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SCIENCE DIRISCT.

A DEGREE 180 SPHERICAL HARMONIC MODEL FOR THE LUNAR TOPOGRAPHY

Jinsong Ping, Kosuke Heki, Koji Matsumoto, and Yoshiaka Tamura

Earth Rotation Division, NAO, Houshigaoka-Machi, 2-12, Mizusawa, Iwate, 023-0861, Japan

ABSTRACT

Using multiple step least squares fitting method and 0.25” griddcd altimetry data (httu://nds-ueophvs.wustl. eduhds) from Clementine, a lunar topography model, NLT18OA, is determined complete to spherical hatmonc degree (n) 180 This model is compared with the GLTM2 model (70x70) and shows good agreement for the long-wavelength component (longer than n=70). Gther important parameters, including the offset between the center of figure (COF) and the center of mass (COM) of the Moon, and the topographical flattening of the Moon, have also been estimated. The estimates are similar to those obtained by other researchers. Possibility to obtain higher (than n=180) degree spherical harmonic model f?om the Clementme gridded data is discussed. Based on NLT180A, the lunar Moho topography and crustal thickness are estimated by assuming a traditional single layered crust on the top of Moho. The method used in this article will benefit the research of the future lunar mission, SELENE ~010gical and w- &cplom). 0 2003 COSPAR. Published by Elsevier Ltd. All rights reserved.

INTRODUCTION

The lunar topography and gravity models have been determined in the form of spherical harmonic expansions. High spatial resolution topographical and gravitational measurements from recent lunar missions, e.g. the Clementine (Zuber et al. 1994) and the Lunar Prospector (Konopliv ct al. 2001) provide au opportunity for high degree and order models by combining data from dif&rcnt mission. For example, the topography model GLTM2 obtained by Smith et al. (1997) is complete to degree n=70 and higher; and the LP165P, a gravity model complete to degree n=165 was developed by Konopliv et al. (2001). At the same time, the NASA’s 0.25”~0.25~ gridded altimetry data with a precision of 1OOm and the LOS Doppler data of NASA’s lunar missions are released by the Planetary Data System (PDS). From the gridded data au attempt has been made to expand spherical harmonic coefficients up to 165 for the lunar global topography by Reindler and Arkani-Hamed (2002). With these new data and models, the internal properties of the Moon have been more extensively studied (e.g. Zuber et al. 1994, Arkani-Hamed 1998, Reindler and Arkani-Hamcd 2002). In this article, by using multiple step least squares fitting method and using the Clementine griddeddata, we develop a lunar topography model complete to degree 180 in spherical harmonics and deduced the topographic potential with Bouguer corrections. Based on the estimated topography model, the first order approximations for lunar Moho topography and crustal thickness are determined. Methodologies used here are expected to be applied to research studies using data from the SELENE mission (Nmiki et al. 1999).

TOPOGRAPHY MODEL

Adv. Space Res. Vol. 3 1, No. 11, pp. 2377-2382.2003 8 2003 COSPAR. Published by Elsevier Ltd. All rights reserved Printed in Great Britain 0273-1177/$30.00 + 0.00

2378 J. Ping ef al

The spherical harmonic expansion of lunar global topography H is defined as

where a is the mean radius of the Moon. The symbols 8 and h represent the co-latitude and the east longitude respectively. YC~S,,,,,(B,h) and H”,‘, are the corresponding even (superscript c) and the odd (superscript s) fully normalized spherical harmonic coefficients for degree n and order m.

The Clementine lidar altimeter measurements with a precision of 1OOm covering between +78” and -78” of latitude range (Smith et al. 1997). These measurements have been filtered and interpolated into a 0.25”xO.25” globally (+/-90 “) gridded data set. The problem of data outage over the pole regions can be overcome by adding the lunar topography data sets at pole regions, e.g., the steroimage data (Cook et al. 2000). Before an effective method being developed to combine different data sets, only the Clementine lidar gridded altimetry data set is used at here. A weighted least squares fitting method is developed, with the weight as the area of the grid so as to expand the griddeddata up to very high degree. The lunar fixed coordinate reference frame in which the altimeter data were measured is adopted in this article. To test the effectiveness and stability of the least squares method, like Smith et al. (1997), the coefficients of low degree models from degree n= 1 to 25 have been estimated together with other lunar tigural parameters, e.g. the mean radii (a+Dr), the offset (Dx, Dy, Dz) between the COF and the COM and the topographical flattening (ll’ of the Moon. The result is given in Table 1. It can be seen that the method used at here is very stable. The coefficients are compared with the GLTM2 model available at http://pds-geophys.wustl.edu/pds/clementine/. However, result in this study is slightly different from Smith et al. (1997). In particular, the offset between the COF and COM along Z-axis, Dz, has an opposite the sign, and the total difference is as large as 500m. Same data set has been used by Smith et al. (1997). The reason for the large difference may come from the difference of estimation methods. Based on our result, the lunar topographical flattening llf=1094.311 is larger than the dynamical flattening of 3234.930, which means the actual figure of the Moon is flatter than its dynamical figure.

Table. 1. Stability of low degree coefficient solutions by using least squares fitting method.(meter)

1x1 2x2 4x4 8x8 16x16 25x25 mean

Dr -865.76441 -865.76315 -865.76365 -865.76399 -865.76388 -865.76381 -865.76381

Dx -1772.173 -1772.173 -1772.173 -1772.173 -1772.173 -1772.173 -1772.173

DY -731.168 -731.168 -731.168 -731.168 -731.168 -731.168 -731.168

Dz -243.715 -243.715 -243.715 -243.715 -243.715 -243.715 -243.715

aHC20 I -710.09434 -710.09545 -710.09621 -710.09598 -710.09581 -710.09554

aHcar I -779.25489 -779.25489 -779.25489 -779.25489 -779.25489 -779.25489

UHS? 1 I -0.48637 -0.48637 -0.48637 -0.48637 -0.48637 -0.48637

aHc22 i 81.53227 81.53227 81.53227 81.53227 81.53227 81.53227

aH*x I 396.51622 396.51622 396.51622 396.51622 396.51622 396.51622

?f I 1094.3 11 1094.311 1094.3 11 1094.3 11 1094.3 11 1094.3 11

An attempt has been made to estimate higher degree coefftcients in one fitting step. Due to the limitation of maximum central memory size of the supercomputer used, to solve more coefficients in one step is not feasible, similar to the problem encountered by Konopliv et al. (2001) for the lunar gravity model LP165P determination. Therefore, a method of multiple steps is introduced into the data analysis procedure. The solution is complete to degree 140, then to remove the components up to degree 140 from the gridded data to get residual gridded data. From the residual gridded data the topography is estimated to degree 180. It is

Degree 180 Spherical Harmonic Lunar Topography Model 2379

called NLTlSOA. Various truncated degrees from 130 through 150 have been set and tested. The results from all of the tests are identified. The gridded data and this model is displayed in Figure 1 together with the bias between the gridded data and two models, GLMT2 and NLT180A. The RMS of the biases is 454Sm and 147.3m, respectively. The topography model is improved largely by NLT180A. 147.3m is a littler larger than the precision of 1 OOm of the gridded data, there is more information can be retrieved from the gridded data beyond degree n=l80.

As comparison, the spherical harmonic coeffkients correlation and admittance between the gravity LP165P and the topography NLTl80A and GLTM2 are shown in Fig.2. It can be seen that below 70 degrees, the results from NLT180A and GLTM2 are almost identical, with only tiny differences at some degrees. The negative correlation around degree 140s may due to the truncate effects on estimating LP165P (Konopliv et al. 2001), it may also be true signal which could be confirmed by future mission, e.g. the SELENE.

O.25oxO.25o Gridded Data Gridded data -NLTI 80A

-8000 -4000 0 4000 801

Topography Topography Residual

Fig. 1. The lunar topography models and their biases from gridded data

E 0.6-l .g m 0.4-

z t: 6 0.2-

0.0 -

1.0,

s 9 :“1: -1 s 0.4- .- E 2 0.2

0.0 -

-0.2. '

-0.4i(sl 0 20 40 60 80 100 120 140 160

Degree

-0.4, 1 (b) I , , , , , , , 0 20 40 60 80 100 120 140 160

Degree

Fig. 2. Correlation (a) and admittance (b) between lunar topography a?d gravity models.

TOPOGRAPHICAL POTENTIAL

2380 J. Ping ef al.

Lunar global gravity field U(r,O,h) can also be expanded in spherical harmonics as

where, all of the symbols are the same as previously denoted, and UC,“, are the spherical harmonic coefficients of the lunar gravitational potential. The symbol I is the radial distance to the observation point over the surface of the Moon, G is the gravitational constant, A4 is the mass of the Moon. According to Arkani-Hamed (1999), assuming the topography as an equivalent surface mass distribution with a surface rock density pC(=2900kg/m3), the topographical potential UH(r,O,h) is read as,

UH = _ 47cp,a3H,(n + 1) nm M(2n + l)(n + 3) ’

- LP165 - - - - Gravity from NLT180A

GLGM-2

- a=3.5x10‘2/nz

-600 -400 -200 0 200 400 600

Topographical Potentia@okm) ” ’ *’ 4o 6o “D:tre? 14’ ‘yb,‘*’

Fig. 3. Topographical potential. (a) free air anomaly at attitude of 30 km; (b), power spectra of gravitational potential from different models

Using Eq.(3), the topographical potential is determined based on NLTlSOA. As examples, the global topographical potential or terrain free air anomalies at 30km is given in Figure 3(a). Following the increase of the altitude, the topographical potential reduces rapidly. The power spectra of the gravitational potential models of GLGM-2 (Lemoine et al. 1997), LP165P (Konopliv et al., 2001), and a Kaula’s constraint law (Kaula, 1966) of o=3.5x1044/n2 together with the obtained topographical potential are displayed in Figure 3(b). A good agreement between LP165P and the topographical potential is kept till the degree n=llO. With a gravitational model of the same degrees of NLTl SOA, the lunar crustal characteristics could be studied in some detail at various scales.

MOHO TOPOGRAPHY AND CRUSTAL THICKNESS

Even without a gravitational model of the same resolution as NLTlSOA, the lunar Moho topography and crustal thickness can also be determined by assuming a single layered crust. A flexure isostatic compensation model, where loads are partially supported by elastic stresses within the lithospheric plate overlying a weak, fluid asthenosphere which has been used by Forsyth (1985) is adopted at here. Assuming that the isostatic response of the lithosphere is isotropic, the isostatic response function Q(k) for a thin elastic plate, which is loaded on the top by the emplacement of topography of density p. is given by (Banks et al. 1977)

Q(k) = -W-W exp(-k z,) / 5, (4 with ~=1+Dk4/Apg and D = E*l;/12(1-v*), where, k, z,, D, g, Ap, and r, are wavenumbers, the depth to the density contrast, the flexural rigidity of crust,

Degree 180 Spherical Harmonic Lunar Topography Model 2381

the gravitational acceleration, the density contrast at crust-mantle interface, and the mean thickness of crust, respectively. Young’s modulus E of lxlO”Nm~* and Poisson’s ratio v of 0.25 are used.

For the Moho topography W, assuming the initial load is p&H-W), if the amplitude of the final elevation is topography H, the amplitude of the deflection of the Moho is (Forsyth, 1985)

W(k) = -p,H(k) / Apt. 0)

Although above algorithm was stated that it was based on the two-dimensional Fourier Transforming (FT), it is by means of a one-dimensional algorithm. In the two-dimensional case, with kl and k2 as the wavenumbers for each dimension, using the method of Banks et al. (1977), the isostatic response function Q(kl,k2) can be rewritten as

Q(k,J,) = -2np,Gexp(-4 z,)exp(-4 z,> / 5~5~ (6)

the amplitude of the deflection of the Moho W(b,kz) is

Wkk,) = -p,Wk,,k,) / APT,& the crustal thickness T is read as

T=H-W+q.

-40000 -20000

Moho To;ogrf;;y 40000

(7)

(8)

km 20 40 60 80 100

Crust Thickness

Fig. 4. Lunar Moho based on NLTl80A. (a) Moho topography from two-dimensional algorithm; (b) different Moho topography between the results from one- and two-dimensional algorithm; (c) thickness of the crust based on from the two-dimensional algorithm.

I I I

-1500 -1000 -500 500 1000 1500

Differential Mpho Topography Above method has been developed primarily for a regional or a local area, we use it to estimate a lunar global Moho topography and crustal thickness as the first order approximations. From NLT180A, the lunar global topography H is expanded into 1024 (in longitude) by 512 (in latitude) gridded data. Both of one- and two-dimensional algorithms have been used in data analysis. Parameters T,=64km (Zuber, et al., 1994), g=1.62 m/s, Ap=500kg/m3 (Arkani-Hamed, 1998) are adopted. .The fast FT results of the gridded data set are introduced into Eq.(S) or Eq.(7) for one- or two-dimensional case, respectively. After the corresponding inverse fast ET, the lunar Moho topography is carried out. Result from two-dimensional algorithm is shown in

2382 J. Ping et crl

Figure 4(a). It is slightly smoother than the result of the one-dimensional algorithm. Difference between the two processing methods is shown in Figure 4(b). The difference is less than 5% of the obtained Moho topography. The mean difference is -137m, and its standard deviation is 800m. The maximum difference appears at the South Pole-Aitken basin and in the high plain area of far side. In fact, various values of TI from 40 to SOlan have been tested in calculation. Difference between the results from them is negligible. Comparing with other parameters, error in T, does not significantly affect the results.

Using Eq.(8) and the two-dimensional algorithm, the lunar crustal thickness is also obtained as in Figure 4(c). The thinnest area occurs at the South Pole-Aitken basin of 25km. At other main basins, the minimum thickness is thicker than 35km. The phenomenon of the near zero thickness of the lunar crust does not appear in our result.

DISCUSSION AND SUMMARY

In this study, we develop a lunar topographical model of NLTl80A complete to degree 180 based on the Clementine altimeter gridded data by using a least squares fitting method using multiple steps. From the residual map of NLTl80, we found that there are present higher frequency information, which can be retrieved in the future to estimated higher degree model than n=180. By assuming a single layered isostatic crust with unified density, simple algorithms as first order approximations are used to determine for the topographical potential, for the Moho topography and for the crustal thickness have been suggested as first order approximation. For higher precision analysis beyond the simple approximation at different spatial scales, methods used here need to be improved, and/or other precise and strict methods should be taken into consideration so as to march the research by using the data of SELENE project.

ACKNOWLEDGEMENTS This Research is supported by JSPS Postdoctral Fellowships for Foreign Researchers. We thank Dr. A. S. Konopliv and an anonymous reviewer for their constructive reviews and helpful comments. We acknowledge help from Profs. C.K. Shum and Mike Duke on the final revision of the manuscript.

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Email address of Jinsong Ping: jsping@miz.nao.ac.jp Manuscript received Sept.26 2002; revised Mar. 12 2003, accepted Mar. 15 2003