CROSSOVER ANALYSIS OF CHANG E-1 LASER ALTIMETER DATA

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1 ISPRS Workshop on Geospatial Data Infrastructure: fro data acquisition and updating to sarter services CROSSOVER ANALYSIS OF CHANG E- LASER ALTIMETER DATA Wenin Hu, Zongyu Yue, Kaichang Di* Institute of Reote Sensing Applications, Chinese Acadey of Sciences P. O. Box 978, Datun Rd, Chaoyang District, Beiing, P.R.China (huw, yuezy, Coission IV, WG IV/7 KEY WORDS: Planetary, Mapping, Chang E-, Laser Altieter, Crossover, Analysis, Adustent, DEM ABSTRACT: This paper presents a preliinary result of crossover analysis and adustent of Chang E-(CE-) Laser Altieter (LAM) data of the Moon for global and regional apping applications. During the operation of Chang E- fro Noveber 28, 27 to Deceber 4, 28, the laser altieter acquired 4 orbital profiles with about 9.2 illion altietric points. In our experient, we derived ore than.38 illion crossovers fro 395 ground tracks covering the entire lunar surface after eliinating outliers of orbits and altietric points. A ethod of least-squares crossover adustent with a series of basis functions of tie (trigonoetric functions and polynoials) is developed to reconcile the LAM data by iniizing the crossover residuals globally. The noral equations are very large but sparse; therefore they are stored and solved using sparse atrix technique. In a test area ( N~6 N, 5 W~ W), the crossover residuals are reduced fro 62. to 32.8, and the quality of the DEM generated fro the adusted LAM data is iproved accordingly. We will optiize the ethod for the global adustent to generate a high precision consistent global DEM, which can be used as absolute control for lunar apping with orbital iages.. INTRODUCTION Chang E- (CE-) is the first lunar exploration ission of China, and one of its ain tasks is to obtain the 3D iages of the lunar surface. The CE- satellite was successfully launched on October 24, 27 fro Xichang Satellite Launch Center, and was placed into a 2-h circular lunar polar orbit on Noveber 7. The spacecraft operated until March 29, when it ipacted the surface of the Moon at 8:3 UTC (Du, 29). As one of the payloads of CE- satellite, the Laser Altieter (LAM) is used to easure the distance fro the laser receiver to the lunar surface by using the laser beas. With the altietry data, the Digital Elevation Model (DEM) of the lunar surface can be derived. Altietric crossovers are the intersection locations of two distinct ground tracks at separate ties. Crossover differences (radial distances or heights) reflect the uncertainties in orbit deterination (i.e., precision of spacecraft position and orientation) and the precision of laser ranging. Crossover analysis, which ais to adust the crossover differences and force the planetary radii or height obtained at crossover locations to be consistent, is known as a powerful approach to iprove orbit deterination and derive ore precise DEM. Crossover analysis has been widely used in satellite altietry for earth observation, especially for the easureent of sea surface height. However, to the best of our knowledge, there is no crossover analysis and corresponding ground track correction for CE- LAM data so far. In this paper, crossover analysis of CE- LAM data is carried out for global and regional apping applications. We first calculate the crossover and crossover difference on each profile, followed by using a series of basis functions of tie (trigonoetric functions and polynoials) to iniize the crossover residuals globally. Finally, the nuerical results and DEM effects are discussed. 2. CROSSOVER DIFFERENCES Figure shows an altietric crossover and crossover difference at the intersection location of two distinct ground tracks at t and t. At a crossover, the height of the sae location is given twice, generally by the observations of an ascending track and a descending track. The crossover difference is the deviation between the two altieter ranges at the crossover point. The two crossing passes provide independent easureents at the sae location while at different ties. Figure. Altietric crossover and crossover difference *Corresponding author 26

2 ISPRS Workshop on Geospatial Data Infrastructure: fro data acquisition and updating to sarter services Figure 3 shows crossover differences as a function of tie for an earth day (May 7, 28). The figure indicates that the crossover differences can be treated as a quasiperiodic signal, occurring once per revolution. Owing to the rotation of the Moon, crossovers occur at all latitudes. Since CE- probe had a polar circular orbit with a dip angle of 88.2 (Li et al., 2), the track of sub-stellar points is basically parallel with the longitude at low latitudes. Therefore, the crossovers are denser at high latitudes than that of iddle and low latitudes. The crossovers in a local area ( N~6 N, 5 W~ W) are extracted for detailed analysis. Figure 4 displays the 2593 crossover locations in this area, with different colors representing different ranges of crossover differences (blue: <2; red: >3; green: 2~3). 3. DATA The laser altieter of CE- fires one narrow pulse of 64n wavelength light per second to the Moon surface. A ground track consists of ~2-diaeter footprints, spaced about.4k apart along track. The distance easuring scope is about 2±25 k and the ranging accuracy is about 5 in aircraft tests (Ping et al., 29). 6 blue:<2 red:>3 green:2~3 5 During the operation of CE- fro Noveber 28, 27 to Deceber 4, 28, the laser altieter acquired 4 orbital profiles with about 9.2 illion altietric points. In this study, ore than.38 illion crossovers are calculated fro 395 ground tracks covering the entire lunar surface after eliinating outliers of orbits and points. Figure 2 shows the global distribution of CE- crossovers in longitude and latitude. The density of crossovers is the greatest at the two poles Figure 4. Distribution of crossovers in a local area ( N~6 N, 5 W~ W) 4. METHOD 4. The procedure of crossover processing The procedure of crossover processing is as follows: Step : Pre-processing of LAM points, including reading the LAM points fro.pds file and eliinating outliers in the data. Step 2: Calculation of crossovers and crossover differences. Because there are generally no direct observations at crossovers locations, these two altietry ranges along their respective ground tracks are interpolated by fitting a quasiherite spline (Akia, 97) with the three nearest points on each side of the crossover point. Calculate the crossover differences and store the tie inforation on the two crossing passes at the crossover. The calculated differences are also called crossover residuals in the subsequent adustent. Step 3: Adustent of crossover residuals with a ethod to be described in Section 4.2. Step 4: Adustent of the original LAM points based on the result of Step 3. Figure 2. Global distribution of CE- LAM crossovers residual, Least squares adustent of crossover differences -4 We adopt the adustent ethod proposed in Neuann et al. (2) with soe odifications in adustent odel orbit nuber Let h(t ) and h(t ') be the heights at the crossover, the crossover residual is defined as Figure 3. Crossover difference versus tie over one earth day 27

3 ISPRS Workshop on Geospatial Data Infrastructure: fro data acquisition and updating to sarter services dtt (, ') = ht () ht (') () where t and t are tie tags of the two crossing traectories. To iniize the discrepancy of the two height values, we assign each crossover two artificial adustents, f (), t f(') t in the residual vector Δ d. At tie t ', Δ dt () =+ dtt (, ') + ft () ft (') Δ dt (') = dtt (,') ft () + ft (') Δ dt (') has the opposite sign with Δ dt (). The value of the adustent at the i th crossover in the th profile is odelled by a tie-dependent polynoial: (2) where ht () = ht () o + ft () (5) h = the corrected altietric value, ht () o = the original altietric value. 5. RESULTS We test the above crossover adustent ethod in a local area ( N~6 N, 5 W~ W). Without adustent, the root-eansquare (RMS) of the original crossover residual is 62., and it is reduced to 36.8 after adustent with a 2-order polynoial for each profile. Figure 3 shows histogras of the crossover residuals before and after adustent. Table lists the range distribution of crossover residuals before and after adustent. Fro Table, 98.7% crossover residuals are under and ost of the are under 3 after adustent, while only about 5% under 3 before adustent. Apparently, the adustent significantly reduced the crossover residuals. 4 n f () t = p + pt+ + p t = G p (3) n i where t is a noralized tie at the crossover, n Gi = [ t t ] and p [ ] T = p p pn (a) For each profile, there is an unknown p atrix with ( n + ) coefficients, Equation (2) has 2( n + ) unknown coefficients to be solved at each crossover. Matrix G is coposed of the crossing tie t. For all crossovers, this leads to a very sparse syste of equations. Because each intersection between one track and crossing tracks can be used, and the nuber of intersections increases as the nuber of cyclic otion increases, there will be enough observations to solve Equation (2). On the other hand, since the values in atrix G are very siilar, the adustent solution of paraeters P can be singular. We set the initial values of P to be zero and calculate the final P by iteration, using an inverse C pp covariance atrix to constrain the iteration so that to avoid singularity. The solution at the ( k + )th iteration is obtained fro (Tarantola and Valette, 982): T T p = k p + k ( G G + Cpp ) ( G Δ d + Cpp pk ) (4) The residual Δd is calculated fro Equation (2) with pk at the kth iteration. With the coefficients P, the altietric observations for each profile can be adusted with the sae odel. For each altietric point, substitute the tie-value t into Equation (3) to get the adusted value f () t. That eans (b) Figure 3. Histogras of altietric residuals at crossovers (a) before and (b) after adustent Crossover Residuals Before adusten t After adusten t > 5~ 3~5 ~3 ~ 9.99% 2.32% 9.9% 3.24% 2.36%.27% 4.7% 7.98% 32.66% 53.9% Table. The range distribution of crossover residuals before and after adustent With the odel paraeters fro the crossover adustent, we estiate the altietric errors for each valid LAM points. Then the adusted LAM points (399,629 points) are used to produce a new DEM by interpolation. Figure 4 displays the k resolution DEMs before and after adustent. It can be clearly observed that the DEM after the crossover adustent is ore consistent with fewer artifacts, indicating the adustent ethod is effective. 28

4 ISPRS Workshop on Geospatial Data Infrastructure: fro data acquisition and updating to sarter services When the crossover adustent odels are applied to all the LAM data globally, the effect is not as significant as above. Figure 8 shows the original DEM and the DEM after a global crossover adustent using 2-order polynoial odel. With the adustent, the RMS of crossover residuals is slightly reduced fro 7.5 to 9.6, and the iproveent for DEM is very liited. Therefore, further investigation is necessary for global crossover adustent and generation of a high precision consistent global DEM. (a) DEM before adustent (b) DEM after adustent Figure 4. DEMs before and after a 2-order polynoial adustent of crossovers Considering that trigonoetric functions have also been used in crossover adustent, we developed the following four adustent odels for coparison purposes, aong which the third odels is the one used above. Model_: f () t = p + pcoswt+ p2sinwt (6) Model_2: f ( t) = p g( t 2 4 ) and g( t) = 8t + 6t (7) 2 Model_3: f () t = p + pt + p2t (8) 2 3 Model_4: f () t = p + pt+ pt + pt (9) 2 3 odels RMS Mean Median Min Max original Model_ Model_ Model_ Model_ Table 2. Crossover adustent results fro the four odels. Table 2 lists the nuerical adustent results using the four odels. Fro the statistics, all the adustent odels have reduced the crossover residuals. To see the effects of the adustent on DEM ore clearly, the DEMs in a saller area (8 N~44 N, 33 W~9 W) are shown in Figure 7. It can be observed that all of the odels reduced the artifacts to soe extents, but generally 2-order and 3-order polynoials perfor better. (a) Before adustent (b) After adustent Figure 6. The lunar nearside DEMs before and after crossover adustent 6. DISCUSSION AND CONCLUTION There are several factors which contribute to the crossover isfit. The first is the errors in the altietric data itself, which is resulted fro the otion of the Moon. Second, the uncertainties of the position and orientation of the spacecraft also contribute to the inconsistencies of the different profiles, especially because ost of the LAM data were acquired without segental arc observation of the orbit (Li et al., 2). In addition, the space resolution of the LAM data is relatively low, and the uncertainties in calculation of crossover locations and height interpolation also contribute to the isfit. The distance between a LAM point and the adacent one is about.4k along track and 7.8k across track at the equator. That eans there is great uncertainty between the values of the true ground elevation and the interpolated, while the latter is used in the odels of crossover adustent. As the result, the crossover adustent in local area deonstrates the effectiveness of the developed ethod in this paper, while the global adustent ethod need to be further studied in future. ACKNOWLEDGEMENTS (a) Before adustent (b) odel_ (c) odel_2 (d) odel_3 (e) odel_4 Figure 5. The DEMs before and after adustent in a local area (8 N~44 N, 33 W~9 W) Funding of this research by National Natural Science Foundation of China (48722, 422) and the National High Technology Research and Developent Progra of China (29AA2Z3) is acknowledged. The CE- LAM data was provided by the Lunar and Deep Space Exploration Science Applications Center of the National Astronoical Observatories (NAOC). We thank Dr. Gregory A. Neuann of Massachusetts Institute of Technology and NASA Goddard Space Flight Center for providing valuable inforation of laser altieter crossover adustent. 29

5 ISPRS Workshop on Geospatial Data Infrastructure: fro data acquisition and updating to sarter services REFERENCES Akia, H., 97. A new ethod of interpolation and sooth curve fitting based on local procedures. Journal of the Association for Coputing Machinery, 7(4), pp Du, G., 29. China's lunar probe Chang'e- ipacts oon. (accessed Sep. 2). Li, C., et al., 2. Laser altietry data of Chang E- and the global lunar DEM odel. Sci. China Earth Sci (in Chinese), 4(3), pp Neuann, G. A., et al., 2. Crossover analysis of Mars Orbiter Laser Altieter data. Journal of Geophysical Research, 6(E), pp Ping, J. et al., 29. Lunar topographic odel CLTM-s fro Chang E- laser altieter. Science in China Series G: Physics, Mechanics & Astronoy, 52(7), pp.5-4. Tarantola, A., Valette, B., 982. Generalized Nonlinear Inverse Probles Solved Using the Least Squares Criterion. Reviews of Geophysics and Space Physics, 2(2), pp