.590	2.642	98.302	99.346
	gt1r	3.252	2.271	3.3489	2.953	3.980	3.020	98.480	99.321
	gt2l	3.128	1.990	3.179	2.609	3.787	2.642	98.786	99.574
	gt2r	3.532	2.498	3.785	3.200	4.313	3.273	97.996	98.865
	gt3l	2.978	2.163	3.270	2.789	3.650	2.812	98.500	99.150
	gt3r	3.097	2.273	3.351	2.910	3.782	2.944	98.429	99.201

. After coordinating correction, the standard deviation (STD) of elevation difference between the photon and the reference terrain decreased by 14.27% and 14.61% in the MDV and ZZ, respectively. The RMSE dropped by 20.99% and 25.00%, respectively. Although the laser tracks in ZZ showed significant improvement, they still have higher STD and RMSE. It can be attributed to the complex vegetation cover, which reduces the geographic positioning accuracy of photons. A large area of exposed rock covered by MDV leads to more accurate matching results.

To further evaluate the effectiveness of our method, we use two alternative methods in the same study site to make a comparative study. Tang’s method utilizes the pyramid iteration method [13]. It determines the window and step sizes according to the terrain characteristics to determine the minimum elevation difference of DSM and improve the geolocation accuracy. Schenk’s method constructs the minimum area element [17]. It uses the least square iteration method to estimate the distance and minimum elevation from the photon to the minimum area element. Notably, both methods treat the laser photon track as a whole, which differs from ours. Figure 9 gives the comparison and verification results of the three terrain-matching methods.

By comparing the matching results calculated by the three methods, it is not difficult to find that the proposed method can accomplish the best matching between photons and the reference terrain. The lower MAV and STD indicate that considering the deformation of laser tracks in terrain matching is necessary.

4.2. Assessment of ICESat-2 Data Accuracy

Initially, 814 laser ground sets are selected, including 582 in MDV and 232 in ZZ. To ensure data quality, we kept 673 sets with high SNR, less affected by clouds and fog. As shown in Figure 10, the statistics are presented as the average offset after matching different beams.

Due to the high accuracy of the selected DSM data (better than 1 m), the average offset can be used as its horizontal error. The average horizontal accuracy of the ICESat-2 laser tracks in MDV and ZZ is 3.86 m and 4.67 m, respectively. The horizontal accuracy of the whole dataset is less than 5 m, which is far lower than the theoretical positioning accuracy requirement of 6.5 m specified in official documents. Additionally, the offset variations along the track direction of different beams are stable. The offset across the track is relatively large, caused by the laser altimetry’s working mode.

Table 4. Comparison of terrain-matching results from different methods.

Data	RGT	Beam	Latitude Range	Tang’s Method [13]		Schenk’s Method [17]		Proposed Method
				Cross-Track (m)	Along-Track (m)	Cross-Track (m)	Along-Track (m)	Cross-Track (m)	Along-Track (m)
19 July 2021	0390	gt1r	−77.718, −77.698	1.96	−5.88	1.77	−5.01	1.64	−4.90
19 July 2021	0390	gt3r	−77.696, −77.676	0	−3.92	0.12	−2.84	0.12	−3.93
19 April 2021	0390	gt1r	−77.718, −77.698	−1.96	1.96	−2.01	1.88	−2.36	1.72
18 October 2021	0390	gt3l	−77.682, −77.662	−1.96	5.88	−1.88	5.34	−1.71	6.87
17 April 2022	0390	gt2r	−77.702, −77.682	−3.92	−3.92	−3.64	−3.94	−3.78	−3.94
30 April 2021	0560	gt1r	34.457, 34.477	−1.01	−1.01	−1.10	−1.02	−1.08	−1.04
30 April 2021	0560	gt1l	34.485, 34.505	0	0	0.01	0.04	−0.13	0.17
30 July 2021	0560	gt2l	34.581, 34.601	−4.06	−1.01	−3.38	1.21	−5.40	0.50
30 July 2021	0560	gt3r	34.630, 34.650	2.03	0	2.21	0.17	2.57	−0.15

shows that the results obtained by the proposed method are the same as those obtained by the other two methods. This consistency further proves the effectiveness of the proposed method.

5. Discussion

5.1. Assessment of ICESat-2 Geolocation Errors

As mentioned in Section 1, our aim is to evaluate the geolocation accuracy and accuracy of ICESat-2 observations. Wang et al. [5] and Liu et al. [33] considered that slope and SNR are the primary factors that affect the positioning accuracy of ICESat-2. The satellite’s flight state constantly changes during operation, and the photon density is affected by the time difference between day and night. These factors can also affect photon positioning. Therefore, using slope, SNR, latitude, and photon density as input features, a random forest model is developed to determine the relative importance of each factor.

The RF model performed well in multivariate data analysis. The increased mean square error percentage (%IncMSE) provided by it can effectively reflect the relative importance of different variables [34]. The %IncMSE is the average increment of the contribution of a variable to the mean square error divided by the variability. The higher the %IncMSE, the more crucial the variable. It is worth noting that due to the limited explanatory power of the regression model, we mainly focus on ordering the variable %IncMSE values rather than their proportions [35]. Figure 11 indicates that slope is the primary factor affecting the positioning accuracy of ICESat-2, followed by SNR. Latitude and photon density can also have little impact on location accuracy.

5.2. Slope

Geographical positioning error caused by slope is regarded as one of the most significant error sources in LiDAR surveys, as terrain undulations can cause deviations in laser echo signals [36]. To verify the influence of slope on the positioning of ICESat-2, we count the slope corresponding to laser tracks in MDV and ZZ and draw box plots with 5° divisions, as shown in Figure 12. It is worth noting that due to the steep terrain of the MDV, 0–10° is merged for the convenience of analysis.

As the slope increases, the STD between the photons and reference terrain increases significantly. When the slope is less than 10°, the STD is low, and the distribution of photons’ elevation difference is concentrated. This indicates that the photon quality is higher at lower slopes. The large terrain slope can alter the path of photon reflection, thus affecting the photon’s positioning accuracy. As a result, future research should prioritize eliminating coordinate errors caused by slope in ATL03 data.

5.3. SNR

According to the random forest results in Section 5.1, SNR is also a crucial factor that affects the positioning accuracy of ICESat-2. Since the laser pulses of the ATLAS system are weak signals, distinguishing the reflected pulses from atmospheric scattering, solar radiation, and the instrument’s noise on the target object’s surface can be challenging. The significant background noise seriously impacts the accuracy of terrain-matching. Although the quadtree segmentation method has been used for photon denoising during data pre-processing, the noise photons cannot be removed completely. To enhance the method’s robustness, we add robust estimation during the least square iteration and select the appropriate estimation method to avoid the influence of gross error on the affine transformation model parameters that need to be solved.

Robust estimation is used to reduce the influence of outliers on estimation results. It uses different equivalent weight functions to affect the adjustment result. The selection of weight functions is based on the relationship between each observation in the sample and the overall sample. The functions will assign higher weights to some observations in the sample, causing these observations to have a greater impact on the least square adjustment. To select a function suitable for our method, we compare the matching results of the Tukey function and the Huber function [37] with different SNRs.

The Tukey function is expressed in Equation (11), and the Huber function can be expressed as follows:

$${\overline{p}}_i=\{\begin{array}{cc}{p}_i& \left|v_i\right|⩽2{\sigma }_0\\ p_i\frac{2{\sigma }_0}{\left|v_i\right|}& \left|v_i\right|>2{\sigma }_0\end{array}$$

(14)

We select a laser photon track and add noise photons to compare the matching results with different equivalent weight functions. The matching result without noise photons is considered the true position of the laser track. As shown in Figure 13, without robust estimation, the matching result deviates significantly from the true position when the SNR is 5%. This indicates that noise photons can easily affect the least square method. As the SNR increases, the error of the matching result tends to be stable, and the RMSE remains around 1 m. After using the equivalent weight function, noise interference on the matching result is significantly suppressed in the case of fewer noise photons. Especially when the SNR is less than 10%, the suppression effect of the Tukey function is particularly obvious. The SNR of denoised data can be kept below 10%. This indicates that the adjustment result can effectively reduce the effect of noise photons after using robust estimation.

6. Conclusions

In this study, we propose a new method for terrain matching to evaluate the geolocation accuracy of ICESat-2 laser altimetry observations. Unlike previous studies, we construct an affine transformation model for laser photon tracks and iteratively correct translation, rotation, and scaling parameters through least square matching to achieve the best match between the laser photon tracks and reference terrain. The method is tested with the ICESat-2 V5 data in McMurdo Dry Valley and Zhengzhou. The results in Section 4.1 show that this method can achieve better matching results. The STD of average elevation difference of MDV and ZZ decreased by 14.27% and 14.61%, respectively, and the RMSE decreased by 20.99% and 25.00%, respectively. Comparing the results of terrain matching, our method considers the deformation of laser tracks.

The average horizontal accuracy of the ICESat-2 laser tracks in MDV and ZZ is 3.86 m and 4.67 m, respectively, which meets the accuracy requirement of 6.5 m. We demonstrate through constructing a random forest model that slope and SNR are the key factors affecting the photon positioning of ICESat-2. Considering the sensitivity of the least square method, we introduce a robust estimation model to reduce the influence of noise photons on the matching results.

The terrain-matching method proposed in this paper can achieve ICESat-2 geolocation accuracy verification and provide a new method for calibrating satellite-based laser radar’s laser point and ranging system. However, this study still has certain limitations. Firstly, we only selected mid-latitude and high-latitude regions for experimentation. Future research will focus on verifying and analyzing the geolocation accuracy of ATLAS data in other parts of the world. Secondly, although we examined the impact of terrain slope on photon positioning, we have yet to propose a corresponding solution. Our next plan will focus on developing a correction model to improve the geolocation accuracy of ICESat-2.