Analysis of the temporal and spatial characteristics of lunar reconnaissance orbiter’s orbit error based on multi-coverage narrow angle camera images

ABSTRACT

Photogrammetric mapping of large area of the lunar surface using high-resolution orbiter images is very challenging due to increasing spatial resolution and lack of a comprehensive understanding of orbit error characteristics. The Lunar Reconnaissance Orbiter (LRO) Narrow Angle Camera (NAC) images are currently one of the highest-resolution images of the Moon available. The study of LRO orbit errors from the perspective of mapping applications is important for high accuracy mapping of large areas using NAC images. This paper proposes a method for orbit error estimation using multi-coverage NAC images. First, available NAC images covering the study regions are orthorectified using ISIS. Then, the impact crater feature matching algorithm is used to match the multi-coverage images. Finally, estimation of the orbit error is achieved through statistical analysis of a large number of homologous points obtained from the matching. The orbit error analyses of nine representative regions on the Moon show that in most cases the refined ephemeris provided by the LOLA/GRAIL team can significantly reduce the orbit error. The orbit errors show a sawtooth variation with a period of about one year, which have been stable for a long time and tend to increase slightly in recent years. There is a slight variation of the orbit error in space, which is even less obvious when using the refined ephemeris. The general tendency is for the far side to be larger than the near side at the same latitude, and for the higher latitudes to be larger than the lower latitudes. The proposed method and analysis results provide a reference for selection and processing of multi-coverage LROC NAC images for high-resolution large-area mapping.

KEYWORDS:

• Orbit error
• LRO
• NAC image
• multiple coverage
• geo-positioning deviation

1. Introduction

High-resolution orbital images are one of the fundamental datasets for lunar exploration and scientific research. These images are usually photogrammetrically processed to produce seamless mosaicked digital orthophoto maps (DOMs) or digital elevation models (DEMs) as mapping products, and applied in high-resolution topographic analyses in regions of interest, lunar landing site selection and assessment, as well as detailed morphological analysis of lunar surface features, e.g. craters (Di et al. 2019a; Li et al. 2023; Nass et al. 2021). The orbit determination accuracies of Earth satellites supported by Global Navigation Satellite Systems (GNSS) are usually very high, e.g. the accuracy of the GNSS-derived CHAMP (CHAllenging Minisatellite Payload) satellite orbit is evaluated to be better than 10 cm by comparison with satellite laser ranging measurements (Qin and Yang 2005), the accuracy can even reach 2–3 cm level using new high precision orbit determination software (Zhao, Liu, and Ge 2006). Due to the vast distance between the Earth and the Moon and lack of GNSS on the Moon, the orbit determination accuracies of lunar orbiters are much lower than that of Earth satellites. As the mapping accuracy of orbital imagery is heavily dependent on the orbit determination accuracy, the orbit errors have to be analyzed and treated carefully in photogrammetric processing of the orbital imagery. When the image resolution is low, the impact of orbit errors on mapping may be acceptable. For instance, with a spatial resolution over 100 m, even orbit errors of tens to a hundred meters only cause deviations equivalent to sub-pixels in the Lunar Reconnaissance Orbiter Camera (LROC) Wide Angle Camera (WAC) images. This has little effect on the seamless mosaic of orthoimages (Speyerer et al. 2011). However, the impact of a ten-to-hundred-meter error in spacecraft position is much greater for high resolution images meeting or exceeding these deviations, such as the meter scale images produced by LROC Narrow Angle Cameras (Di et al. 2019b). Therefore, it is crucial to carefully consider orbit errors when mapping high-resolution lunar orbiter images for large regions of the lunar surface.

The Lunar Reconnaissance Orbiter (LRO) has been in orbit for over 14 years since its launch in 2009. It carries seven payloads, including the LROC Narrow Angle Camera (NAC) system, which has imaging spatial resolution up to 50 cm (Robinson et al. 2010), and the Lunar Orbiter Laser Altimeter (LOLA), which has a footprint diameter of 5 m and a vertical ranging precision of 10 cm (Smith et al. 2010). These payloads have continuously provided high-precision and high-resolution lunar imaging and elevation data for subsequent lunar explorations and scientific researches (Vondrak et al. 2010). The LROC NAC acquires images mostly with a resolution of 0.5 m to 2 m, making it one of the highest resolution sources of lunar surface data available. In-orbit geometric calibration has improved the internal accuracy of NAC images by providing more precise camera orientation parameters (Speyerer et al. 2016). NAC images have been used extensively to produce high-resolution DOMs and DEMs (Henriksen et al. 2017; Liu and Wu 2020; Liu et al. 2022), and to extract small features on the lunar surface (Bickel et al. 2019; Fairweather et al. 2022; Jia et al. 2023). After collecting data for a prolonged period, NAC images have now covered almost the entire lunar surface. Many studies have achieved encouraging results from large-scale mapping using NAC images (Archinal et al. 2023; Di et al. 2019b; Henriksen et al. 2015; Klem et al. 2014; Wagner et al. 2022). Orbit errors have been found to be a major source of positional discrepancies between overlapping images. Constrained by orbit errors and challenges of image matching under different illumination conditions, producing large-scale controlled mosaic using LROC NAC images remains a complex task.

The orbit determination (OD) of the LRO spacecraft has been achieved through analysis of its radio tracking data. Typically, most LRO tracking data consist of radiometric Doppler and range measurements from various ground stations (Mazarico et al. 2012). Many OD techniques have been developed to improve LRO spacecraft position accuracy. The lunar gravity field developed using LRO radio tracking data improves the accuracy of radio-only OD to 20 m and 14 m in combination with LOLA altimetric crossovers (Mazarico et al. 2012). The high-accuracy gravity fields from Gravity Recovery and Interior Laboratory (GRAIL), which have a spatial resolution of less than 10 km, significantly improved the quality of radio-only orbit reconstruction to less than 10 m (Mazarico et al. 2013). One-way laser ranging data from ground-based observations has also been utilized for OD, but its effectiveness is considered to be moderate (Loecher and Kusche 2019). Although there are other attempts to improve the accuracy of OD, so far the two ephemerides provided by the LRO Mission Operations Center and the LOLA/GRAIL team are the most commonly used for mapping applications (Löcher and Kusche 2018; Maier and Baur 2016; Mazarico et al. 2018). The analysis of overlapping arc segments is a common means of evaluating the performance of the OD method described above, but yields only the precision of the internal conformity of the track measurement data to the orbital dynamics model, not the accuracy of the LRO position itself, which is what is really wanted to be understood in the mapping process. Therefore, by using NAC images with sufficient resolution to observe the Lunar Laser Ranging Retro-Reflectors (LRRRs) on the lunar surface as a bridge between the orbit and the absolute control points, the five LRRRs on the lunar nearside (whose positional accuracy is in the sub-meter level due to long-term ground-based ranging observations and which serve as absolute control points on the lunar surface) can be used to evaluate the absolute accuracy of the OD. However, the limited number of LRRRs makes it difficult to establish a more comprehensive understanding of LRO orbit errors in time and space, and the assessment results can seldom be practically applied in large-area mapping. Therefore, it is highly desirable to develop a more comprehensive approach to analyzing orbit errors for large-area and global mapping of the Moon.

Multiple coverage (multi-coverage) NAC images exist for most regions of the lunar surface, with tens of observations for some important target areas. Using multi-coverage NAC images, the points of interest on the lunar surface can be accurately localized based on the assumption of orbit error randomness (Liu et al. 2015; Wagner et al. 2017). Based on multi-coverage NAC images, this paper proposes a method for orbit error estimation using the geo-positioning deviations between corresponding points of multiple images. The orbit errors of two common LRO ephemerides are compared by estimating the orbit errors of more than 10,000 NAC images in nine representative regions of the lunar surface, and the temporal and spatial trends of the orbit error are then analyzed, respectively. The paper is organized as follows: Section 2 introduces the data used in this paper, including images, tracking and auxiliary data, and elevation data; Section 3 describes the detailed process of image processing, matching and statistical analysis involved in the method; Section 4 shows the results of the statistical calculation of the orbit errors and analyses and discussions in time and space. The findings are summarized in Section 5.

2. Data

2.1. LROC NAC images

The LROC NAC consists of two identical cameras, NAC-Left (NAC-L) and NAC-Right (NAC-R). Each NAC is equipped with a telescope with a focal length of 700 mm and a linear array charge-coupled device (CCD) with 5064 pixels. The NACs are mounted at an offset of 1.39° from the nadir, and the combined field of view across-track is 5.67° (~10000 pixels) with an overlap of about 135 pixels between the image pair from NAC-L and NAC-R. Panchromatic images with a resolution of 0.5 m can be acquired at an orbital altitude of 50 km, corresponding to a total ground width of about 5 km for both cameras (Robinson et al. 2010). The NAC is a typical line-array push-broom imager that uses the motion of the orbiter to construct a two-dimensional image; a typical NAC image is 5064 columns by 52,224 rows (Humm et al. 2016). Currently, the LRO is in the extended science mission phase, and the NACs are still collecting data on an ongoing basis.

After collecting data for 14 years, NAC images now provide almost complete global coverage of the Moon. Additionally, many regions have a significant number of repeated observations. For example, the Apollo 15 LRRR has been observed over 40 times. The workload of processing all images of the lunar globe would be enormous, therefore, nine representative regions of interest were selected in this study, as shown in Figure 1. The regions were selected based on their global representativeness, including both the nearside and far side of the Moon, as well as regions of varying latitude. The principle of image selection in each region is to avoid rejecting images to enable as many repeat observations as possible, unless the observing conditions are poor. Images with an emission angle greater than 10° and a solar incidence angle less than 5° or greater than 85° (90° for the region E near the south pole) were rejected. More than 1000 NAC images for each region were selected and downloaded from the Planetary Data System (PDS) Geosciences Node Lunar Orbital Data Explorer website (https://ode.rsl.wustl.edu/moon/index.aspx), including PDS experiment data record (EDR) files and their corresponding image metadata, the details of which are given in Table 1. A total of 14,527 NAC images were used in this study.

Figure 1. Distribution of selected regions on the lunar surface. Nine regions are marked as white rectangles on the LROC Wide Angle Camera (WAC) mosaic. The left and right figures for nearside and far side of the Moon are in orthographic projection with a projection center latitude of 0° and projection center longitudes of 0° and 180°, respectively.

Table 1. Detailed information of the selected regions.

	Western Most Lon.(°)	Min Lat.(°)	Eastern Most Lon.(°)	Max Lat.(°)	No. of NAC images	Area （km^2 ）	Start epoch	Stop epoch
A	−56	41	−48	45	1285	21531	17 July 2009	14 June 2023
A1	124	41	132	45	1366	21531	31 July 2009	20 April 2023
B	−7.5	9	−2	14	1180	24787	14 July 2009	12 May 2022
B1	172.5	9	178	14	1120	24787	27 July 2009	25 January 2023
C	22	−45	30	−41	1484	21531	11 July 2009	7 June 2023
C1	−158	−45	−150	−41	1442	21531	15 October 2009	12 May 2023
D	−6.5	−65.5	6.5	−61.5	2598	21374	13 July 2009	9 June 2023
D1	173.5	−65.5	−173.5	−61.5	2467	21374	27 July 2009	14 May 2023
E	110.69	−89.04	140.26	−88.35	1585	225	30 June 2009	11 November 2013

2.2. SPICE kernels

The imaging geometry model of the orbital imagery provides a one-to-one correspondence between each pixel in the image and its corresponding geographic position on the lunar surface. The parameters of the model consist of the position and attitude of the spacecraft during imaging, camera parameters, coordinate system transformation parameters and more. The Navigation and Ancillary Information Facility (NAIF) at NASA’s Jet Propulsion Laboratory archives and updates the parameters for the NAC images. These parameters are publicly available as a series of binary and text based Spacecraft, Planet, Instrument, C-Matrix and Events (SPICE) kernels (Acton 1996). For this study, all SPICE kernels for LRO were downloaded from the NAIF website (https://naif.jpl.nasa.gov/pub/naif/). The integrated software for imagers and spectrometers (ISIS) can invoke these parameters during the ortho-rectification process of the images.

SPICE kernels contain exterior orientation (EO) parameters and interior orientation (IO) parameters for the cameras. During image acquisition, the EO parameters change significantly, whereas the IO parameters ideally remain constant throughout the mission. The EO parameters describe the position and attitude of the camera relative to the Moon. In the NAIF archive, the spacecraft position (ephemeris) and attitude information are provided by the LRO Mission Operations Centre (MOC). In addition, a refined ephemeris is also provided by the LOLA and the GRAIL science team. The EO information is stored in a series of kernel files, including Spacecraft Position Kernels (SPKs), C-Matrix Kernels (CKs), and a single Frames Kernel (FK). Additionally, a planetary and lunar ephemeris (de421) kernel, planetary constants kernel for the Moon (moon_pa_de421_19002050), and lunar frames kernel (moon_080317 and moon_assoc_me) are also provided to obtain the relative position and attitude of the Moon. The IO parameters are determined by the physical characteristics of the optics and sensor system, including focal length, principal points and lens optical aberrations, which are stored in the Instrument Kernel (IK).

To ensure accurate estimation of orbital errors and minimize the impact of other sources of error, we maintain consistency in all other parameters and use the most accurate data possible when processing the images independently using the two LRO ephemerides.

2.3. Lunar digital elevation models

The NAC is not designed for stereo imaging. Although the spacecraft roll can provide a small amount of stereo information, the intersection angles between most multi-coverage images are minimal. Therefore, image geo-positioning and ortho-rectification both depend on external elevation data. Efforts are made to avoid using Digital Elevation Models (DEMs) with significantly different accuracies in different regions, despite the relatively small horizontal positioning deviation caused by differences in the vertical direction (Toutin 1995). High-precision DEM is utilized for each region to minimize geometric distortions caused by elevation uncertainties. SLDEM is uniformly used for 60°N and 60°S, while LDEM is used for the rest. LDEM was generated by interpolating the LOLA data. The laser points collected by LOLA are more densely populated at the poles, providing a higher accuracy and resolution of LDEM in the polar regions (Smith et al. 2010). SLDEM is a topographic dataset created by merging the DEM generated by SELENE Terrain Camera images with the LOLA data in the range of 60°N to 60°S. The product resolution is up to 512 pixels per degree (~60 m in the equatorial region), and the vertical accuracy is approximately 3–4 m (Barker et al. 2016). The lunar shape model used in ISIS is generated by adding the elevation of the DEM to the radius of the moon, which is 1,737,400 m.

3. Methods

This paper proposes a method to evaluate the orbit error by using the geographic position deviations among the multi-coverage NAC images. The LRO orbit error is estimated by utilizing the characteristics of NAC images with many repeated observations and high resolution. Subsequently, the temporal and spatial characteristics of the orbital error are analyzed. Figure 2 illustrates the overall process.

Figure 2. The flow chart of orbit error analysis using multi-coverage NAC images.

First, we use the USGS Integrated Software for Imagers and Spectrometers (ISIS) to orthorectify all the NAC images (see Section 3.1 for details); then we conduct image matching of all the orthophotos using the in-house developed impact crater feature matching algorithm (see Section 3.2 for details) to obtain the corresponding points among the multi-coverage images. Finally, the corresponding orbit error of the image is estimated by statistically calculating the geo-positioning deviations of the corresponding points. The results of all regions are then combined to complete the analysis of the spatial and temporal characteristics of the orbit error (see Section 3.3 for details).

3.1. Orthophoto generation using ISIS

Orthophotos accurately describe the geo-position of each pixel in the corresponding region of the lunar surface and represent natural shape, and are often used as basemaps for precise geometric measurements. In this research, orthophotos are beneficial for subsequent impact crater matching and the precise and efficient calculation of geographic positioning deviations among multi-coverage images. Therefore, we first have generated orthophotos of all the selected images. The Astrogeology Science Center of the United States Geological Survey (USGS) has developed a free and open-source software ISIS for processing planetary images (Anderson et al. 2004). ISIS is one of the most professional tools for handling NAC images. ISIS is capable of processing NAC image EDR data (mentioned in Section 2.1), incorporating specified NAC position and pointing information (SPICE kernels, mentioned in Section 2.2), and utilizing a lunar shape model generated from specified elevation data (mentioned in Section 2.3), creating orthophotos that eliminate image distortions caused by camera or spacecraft motion and topography relief. The process of generating each orthophoto using ISIS version 7.2.0 (Laura et al. 2023) is summarized as follows:

1. Convert NAC EDR files to ISIS cubes using program “lronac2isis”, which preserves all the necessary keywords for proper image calibration.

2. Attach the SPICE kernels for camera position and attitude during imaging, camera parameters, etc. and the specified lunar shape model to the cube file using the “spiceinit” program. Ephemeris information is controlled by adding “Smithed = yes”, which gives priority to the refined ephemeris provided by the LOLA/GRAIL team, otherwise the ephemeris provided by the MOC is used. It is to be noted that the refined ephemeris is updated with a delay, and even if “Smithed = yes” is added when processing recently acquired images, the ephemeris actually used may still be the one provided by the MOC and hence needs to be checked. This paper generates orthophotos using the two ephemerides separately. To specify the DEM used, add “shape = user model = Path of a shape model” to the command.

3. Convert DN values in EDR files to irradiance (I/F) using the “lronaccal” program.

4. Correct artifacts due to the imaging characteristics of the NAC camera using the “lronacecho” program.

5. Generate orthophotos according to the specified projection and spatial resolution using the “cam2map” program, based on the information attached in step 2). Mercator projection (polar stereographic projection for the region E near the south pole) with the latitude and longitude of the center of the region as the center of projection is used for the projection of each region image in this paper to reduce the impact of projection distortion. The spatial resolution of the output orthophoto was uniformly set at 2 m per pixel to reduce data processing time as much as possible without sacrificing analysis accuracy.

6. Convert the ISIS cube format to the more general GeoTIFF format by using the open-source GDAL library for subsequent use. Since the left and right NACs have the same ephemeris, the geo-positional inconsistency between them is reduced to a low level after on-orbit geometric calibration (Speyerer et al. 2016), and so in this paper the left and right NACs are regularly geographically coupled into a single image before matching.

3.2. Regional image matching among multi-coverage orthophotos

Taking full advantage of multi-coverage images is the core of this work. The corresponding points (also called homologous points) obtained by image matching can tie the multi-coverage images for subsequent geo-positioning deviation analysis. Non-linear radiometric differences in the image caused by different lighting conditions still make it challenging to obtain homologous points in multi-coverage images. In this work, an in-house developed image matching algorithm based on small impact craters (typically 5 to 30 pixels) is used (Xie et al. 2023), which can better adapt to illumination variations and achieve more robust matching of regional multi-coverage images. The specific image matching process includes image overlapping area calculation, impact crater detection, feature description vector construction using K-nearest neighbor structure, and feature matching using cosine distance as the metric and Nearest Neighbor Distance Ratio (NNDR) as the threshold. With this matching result, the Iterative Closest Point (ICP) method can be used to further increase the number of matched homologous points in the result. The crater feature detection part is based on a high-performance deep learning target detection algorithm, which is trained using a manually labeled crater dataset of the NAC images in a variety of illumination conditions, and is able to achieve crater detection under most illumination conditions with a reasonable level of precision and recall. The matching part then fully exploits the K-nearest neighbor structure information of the craters on the orthophoto to reduce the effect of non-linear radiation differences on the descriptors. The number of homologous points matched between two images is generally huge, so a grid with an interval of 250 m is used for sparse sampling, which reduces the computation time in the subsequent analysis and also ensures a uniform distribution of homologous points. Figure 3 shows the results of the crater feature matching after grid-based sampling for two common NAC images with different illumination conditions in region D1.

Figure 3. Results of the crater feature matching for two common NAC images with different illumination conditions. The image IDs for (a) and (b) are M1097495328 and M1139912969, respectively. The green cross represents the point of matched crater’s center. Zoomed-in images of the red boxes in a) and (b) are shown in (c) and (d), respectively. Zoomed-in images of the red boxes in (c) and (d) are shown in (e) and (f), respectively.

3.3. Orbit error estimation and statistical analysis

3.3.1. Geo-positioning deviation calculation

There are a number of factors that contribute to geo-positioning errors during the imaging process, and the LROC team has essentially reduced the positioning errors caused by the characteristics of the NAC camera to a low level through in-orbit geometric calibration of the narrow angle camera (Speyerer et al. 2016). The almost nadir-looking imaging characteristics of most NAC images, coupled with the use of high accuracy DEMs, make the planimetric positioning errors caused by the vertical errors of the DEMs of a very small order of magnitude. Therefore, the attitude and position measurement errors of LRO are the main source of the NAC images geo-positioning errors. The LRO system is known to have an attitude measurement frequency of 5 Hz, with a measurement error of less than 30 arc seconds (Calhoun and Garrick 2007), and to cause a displacement of up to 7 m in a 50 km orbit. Over a period of seconds to minutes of NAC imaging, the total image position offset due to attitude measurement errors should be insignificant, and generally be smaller compared with the total image position offset due to orbit errors. Accordingly, in this paper, we assume that the geographic position deviations between the homologous points of the two images comes from two parts: one part is a constant deviation caused by the superposition of the orbit errors, which does not change with the rows or columns of the images, i.e. the orbit errors are considered as a fixed value for a short period of the imaging. The other part is the fluctuation of the offset caused by the accumulation of many other errors, which might reflect the internal deformation of the image. The above assumption has been confirmed by extensive visual inspection and statistics, which are described in more detail in Section 4.

Using the large number of homologous points matched from the two images, the constant deviation components $\mathrm{d}\mathit{X}$ and $\mathrm{d}\mathit{Y}$ along the $\mathit{X}$ and $\mathit{Y}$ directions of the projected coordinate system can be calculated based on the least squares method using Equations (1)(1) $\mathrm{d}\mathit{X},\mathrm{d}\mathit{Y}=\underset{\mathrm{d}X,\mathrm{d}Y}{argmin}\sum _{\mathit{i}=1}^n\left[{\left(X_i+\mathrm{d}\mathit{X}-X_i^{\prime }\right)}^2\text{break}+{\left(Y_i+\mathrm{d}\mathit{Y}-Y_i^{\prime }\right)}^2\right]$(1) and (2(2) $\mathit{\sigma }=\underset{i}{max}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]-\underset{i}{min}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]$(2) ), and then the metric $\mathit{\sigma }$, which corresponds to the scale of the fluctuating part of the deviations, can also be calculated using Equation (2)(2) $\mathit{\sigma }=\underset{i}{max}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]-\underset{i}{min}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]$(2) :

(1) $\mathrm{d}\mathit{X},\mathrm{d}\mathit{Y}=\underset{\mathrm{d}X,\mathrm{d}Y}{argmin}\sum _{\mathit{i}=1}^n\left[{\left(X_i+\mathrm{d}\mathit{X}-X_i^{\prime }\right)}^2\text{break}+{\left(Y_i+\mathrm{d}\mathit{Y}-Y_i^{\prime }\right)}^2\right]$(1)

(2) $\mathit{\sigma }=\underset{i}{max}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]-\underset{i}{min}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]$(2)

where $\left(X_i,Y_i\right)$ and $\left(X_i^{\prime },X_i^{\prime }\right)$ are the projected coordinates of a pair of homologous points, $\mathit{i}$ is the serial number of the homologous points, and $\mathit{n}$ is the total number of homologous points. It is also necessary to eliminate the homologous points whose offset values are more than three times the standard deviation in order to reduce outliers.

3.3.2. Orbit error estimation

After 14 years of data collection by NAC, most of the lunar surface has been repeatedly observed many times, and due to the amount of area available for imagery for a given latitude band decreasing as latitude increases, the coverage of image repetition generally increases with increasing latitude for a camera operating in polar orbits. Figure 4 shows the multiple coverage of the NAC images in Region B near the lunar equator, with most areas having more than 10 repeated observations. Under this condition, combined with the randomness nature of the errors, multi-NAC images have been used to locate targets on the lunar surface such as the LRRRs with high accuracy and verified that the average of the results of multiple localizations can estimate the true position well (Liu et al. 2015; Wagner et al. 2017). This paper follows a similar idea to estimate the orbit error of a single image. The homologous points between the image and all other overlapping images are obtained using the method described in Section 3.2, and the corresponding constant deviations can be calculated according to Section 3.3.1. Since the orbit errors are considered to be random and normally distributed, constant deviations of the image to multiple images are averaged to counteract the component of other image orbit errors, and finally an evaluation of the orbit error for that image is left. Similarly, multiple fluctuating deviations are directly averaged as $\bar{\sigma }$ and used to express the image internal distortion. Equation (3)(3) $\mathit{OrbitError}=\frac{\sqrt{{\left[{\sum }_{j=1}^m\mathrm{d}{X}_j\right]}^2+{\left[{\sum }_{j=1}^m\mathrm{d}{Y}_j\right]}^2}}{\mathit{m}}$(3) and (4(4) $\bar{\sigma }=\frac{{\sum }_{j=1}^m{\sigma }_j}{\mathit{m}}$(4) ) show the specific calculations:

Figure 4. Repeat observations of NAC images in region B.

(3) $\mathit{OrbitError}=\frac{\sqrt{{\left[{\sum }_{j=1}^m\mathrm{d}{X}_j\right]}^2+{\left[{\sum }_{j=1}^m\mathrm{d}{Y}_j\right]}^2}}{\mathit{m}}$(3)

(4) $\bar{\sigma }=\frac{{\sum }_{j=1}^m{\sigma }_j}{\mathit{m}}$(4)

where $\mathrm{d}{X}_j,\mathrm{d}{Y}_j$ are the constant deviations between the image and its overlapping image $\mathit{j}$, calculated using Equation (1)(1) $\mathrm{d}\mathit{X},\mathrm{d}\mathit{Y}=\underset{\mathrm{d}X,\mathrm{d}Y}{argmin}\sum _{\mathit{i}=1}^n\left[{\left(X_i+\mathrm{d}\mathit{X}-X_i^{\prime }\right)}^2\text{break}+{\left(Y_i+\mathrm{d}\mathit{Y}-Y_i^{\prime }\right)}^2\right]$(1) , ${\sigma }_j$ is the fluctuating deviation calculated using Equation (2)(2) $\mathit{\sigma }=\underset{i}{max}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]-\underset{i}{min}\left[{\left(X_i-X_i^{\prime }\right)}^2+{\left(Y_i-Y_i^{\prime }\right)}^2\right]$(2) , and $\mathit{m}$ is the number of overlaps. Values exceeding three times the standard deviations should be excluded from the averaging calculation to minimize the adverse effects of images with unusually large orbital errors or severe internal distortions. To make the estimation as accurate as possible, only when the number of overlapping images after rejecting gross errors is greater than 10, Equations (3)(3) $\mathit{OrbitError}=\frac{\sqrt{{\left[{\sum }_{j=1}^m\mathrm{d}{X}_j\right]}^2+{\left[{\sum }_{j=1}^m\mathrm{d}{Y}_j\right]}^2}}{\mathit{m}}$(3) and (4(4) $\bar{\sigma }=\frac{{\sum }_{j=1}^m{\sigma }_j}{\mathit{m}}$(4) ) are used for calculation.

3.3.3. Statistics and analysis

The orbit error corresponding to an image with sufficient overlaps can be calculated using the above method. To analyze the pattern of change in orbit error over time, the start time of imaging is obtained from the image metadata. The mean and standard deviation of the image orbit errors are calculated for each region and all regions together, respectively, and are used to analyze the influence of ephemeris and space on the orbit errors. In addition, the SPICE toolkit (Annex et al. 2020) is used to calculate the orbit altitude, the solar beta angle (the angle between the Sun and the LRO orbital plane) and the earth beta angle (the angle between the Earth and the LRO orbital plane) at the time of imaging, which are used to analyze the possible effects of orbit altitude, illumination and orbital tracking geometry on orbit determination. Finally, the true geo-positioning deviations of the images before and after orbit error correction are compared using Apollo15 LRRR, to illustrate the reliability and effectiveness of the orbit errors estimated in this paper. The specific results and discussions of the above analyses are described in detail in the next section.

4. Results and discussions

Using the data mentioned in Section 2 and the methods mentioned in Section 3, we have completed the generation of orthophotos and image matching for all 8 regions and both ephemerides. We have visualized the deviations between homologous points of all images, and found that the results are consistent with the hypothesis that the deviation between homologous points of most images does indeed exhibit both constant and fluctuating deviations. The deviations between two images do not vary with the rows or columns with a significant tendency, and show randomness. A typical example of the distribution of geo-positioning deviations of homologous points using different ephemerides is shown in Figure 5. The reference image is in region C1 with an ID of M1097495328. The number of images successfully matched with it are 29 (LOLA/GRAIL ephemeris) and 30 (MOC ephemeris). This specific example also clearly demonstrates that the LOLA/GRAIL ephemeris provides much better geo-positioning precision than the MOC ephemeris. The statistics of the fluctuating part (Figure 6) show that the magnitude of the fluctuating deviations is mostly smaller than that of the constant deviations, and its magnitude is less dependent on the type of ephemeris used, and is most likely caused by internal deformation of the image. However, when the overlapping area between images is particularly small, the fluctuating deviations does not reflect the internal deformation of the whole image well, and the influence of the internal deformation of the image on the estimation of the constant deviations increases, which is why images with a large fluctuation offset are excluded.

Figure 5. Distribution of geo-positioning deviations of homologous points using the two ephemerides. Each point represents a geo-positioning deviation from a pair of homologous points. The use of different colors indicates that the homologous points were obtained by matching with different images. The reference image is in region C1 with an ID of M1097495328.

Figure 6. Histograms of $\bar{\sigma }$ for all images with the two ephemerides, respectively. The magnitude of internal deformation in the images are distributed centrally between 3 m and 7 m.

4.1. Orbit error using different ephemeris

The results of the LRRR-based assessment show that the orbital errors of the precision ephemeris provided by the LOLA/GRAIL team are much smaller than those provided by the MOC (Speyerer et al. 2016), and this paper estimates the orbit errors of the two ephemerides for a wider range of regions and times and reaches similar conclusions. We have calculated a total of 6908 orbit errors based on the LOLA/GRAIL ephemeris and 7369 orbit errors based on the MOC ephemeris, as shown in Figure 7. The refined ephemeris provided by the LOLA/GRAIL team is able to reduce the orbit error significantly. The median error from the LOLA/GRAIL is 16 m, the mean error is 23 m. There are still a few images where the use of refined ephemeris does not significantly reduce the orbital error, and in some cases, it even increases abnormally. If the images with abnormal errors are rejected, the mean error is reduced from 23 m to 20 m. This highlights the importance of the orbit error analysis, which can help avoid the use of such anomalous images in mapping.

Figure 7. Statistical results of orbit errors corresponding to different ephemerides. Each point represents the orbit error corresponding to an image. The horizontal line in the box plot represents the median, while the solid black square represents the mean. Note that the scales of the vertical axis below and above the horizontal line at 200 are different for convenience of visualization.

4.2. Temporal characteristics of orbit error

By correlating the orbit error with the corresponding imaging time, we can observe the temporal characteristics of the orbit error. Due to the short imaging time of the image, the orbit error is considered to be almost constant during the imaging period. As illustrated in Figure 8, each point represents an orbit error at a moment (referring to the moment when the image begins to be captured). The orbit errors corresponding to the MOC ephemeris are shown above, and the orbit errors corresponding to the LOLA/GRAIL ephemeris are shown below, and in general the orbit errors from both ephemerides show a jagged fluctuation over time, which has been relatively smooth for a long time, with a fluctuating upward trend from about year 2018 onwards. The orbit of LRO underwent several adjustments throughout the mission. During the commissioning phase, LRO was initially placed in a 30 × 200 km quasi-stable polar orbit with perilune over the south pole, followed by a 50 km circular orbit from 15 September 2009. To conserve fuel, LRO was then returned to a more stable 30 × 180 km elliptical orbit on 11 December 2011 (Mazarico et al. 2018). The two vertical gray lines in the figures correspond to the two orbit adjustments, which seem to have not caused noticeable change of orbit error. The sawtooth fluctuation has a period of one year, which is similar to the period of variation of the solar beta angle. Hence, it may be due to temperature changes caused by illumination leading to periodic variation of camera orientation parameters, or to the periodic effect of sunlight pressure on the orbit.

Figure 8. Distribution of orbit errors over time. Each point represents the orbit error corresponding to an image. Different colors are used for different regions. The solar beta angles at the time of imaging are indicated by the red crosses in the lower figure. Note that the scales of the vertical axis below and above the horizontal line at 200 are different for convenience of visualization.

4.3. Spatial characteristics of orbit error

Due to tidal locking between the Earth and the Moon, the Moon always faces Earth on one side. It is unable to directly track the orbit from the Earth tracking stations when the orbiter travels to the far side of the Moon and the orbiter’s position on the near side also affects the geometry of the Earth tracking observations, which may have an impact on orbit errors. To reveal the potential inconsistencies of orbit errors in different regions of the near side and far side of the Moon, we conducted a spatial analysis, with the results depicted in Figure 9. At the same latitude, the orbit error of the LRO at lunar near side is slightly smaller than that at lunar far side. At various latitudes, there is a tendency for orbit errors to decrease and then increase from north to south. i.e. orbital errors at mid-latitudes are slightly smaller than those at higher latitudes. These spatial differences have been reduced to lower levels in the refined ephemeris.

Figure 9. Spatial distribution of orbit errors. Each point represents the mean value of the orbital error for a specific region. The MOC and LOLA/GRAIL ephemerides are shown as dotted and solid lines, respectively. The solid square represents the near side of the moon, while the solid circle represents the far side. The standard deviation is indicated by the error bar for each point.

4.4. Assessment of geo-positioning accuracy improvement using A15 LRRR

Through the above temporal and spatial analyses of the orbit error, we have gained a more comprehensive understanding of the orbit error of the LRO. Furthermore, we can directly improve the geo-positioning accuracy of the image by correcting the orbit error, whose effectiveness is assessed as follows using Apollo 15 LRRR. The orbit errors of NAC images in the A15 LRRR region using LOLA/GRAIL ephemeris are estimated and used for image geo-positioning correction. By manually determining the coordinates of the LRRR on 42 NAC images before and after the orbit correction, and using the LRRR coordinates of the ground-based observations as ground truth, the planimetric deviations of the image positioning were calculated, respectively. The results are shown in Figure 10, which demonstrates that the orbit error correction can reduce the geo-positioning errors of the images to a mean of 2.25 m, with a maximum of 7.17 m. This example demonstrates that using multiple coverage images and the proposed method, the orbit error can not only be evaluated but also be corrected so that the geo-positioning accuracy is significantly improved.

Figure 10. Geo-positioning deviation of images before and after orbit error correction. Blue line indicates before correction, and red line indicates after correction. Note that the scales of the vertical axis below and above the horizontal line at 60 are different for convenience of visualization.

5. Conclusions

This paper proposed a method for orbit error estimation using multiple-coverage images and analyzed the temporal and spatial characteristic of LRO’s orbit error by using 14,527 NAC images acquired from year 2009 to 2023 in nine representative regions of the Moon. The following conclusions can be drawn from the experimental results.

1. The refined ephemeris of the LRO provided by the LOLA/GRAIL team has significantly reduced the orbit error to 16 m (represented by the median of geo-positioning deviations from multi-coverage images acquired from year 2009 to 2023), although there are still some moments of high orbit error.

2. The LRO’s orbit errors have remained relatively stable over long periods of time, with periodic sawtooth fluctuations that may tend to increase slightly in recent years.

3. Spatially, the orbital errors show a tendency for the far side of the Moon to be slightly higher than the near side, and for high latitudes to be slightly higher than low latitudes. These spatial differences have been reduced to lower levels in the refined ephemeris.

4. Error estimation for spacecraft position using overlapping images is effective. It can be further applied to improve geo-positioning accuracy of multi-coverage orbital images and significantly contribute to large area mapping using high-resolution orbital images.

The paper aims to complement and verify existing orbit determination results from the perspective of mapping applications. The temporal and spatial characteristics of the orbit error revealed in this research provide a reference for selection and processing of multi-coverage LROC NAC images for high-resolution large-area mapping. Moreover, the proposed method could be combined to other existing orbit determination methods in the future to further refine the ephemeris of LRO. The method should be also applicable to other orbiters as long as high-resolution multi-coverage images are available.

Acknowledgments

We are grateful to the LRO mission teams and all those who worked on the Planetary Data System archive for their tireless work in making the LROC images, SPICE kernels, LDEM and SLDEM2015 publicly available.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The IDs of the LROC NAC images used in this research are included in a spreadsheet file and can be found at https://zenodo.org/records/10450358. The corresponding images can be downloaded from the Planetary Data System (PDS) Geosciences Node Lunar Orbital Data Explorer website (https://ode.rsl.wustl.edu/moon/index.aspx) according to their IDs.

Additional information

Funding

This work was supported by the National Key Research and Development Program of China [Grant No.2022YFF0503100].

Notes on contributors

Bin Xie

Bin Xie is a PhD student at Aerospace Information Research Institute (AIR), Chinese Academy of Sciences. His research interests include planetary image feature matching and high-precision mapping.

Bin Liu

Bin Liu is an associate professor at AIR. His research interests include geometric modeling orbital imagery and high-precision topographic mapping.

Kaichang Di

Kaichang Di is a professor at AIR. His research interests include planetary photogrammetry and remote sensing, visual localization and navigation, and planetary science.

Yifan Zhang

Yifan Zhang is a Master’s student at AIR. His research interest is Planetary remote sensing and photogrammetry.

Biao Wang

Biao Wang is a PhD student at AIR. His research interests include planetary geology and planetary mineral retrieval using hyperspectral data.

Chenxu Zhao

Chenxu Zhao is a PhD student at AIR. His research interests include planetary remote sensing and planetary science.