The Gravity Recovery and Interior Laboratory (GRAIL) mission was designed to map the structure of the lunar interior from crust to core and to advance the understanding of the Moon's thermal evolution by producing a high-quality, high-resolution map of the gravitational field of the Moon. The mission consisted of two spacecraft, which were launched in September 2011 on a Discovery-class NASA mission. Ka-band tracking between the two satellites was the single science instrument, augmented by tracking from Earth using the Deep Space Network (DSN)

Chinn, Douglas S.

Goossens, Sander

Lemoine, Frank G.

Loomis, Bryant D.

Mazarico, Erwan

Neumann, Gregory A.

Nicholas, Joseph B.

Rowlands, David D.

Sabaka, Terence J.

Smith, David E.

Zuber, Maria T.

NASA Technical Reports Server

GLOBAL AND LOCAL GRAVITY FIELD MODELS OF THE MOON USING GRAIL PRIMARY AND 
EXTENDED MISSION DATA.  Sander Goossens1,2, Frank G. Lemoine2, Terence J. Sabaka2, Joseph B. Nicho-
las2,3, Erwan Mazarico2, David D. Rowlands2, Bryant D. Loomis2,5, Douglas S. Chinn2,5, Gregory A. Neumann2, 
David E. Smith2,4, Maria T. Zuber4. 1Center for Research and Exploration in Space Science and Technology, Uni-
versity of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore MD 21250 U.S.A. (email: sand-
er.j.goossens@nasa.gov), 2NASA GSFC, Code 698, 8800 Greenbelt Road, Greenbelt MD 20771 U.S.A.); 3Emergent 
Space Technologies, 6411 Ivy Lane, Greenbelt, MD 20770, U.S.A; 4Massachusetts Institute of Technology, MIT 54-
518, 77 Massachusetts Avenue, Cambridge MA 02139 USA;  5Stinger Ghaffarian Technologies, 7701 Greenbelt 
Road, Greenbelt, MD 20770 U.S.A. 
 
 
Introduction:  The Gravity Recovery and Interior 
Laboratory (GRAIL) mission was designed to map the 
structure of the lunar interior from crust to core and to 
advance the understanding of the Moon's thermal evo-
lution by producing a high-quality, high-resolution 
map of the gravitational field of the Moon [1]. The 
mission consisted of two spacecraft, which were 
launched in September 2011 on a Discovery-class 
NASA mission. Ka-band tracking between the two 
satellites was the single science instrument, augmented 
by tracking from Earth using the Deep Space Network 
(DSN) [2].  
The primary mapping mission for GRAIL com-
menced on March 1, 2012 and continued until May 29, 
2012. During the primary mission, the altitude of the 
spacecraft was on average 55 km above lunar surface. 
GRAIL's extended mission initiated on August 30, 
2012, and was successfully completed on December 
14, 2012. The average altitude during the extended 
mission was 23 km above lunar surface, but the lowest 
altitudes achieved during the extended mission are 
much lower, with altitudes above topography as low as 
2 km. This allows the estimation of global gravity field 
models at finer and finer resolutions, up to and beyond 
degree and order 900 (a block-size of 6 by 6 km) [3,4].  
In addition to the high-resolution global models, 
local models have also been investigated [5]. Due to 
varying spacecraft altitude and ground track spacing, 
the actual resolution of the global models varies geo-
graphically [3]. Information beyond the current resolu-
tion is still present in the data, as indicated by relative-
ly worse fits in the last part of the extended mission 
[4], where the satellites achieved their lowest altitude 
above lunar surface. Local models of the lunar gravita-
tional field at high resolution were thus estimated to 
accommodate this signal. Here, we present the current 
status of GRAIL gravity modeling at NASA/GSFC, for 
both global and local models. 
Methods:  Our processing relies on a dynamical 
approach called precision orbit determination, in which 
the satellite orbits are integrated over a certain time-
span (called an arc), using high-precision force models. 
In addition, measurements are modeled at high preci-
sion as well and they are compared to actual observa-
tions, resulting in data residuals from which model 
parameters are estimated iteratively. Precision orbit 
determination for the GRAIL satellites is done with the 
GEODYN II software [6]. 
For our global models, the data used in our pro-
cessing are 2-way tracking data from the DSN at the S-
band frequency, and the precise KBRR data. We weigh 
the DSN data at 0.12 mm/s (close to its expected noise 
level of 0.1 mm/s). KBRR data for the primary mission 
are weighted at 0.03 micron/s, and those for the ex-
tended mission at 0.05 micron/s (data prior to Novem-
ber 17), or 0.1 micron/s (data after November 17). 
DSN data has a sample time of 10 s, that of primary 
mission KBRR data is 5 s, and that of extended mis-
sion KBRR data is 2 s. Arc lengths are on average 2.5 
days. 
The force models used for integrating the satellite 
orbits include a lunar gravity field model, degree-2 
potential Love numbers, third-body perturbations, and 
solar and indirect (planetary) radiation pressure. We 
also forward-model dissipation in the lunar interior, as 
derived from lunar laser ranging [7,8]. The measure-
ment modeling uses high-precision corrections for 
relativity, station motion, and troposphere and iono-
sphere-induced media delays. The estimated parame-
ters are divided into those affecting all measurements 
(termed common parameters) and those affecting only 
those measurements within an arc (termed arc parame-
ters). Our common parameters are the spherical har-
monic coefficients of the selenopotential, tidal poten-
tial Love numbers k2m and k3m, and the product of the 
lunar mass and the gravitational constant, GM. Our arc 
parameters are the initial position and velocity vectors 
of each satellite, a KBRR measurement bias, and em-
pirical accelerations with time-constraints [9]. 
The local models are determined using a short-arc 
approach [5]. Orbits determined with a background 
global model are further refined using short-arcs with 
only KBRR data over the area of interest. A baseline 
representation of the twelve parameters describing the 
difference vector between the two satellites and the 
state vector of the midpoint of the satellites is used 
1395.pdf46th Lunar and Planetary Science Conference (2015)
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[10], and only three of the twelve parameters to which 
KBRR data are mostly sensitive are estimated (the 
pitch of the GRAIL A-B position baseline, the magni-
tude of the GRAIL A-B velocity baseline, and the 
pitch of the GRAIL A-B velocity baseline). Gravity is 
expressed as gridded gravity anomalies with respect to 
the background model. We apply neighbor smoothing 
[11,12] to the full model to obtain the local adjustment. 
The high degree and order global models as well as 
the high-resolution local models that we develop from 
the GRAIL data require the estimation of a large num-
ber of parameters. We have therefore turned to using 
the supercomputers of the NASA Center for Climate 
Simulation (NCCS) at NASA/GSFC for the inversions. 
 
 
Figure 1 Power and error spectrum of various 
global models. 
Results: We have processed all primary mission 
and extended mission data, resulting currently in a 
global model of degree and order 1080. For this model, 
we used a Kaula rule of 25x10-5/l2 for degrees l larger 
than 600. Fig. 1 shows the power and error spectra of 
this solution, along with those of the GRGM900C 
model [4]. For both models shown in Fig. 1, the error 
curves intersect the power curves. We stress that both 
models are calibrated in such a way that the formal 
residual statistics from the covariance matrix match the 
observed statistics. This was done using a scaling fac-
tor derived from the square-root information filter 
[9,13]. This scaling factor was 1.93 for GRGM900C, 
and it decreased to 1.82 for the new 1080 model.  
We have also used the extended mission data only 
to estimate a local model over the lunar south pole. 
Our model has a resolution of 0.15° by 0.15° (4.5 km 
block size), which is equivalent to that of a degree and 
order 1200 expansion in spherical harmonics. The 
background model is a global degree and order 1080 
model which was also the starting model for our new 
1080 model. Fig. 2 shows localized correlations (using 
a cap radius of 30° and a windowing function with 
Lwin=13 [14]) with LOLA topography [15] for both 
global and local models. The local model improves the 
correlations of the background model, and also shows 
higher correlations with topography for the higher de-
grees when compared with the updated 1080 global 
model.  
 
 
Figure 2 Localized correlations with topography for 
various global and local models. 
The fits for the late extended mission are still rela-
tively worse using the current global 1080 model, and 
the updated local model shows further improvements 
in correlations with topography. We will thus continue 
to analyze the GRAIL data to extend the resolution of 
our models. 
 
References: [1] Zuber  M. T. et al. (2012) Science 
339, doi: 10.1126/ science.1231507. [2] Asmar S.W. et 
al. (2013) Space Sci. Rev., doi: 10.1007/s11214-013-
9962-0. [3] Konopliv A. S. et al. (2014) Geophys. Res. 
Lett. doi: 10.1002/2013GL059066. [4] Lemoine F. G. 
et al. (2014) Geophys. Res. Lett., doi: 
10.1002/2014GL060027. [5] Goossens S. et al., Ge-
ophys. Res. Lett., doi: 10.1002/2014GL060178. [6] 
Pavlis D. E. et al. (2013), GEODYN Operations Man-
ual. [7] Konopliv A. S. et al. (2013) J. Geophys. Res. 
Planets 118, doi: 10.1002/jgre.20097. [8] Williams J. 
G. (2014) J. Geophys. Res. Planets, doi: 
10.1002/2013JE004559. [9] Lemoine F. G. et al. 
(2013) J. Geophys. Res. Planets 118, doi: 10.1002/ 
jgre.20118. [10] Rowlands D.D. (2002) J. Geod., doi: 
10.1007/s00190-002-0255-8. [11] Rowlands D.D. et 
al. (2010) J. Geophys. Res., doi: 
doi:10.1029/2009JB006546. [12] Sabaka T. J. et al., J. 
Geophys. Res., doi:10.1029/2010JB007533. [13] 
Bierman G. J. (2006) Factorization Methods, Dover 
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(2005), Geophys. J. Int., doi:10.1111/j.1365-
246X.2005.02687.x. [15] Smith D. E. et al. (2010) 
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1395.pdf46th Lunar and Planetary Science Conference (2015)


Global and Local Gravity Field Models of the Moon Using GRAIL Primary and Extended Mission Data

Global and Local Gravity Field Models of the Moon Using GRAIL Primary and Extended Mission Data

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