highdegree gravity models from grail primary …lemoine et al.: high-degree grail gravity models 0...

JOURNAL OF GEOPHYSICAL RESEARCH: PLANETS, VOL. 118, 1676–1698, doi:10.1002/jgre.20118, 2013

High-degree gravity models from GRAIL primary mission dataFrank G. Lemoine,1 Sander Goossens,1,2 Terence J. Sabaka,1 Joseph B. Nicholas,1,3

Erwan Mazarico,1,4 David D. Rowlands,1 Bryant D. Loomis,1,5 Douglas S. Chinn,1,5

Douglas S. Caprette,1,5 Gregory A. Neumann,1 David E. Smith,4 and Maria T. Zuber4

Received 5 June 2013; revised 25 July 2013; accepted 1 August 2013; published 23 August 2013.

[1] We have analyzed Ka-band range rate (KBRR) and Deep Space Network (DSN)data from the Gravity Recovery and Interior Laboratory (GRAIL) primary mission(1 March to 29 May 2012) to derive gravity models of the Moon to degree 420, 540,and 660 in spherical harmonics. For these models, GRGM420A, GRGM540A, andGRGM660PRIM, a Kaula constraint was applied only beyond degree 330.Variance-component estimation (VCE) was used to adjust the a priori weights and obtaina calibrated error covariance. The global root-mean-square error in the gravity anomaliescomputed from the error covariance to 320 � 320 is 0.77 mGal, compared to 29.0 mGalwith the pre-GRAIL model derived with the SELENE mission data, SGM150J, only to140 � 140. The global correlations with the Lunar Orbiter Laser Altimeter-derivedtopography are larger than 0.985 between ` = 120 and 330. The free-air gravityanomalies, especially over the lunar farside, display a dramatic increase in detailcompared to the pre-GRAIL models (SGM150J and LP150Q) and, through degree 320,are free of the orbit-track-related artifacts present in the earlier models. For GRAIL, weobtain an a posteriori fit to the S-band DSN data of 0.13 mm/s. The a posteriori fits to theKBRR data range from 0.08 to 1.5 μm/s for GRGM420A and from 0.03 to 0.06 μm/s forGRGM660PRIM. Using the GRAIL data, we obtain solutions for the degree 2 Lovenumbers, k20=0.024615˙0.0000914, k21=0.023915˙0.0000132, and k22=0.024852˙0.0000167, anda preliminary solution for the k30 Love number of k30=0.00734˙0.0015, where the Lovenumber error sigmas are those obtained with VCE.Citation: Lemoine, F. G., et al. (2013), High-degree gravity models from GRAIL primary mission data, J. Geophys. Res. Planets,118, 1676–1698, doi:10.1002/jgre.20118.

1. Introduction[2] The pair of spacecraft comprising the NASA Dis-

covery Gravity Recovery and Interior Laboratory (GRAIL)mission successfully mapped the gravity field of the Moonfrom a mean altitude of 55 km between 1 March and 29May 2012 [Zuber et al., 2013a]. The GRAIL mission useda modified version of the precision intersatellite rangingsystem used on the Gravity Recovery and Climate Experi-ment (GRACE) mission [Tapley et al., 2004a]. The GRAIL

Additional supporting information may be found in the online versionof this article.

1NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.2CRESST, University of Maryland, Baltimore County, Baltimore,

Maryland, USA.3Emergent Space Technologies, Greenbelt, Maryland, USA.4Department of Earth, Atmospheric and Planetary Sciences,

Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.5Stinger Ghaffarian Technologies Inc., Greenbelt, Maryland, USA.

Corresponding author: F. G. Lemoine, Planetary Geodynamics Labora-tory, Code 698 NASA Goddard Space Flight Center, Greenbelt, MD 20771,USA. ([email protected])

©2013. American Geophysical Union. All Rights Reserved.2169-9097/13/10.1002/jgre.20118

Lunar Gravity Ranging System (LGRS) measures preciselythe range between the two co-orbiting spacecraft [Klipsteinet al., 2013]. The GRAIL mission is the latest and themost comprehensive effort to map the lunar gravity field.Asmar et al. [2013, Table 1] give a detailed summary of thedifferences and similarities between GRACE and GRAIL.

[3] The earliest efforts to determine the lunar gravity fielddate to the 1960s and 1970s and used the S-band Dopplertracking of the Lunar Orbiters 1–5, the Apollo CommandModules, and the Apollo 15 and 16 subsatellites. Muller andSjogren [1968] demonstrated the existence of mascons overthe large lunar maria from an analysis of Lunar Orbiter data.The data from the Apollo Command Module and Apollo16 subsatellite were acquired from very low altitude orbits(12–30 km altitude); however, the tracking coverage pro-vided only localized sampling of the lunar gravity field usingS-band Doppler [Gottlieb et al., 1970; Sjogren et al., 1972,1974; Phillips et al., 1978]. The S-band Doppler of thisera had a precision of a few mm/s, while the GRACE andGRAIL Ka-band range rate (KBRR) data have a precision of0.1 μm/s or better. The Lunar Orbiter and Apollo subsatel-lite data provided low-altitude coverage over the equatorialregions to˙30ı latitude. In the early 1990s, the S-band datato the Apollo-era lunar orbiters were reanalyzed with the

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development of updated spherical harmonic models such asLun60d to 60 � 60 [Konopliv et al., 1993] and GLGM2 to70 � 70 [Lemoine et al., 1997], where GLGM2 incorpo-rated data from the Clementine mission. The Soviet Unionsent a series of spacecraft to the Moon, which occupied avariety of inclinations and periapse altitudes. The trackingdata from the Luna satellites were combined into a 16 � 16model by Sagitov et al. [1986] and later reprocessed by Akimand Golikov [1997]. So far as we are aware, logistical andtechnical efforts appear to have stymied efforts to combinetracking data from the Soviet Luna orbiters and the U.S.historical orbiters into a single combined solution for thelunar gravity field.

[4] The tracking data from Lunar Prospector provideda substantial leap in knowledge due to the low-altitude,near-polar orbit of the spacecraft. Whereas during the pri-mary mission (January to December 1998) the spacecraftorbited the Moon at an average altitude of 100 km, duringthe extended mission (January to July 1999) the spacecraftaltitude was lowered to an average of 30 km. The LunarProspector data, in combination with the previous lunar mis-sion data, were used to develop the LP150Q [Konopliv,2000; Konopliv et al., 2001] and GLGM3 [Mazarico et al.,2010] models to degree 150 in spherical harmonics.

[5] There is evidence that the Lunar Prospector trackingdata contain information to higher degree than can be reli-ably extracted from the tracking data. Han et al. [2011]used the Lunar Prospector data tracking data residuals toderive a solution to degree 200 with localized harmonicsthat increased the resolution of nearside features, while notchanging the a priori farside gravity model. However, due tothe synchronous rotation of the Moon, none of the historicalmissions just mentioned could obtain direct Doppler track-ing over the lunar farside. As a result, the farside gravityfield in those models was poorly determined, as evidenced,for example, by the errors in the gravity anomalies com-puted using the error covariance of the GLGM3 model[cf. see Mazarico et al., 2010, Figure 4].

[6] A variety of mission concepts were proposed overseveral decades to try and obtain direct tracking data over thelunar farside. Floberghagen [2002] reviews some of thesemission proposals which relied on high-low and low-lowtracking concepts. The first mission to collect direct space-craft tracking data over the lunar farside was the Selenolog-ical and Engineering Explorer (SELENE/Kaguya) mission[Kato et al., 2010]. It included an orbiter in a low-altitude,polar, 100 km orbit, and a relay satellite which at times hadvisibility of the lunar orbiter over the farside and Earth-based tracking stations. The four-way Doppler data that wereobtained through the Kaguya relay satellite, RSTAR, pro-vided the first direct sampling of the lunar farside gravityfield [Namiki et al., 2009; Matsumoto et al., 2010; Goossenset al., 2011a; Goossens et al., 2011b; Goossens et al.,2011c]. The Kaguya mission data, historical data from LunarProspector, and the earlier historical data were combined inthe SGM150J, a degree and order 150 solution, which weused as the a priori model to analyze GRAIL data [Goossenset al., 2011a].

[7] The Lunar Reconnaissance Orbiter (LRO) waslaunched into a low-altitude, near-polar orbit of the Moonin June 2009 to provide a detailed high-resolution survey ofthe planet using a variety of instruments [Chin et al., 2007].

The Lunar Orbiter Laser Altimeter (LOLA) was designed tomap the Moon with 5 m footprint and a vertical resolution of10 cm. To satisfy the LRO orbit determination requirementsand aid in the accurate geolocation of the laser altimeter data,Mazarico et al. [2012] developed a tuned gravity model ofthe Moon to degree 150 that combined LRO S-band trackingdata, altimetric data in the form of altimeter crossovers, andthe historical U.S. Lunar spacecraft data into the 150 � 150model, LLGM-1. This model resulted in LRO orbit accura-cies of 20 m, a substantial improvement over what could beachieved with respect to the untuned models (e.g., GLGM3).For GRAIL though, we did not notice any benefit of usingLLGM-1 and thus preferred to use SGM150J as a priori.

[8] Detailed simulations showed that the intersatellitetracking data obtained from GRAIL over the 3 months of theprimary mission of GRAIL would provide a factor of 100 to1000 improvement over gravity models based on the earlierdata [Park et al., 2012]. As we will show in this paper, theresults we obtain substantiate and in some respects exceedthese pre-mission expectations.

[9] Under the auspices of the GRAIL Science Team, twogroups at the NASA Goddard Space Flight Center (NASAGSFC) and at the Jet Propulsion Laboratory (JPL) weretasked to analyze the Level 1B tracking and produce geopo-tential models as Level 2 products [Zuber et al., 2013a]. Theresults of the JPL analyses are discussed by Konopliv et al.[2013], including the development of models to 660 � 660in spherical harmonics. The model GL420A model devel-oped by the JPL team was presented in Zuber et al. [2013b]in the first publication of GRAIL results from the primarymission. This paper focuses on the development of the lunargravity models from GRAIL primary mission data by theNASA GSFC team and discusses the development of modelsto 420 � 420, 540 � 540, and 660 � 660 in spherical har-monics. This paper is structured as follows: In section 2, wesummarize the GRAIL mission and describe the Deep SpaceNetwork (DSN) and KBRR data; in section 3, we describethe force and measurement models applied; in section 4,we describe the strategy used to analyze the tracking dataand the inversions strategies that were employed to obtainthe geopotential solutions; in section 5, we present the mainsolution results including a characterization of the modelGRGM660PRIM to 660 � 660 in spherical harmonics; andin section 6, we provide a summary and concluding remarks.

2. Satellite and Data Description2.1. The GRAIL Satellites

[10] The GRAIL mission consisted of two satellites,named “Ebb” and “Flow,” but throughout this paper referredto as GRAIL-A and GRAIL-B. Both satellites are nearlyidentical so as to reduce costs and simplify the testing pro-cess [Zuber et al., 2013a]. The spacecraft are three-axisstabilized and have a rectangular body with two fixed solarpanels. There were only two payloads on the satellites:the Lunar Gravity Ranging System (LGRS) and an edu-cation/public outreach instrument named Moon KnowledgeAcquired by Middle School Students (MoonKAM). Otherinstruments that were also used in the analysis presented inthis paper included the star trackers that recorded the satel-lite altitude in inertial space and the radio science antennaethat were used for tracking the spacecraft.

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Figure 1. (a) Periapse and apoapse heights, referred to aradius of 1738 km, and (b) intersatellite distance for theduration of the primary mission.

[11] The GRAIL satellites were inserted into lunar orbiton 31 December 2011 (GRAIL-A) and 1 January 2012(GRAIL-B). After a series of maneuvers [Zuber et al.,2013a], they attained their science orbits, which were near-circular, polar orbits at an average altitude of 55 km abovethe lunar surface. Both the spacecraft altitude and the dis-tance between the two satellites are important parametersthat influence the spatial resolution of the lunar gravity fieldmodels that can be obtained. The lower the altitude andthe smaller the distance between the satellites, the higherthe resolution of the gravity field models that can be deter-mined [e.g., Douglas et al., 1980; Floberghagen, 2002].Because of instrument constraints, this meant that the space-craft needed to fly in precise formation at low altitude [Zuberet al., 2013a]. In Figure 1, both the periapse and the apoapseheights and the intersatellite distance between GRAIL-A

and B are shown. Figure 2 shows the lowest altitude ofthe GRAIL satellites above the lunar topography. Over thecourse of the primary mission, which lasted from 1 Marchto 29 May 2012, the periapse height varied between 15 and53 km and the apoapse height varied between 56 and 94 km.Figure 2 shows that lowest minimum altitudes were obtainedover the low-latitude regions of the farside, while the min-imum altitude is highest over the poles. In Figure 1b, weshow that the intersatellite distance varied linearly, with abreakpoint on 30 March when a maneuver was performed toreverse the increasing trend in the separation distance. Thisvariation was part of the mission planning, with the shorterdistances aimed at resolving local and regional features, andthe longer distances aimed at resolving global features suchas the potential degree 2 Love numbers and time-varyingdegree 2 coefficients [Asmar et al., 2013]. The maximumdistance obtained was roughly 217 km, while the minimumdistance was 82 km.

[12] Because the primary mission lasted 3 months, threefull mapping cycles (with a period of 27.3 days, the lunarsidereal period) were completed, with six full coverages ofthe lunar surface (in a near-circular orbit, one full coverageoccurs every 2 weeks). At the end of the primary mission,the ranging system was turned off (but Earth-based trackingcontinued), and the satellite orbits were raised to on average90 km above lunar surface, in preparation for an extendedmission.

2.2. Doppler Tracking Data[13] Tracking between the GRAIL satellites and Earth-

based stations was performed using two different modesof tracking: two-way tracking from station to satellite andback again (to the transmit station), and one-way trackingfrom satellite to tracking station. Some three-way tracking(where the transmit and receive stations are different) wasalso performed, but much less than two-way tracking. Wedid not include the three-way Doppler data in our processing.The one-way tracking used a reference frequency in X-band(8.4 GHz), while two-way tracking used S band (2.3 GHz).For lunar orbiters, the precision for X band is estimated tobe roughly 0.03 mm/s (assuming an integration time of 10s), based on experience with the X-band performance ofthe Mars Global Surveyor spacecraft [Asmar et al., 2013].S-band data will be more susceptible to ionosphere and solarplasma effects. Based on our experience with the GRAILS-band data, we estimate the noise to be close to 0.1 mm/s.

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Figure 2. Lowest altitude of the GRAIL satellites above the LOLA-derived topography [Smith et al.,2010]. The map uses a Mollweide projection centered on 270ıE, with the farside on the left and thenearside on the right.

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Table 1. Antenna Offsets, in Coordinates With Respect to theSpacecraft Frame, for the Radio Science Beacons (RSB) Used forOne-Way Tracking and for the Low-Gain Antennae (LGA) Usedfor Two-Way Trackinga

Antenna Name X (m) Y (m) Z (m)

LGA +X 1.192 –0.178 0.0LGA –X –0.082 0.152 –0.810RSB +X 1.192 0.207 0.234RSB –X –0.082 0.318 0.762

aThe offsets are the same for both GRAIL-A and GRAIL-B.

[14] The Earth-based component used the Deep SpaceNetwork (DSN), with tracking stations in Goldstone(California), Madrid (Spain), and Canberra (Australia). Thelocations of these antennae are known with an accuracyof centimeters, determined through various space-geodetictechniques [Folkner, 1997]. The satellite component usesdifferent antennae for one-way and two-way data. One-waydata used the Radio Science Beacons (RSB), whereas two-way data used low-gain antennae. For reference, the antennalocations on the spacecraft are given in Table 1. There aretwo of each antenna on each spacecraft, designated by +Xand –X. Switches from +X to –X and vice versa occurredevery 2 weeks and were dependent on the orbital geometry(i.e., which side of the spacecraft was visible from Earth toestablish the tracking link).

[15] Tracking to the GRAIL satellites using the DSN wasdense though not continuous. On average, about 6 h per dayof two-way data were collected (per satellite), whereas forthe one-way data, about 8 h per day were collected. The two-way data had an integration time of 10 s and the one-waydata of 1 s. In our initial processing, we used both the one-way and two-way data for orbit determination. A few weeksinto the mission, we decided to discard the one-way Doppler.The main reason was a solar flare that occurred on 7 March2012 and disrupted the ultrastable oscillator (USO), as couldbe seen in the estimates for the frequency biases that wereestimated in the orbit determination solutions for the one-way data. The project decided to release updated versionsof the one-way Doppler data at a later date that would bet-ter account for these USO frequency changes, but it meantthat the more conservative approach was to forego inclusionof the one-way data in the orbit determination. A secondaryreason was that the 1 s integration interval made the pro-cessing of these data somewhat laborious, even if we diddecimate the data in the gravity field estimation.

2.3. Intersatellite Ranging[16] Like the highly successful GRACE mission, the

GRAIL mission is based on the observations of the range-rate between two satellites offset from each other in thesame polar orbit. The Ka-band range rate (KBRR) obser-vations are formed between two satellites that are in nearlyidentical circular orbits with nearly identical orbital ener-gies. It follows from the equation of conservation of orbitalenergy that each KBRR observation is approximately pro-portional to the difference in the gravitational potential atthe two satellite locations. The intersatellite range rate dataare powerful measurements for gravity recovery but only ifthe trajectories of the satellites have been well determined.

Furthermore, the intersatellite range rate data are only ableto resolve at most 4 of the 12 components of the initial statevectors of the dual satellite system [Rowlands et al., 2002].While on GRACE, satellite orbit determination is supportedby the Global Positioning System (GPS), on GRAIL theDSN Doppler data in combination with the KBRR data areused to determine the GRAIL satellite trajectories.

[17] The LGRS provided KBRR data from 1 March2012, 16:30:35, to 29 May 2012, 17:07:30, returning morethan 99.99% of the possible data that could be collected.Although the KBRR data were continuous through the orbittrim maneuver on 30 March 2012, the regular angularmomentum desaturation (AMD) maneuvers, and the six pri-mary mission Ka-boresight calibration maneuvers, by designwe excluded the KBRR over those short time spans so as notto deleteriously affect the GRAIL geopotential solutions.

3. Force and Measurement Models[18] The orbit determination of the GRAIL satellites

was performed with the NASA GSFC GEODYN II OrbitDetermination and Geodetic Parameter Estimation package[Pavlis et al., 2013]. The strategy we applied uses a dynami-cal approach which relies on the precision modeling of boththe forces acting on the spacecraft and the measurementsused in the orbit determination process. Orbit determinationrelies on least squares via batch estimation [Montenbruckand Gill, 2000; Tapley et al., 2004b]. GEODYN has previ-ously been used to analyze planetary orbiter tracking data tothe Moon and Mars [Lemoine et al., 1997, 2001; Mazaricoet al., 2009, 2010; Matsumoto et al., 2010; Goossens et al.,2011a]. GEODYN has also been used previously to analyzeKa-band range rate data for GRACE [Rowlands et al., 2010;Sabaka et al., 2010]. We modified GEODYN to adapt theKBRR observation model for lunar orbiting satellites.

[19] In this section we describe the force and measure-ment models pertinent to the analysis of GRAIL data, focus-ing in particular on the nonconservative force modeling thatwas required. We also provide a description of the precisionmodeling of the Doppler and the KBRR observables.

3.1. Lunar Gravity Field[20] The gravity field potential due to the Moon is

modeled as a spherical harmonics series [Kaula, 1966;Heiskanen and Moritz, 1967]:

U =GM

r+

GMr

1X`=2

Xm=0

�Re

r

�`P`m(sin � )�

C`m cos(m') + S`m sin(m')�

, (1)

where G is the gravitational constant, M is the mass of theMoon, P`m are the normalized associated Legendre polyno-mials of degree ` and order m, Re is the reference radius(1738 km), and ', � , and r are the longitude, latitude, andradius at the evaluation point. C`m and S`m are the normal-ized Stokes coefficients, the main parameters of interest ofour work.

[21] Because this is the first time that GEODYN is usedfor such high expansions in spherical harmonics (up todegree 660), we verified the evaluation of the associatedLegendre polynomials up to degree and order 700 in twoways: (1) We compared the computations in GEODYN

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(using 64-bit arithmetic) to the same computations using128-bit arithmetic and (2) we compared the GEODYN com-putations with results generated using SHTOOLS (http://shtools.ipgp.fr). Both tests yielded the same results (towithin machine precision), giving us confidence in the sta-bility of high degree and order expansions computed withGEODYN.

3.2. Solid Tides[22] The solid tides raised on the Moon by the Earth and

the Sun are modeled according to the International EarthRotation Service (IERS) conventions [McCarthy and Petit,2004]. This representation has Love number parameters thatare dependent on the degree and order (`, m) of the forc-ing and response terms. We only consider the nondissipativetidal response, and identical forcing and response degreesand orders. For degree 2, we estimate, with no a prioriconstraint, the following: (2,0) forcing and (2,0) response;(2,1) forcing and (2,1) response; and (2,2) forcing and (2,2)response. The resulting three parameters are called k20, k21,and k22. In earlier solutions using satellite data only, onedegree 2 Love number was estimated [e.g., Konopliv et al.,2001; Goossens and Matsumoto, 2008; Matsumoto et al.,2010; Goossens et al., 2011a]. We also solve for the degree3 Love number, k30, describing the (3,0) response to a (3,0)forcing, also with no a priori constraint.

3.3. Nonconservative Force Modeling[23] The main nongravitational forces that affect the

spacecraft trajectory are the radiation pressure accelerationsdue to the Sun (e.g., direct solar radiation) and the Moon(both reflected albedo short wavelength radiation and ther-mal planetary long wavelength radiation). Accelerations dueto the thermal radiation pressure emitted by the spacecraftalso exist and have been modeled by Fahnestock et al.[2012], but we do not consider these in our present work.3.3.1. Spacecraft Box-Wing Model

[24] We use the box-wing approach to compute the effectof the radiation pressure forces on the spacecraft trajectory[Marshall and Luthcke, 1994]. The GRAIL spacecraft ismodeled as a series of plates, where each plate is describedby a surface area, an orientation in the spacecraft-fixed ref-erence frame, and two reflectivity parameters for diffuseand specular reflectivity. At any given time, the plate ori-entation in inertial space is specified using the spacecraftattitude information, which are available as quaternions sup-plied with the Level 1B tracking data. The contributionsof each plate are summed vectorially to obtain the totalacceleration. Previously, we have developed algorithms totake into account possible self-shadowing of some plates byother parts of the spacecraft [e.g., Mazarico et al., 2009],but we did not find this more complex modeling necessaryfor GRAIL because of the lack of movable solar panels onboth spacecraft and because GRAIL follows a simple atti-tude control strategy with respect to the Sun. We adopted a17-plate macromodel developed by Fahnestock et al. [2012]to model the GRAIL satellites, which includes separatereflectivity coefficients for GRAIL-A and GRAIL-B.3.3.1.1. Solar Radiation Pressure

[25] For the solar or planetary radiation pressure, thetotal acceleration (for solar radiation or planetary radiation

pressure) is given by

aRP =ˆ

cCR

NiXi=1

Ai cos(�i)msc

��23ıi + 2ˇi cos(�i)

�Oni

+(1 – ˇi)Os�

, (2)

where ˆ is the radiation flux at the satellite, c is the speedof light, Ai is the area of the ith (of Ni) surface plate, Oniis the surface normal unit vector, Os is the unit vector alongthe direction from the satellite to the radiation source (eitherthe Sun or a surface element of the planet), ‚i is the anglebetween the plate normal Oni and the radiation source direc-tion Os, ˇi is the specular reflectivity (percent of the totalincident radiation) of the ith plate, ıi is the diffuse reflectivity(percent of incident radiation), CR is the reflectivity coeffi-cient (which is generally estimated per each satellite arc),and Lambert’s cosine law of diffuse reflection is assumed[Milani et al., 1987].

[26] The flux ˆ is typically calculated within GEODYNusing the reference ellipsoid for the shape of the centralbody. While this allows for penumbra (0 < ˆ < 1) periods,it of course does not account for topographic variations ofthe surface. In our previous work on Clementine [Lemoineet al., 1997], Lunar Prospector [Mazarico et al., 2010],SELENE [Goossens et al., 2011a], and the Lunar Recon-naissance Orbiter [Mazarico et al., 2012], this did not causeany noticeable artifacts or spurious accelerations. However,the high precision of the GRAIL KBRR data and the dif-ferential nature of the measurement led to noticeable effectsof the subtle mismodeling of the entry into or exit fromshadow. The effect is on the order of 0.4 μm/s, so that we didnot notice this effect until we had developed interim gravitymodels that produced KBRR residuals fits of the order of afew μm/s or less.

[27] Figure 3a illustrates the signature of the eclipse tim-ing modeling errors observed in the KBRR residuals afterconvergence of an arc in May 2012, although we notedsimilar artifacts over many days in the primary mission. Aband of anomalously high residuals is visible near 60ıNover the lunar nearside. It occurs at or near the transitionpredicted from the simple ellipsoid shape, and its widthmatches the separation distance between the two spacecraft.The residuals grow slowly at first, reach a maximum, andthen decrease to typical background amplitudes. This is con-sistent with the solar radiation modeling being responsiblefor the artifacts, as the mismodeling of the leading spacecraft(for instance, modeled to still be in sunlight when in fact ithad already entered penumbra or umbra) imparts a spuriousrelative acceleration between the pair until the trailing space-craft experiences the same mismodeling, at which point theobserved residual amplitude starts to decrease. It is not untilboth spacecraft are correctly modeled (both in umbra in themodel and in reality for the same example) that the effectdisappears from the KBRR residuals. A chronological plotof the residuals, over each orbit track, is shown in Figure 3b.The high-signal periods do not actually appear in the KBRRdata but are introduced by the simplified (ellipsoid-based)eclipse modeling. For this reason, we improved the model-ing of the direct solar flux ˆ in GEODYN. We processedthe spacecraft telemetry of the voltage output of the solararrays to obtain the true eclipse timings. This information

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was passed on to GEODYN for use in the computation ofthe solar radiation pressure accelerations.

[28] As indicated in Figures 3c and 3d, this led to sig-nificant improvements and the near-complete removal ofartifacts in the KBRR residuals. Because the solar ˇ anglevaried during the primary mission, from 42.5ı (1 March) to90ı (17 April) and back to 49.6ı (29 May), the resulting arti-fact bands do not reappear when the spacecraft fly over thesame area at a later date. We note that during the period ofhigh solar ˇ angle (April 2012), the GRAIL spacecraft werein full Sun and so the detailed modeling of eclipse timingswas not an issue.

[29] Although we relied on the telemetry data for the solu-tions presented here, we also developed a standalone toolto create time series of ˆ for any spacecraft trajectory andsurface topography. For the Moon, we use the LOLA data[Smith et al., 2010] archived in the Planetary Data System(LOLA/LRORS PDS Data Node, http://imbrium.mit.edu).This approach enables more frequent sampling (the teleme-try is delivered every 5 s, which is longer than our integrationtime step) and circumvents the need to detrend and to correctthe telemetered voltage data. The voltage data are affectedby the variability in the solar ˇ angle and in the systemresponse of the solar arrays. We compared the telemetryand forward modeled time series to determine outstanding

differences that could indicate spurious telemetry data orincorrect modeling from topography. In terms of eclipse tim-ing, the differences were typically less than 3 s (compared toup to 40 s with the initial perfect ellipsoidal approximation),which is reasonable given the 5 s sampling frequency of thetelemetry. Just before and after the “full Sun” periods (highˇ angle), the GRAIL spacecraft see the Sun grazing the hori-zon, leading to very long transitions from or to darkness,which are most sensitive to the surface relief. We observethat the resulting high-frequency variations in the percentageof Sun visible from each spacecraft match remarkably wellbetween the telemetry and calculations based on the LOLAdata, giving confidence to our modeling of the eclipses.

[30] Figure 4 summarizes the terminator crossing timesover the primary mission. The irregular and asymmetricalaspects are the result of the topography. The more significantdepartures from the “billiard ball” expectations are foundnear the “full Sun” period in the middle of the primary mis-sion, because of the longer penumbra transitions and theeffects of the Sun grazing the topography.3.3.1.2. Planetary Radiation Pressure Modeling

[31] In addition to the direct solar radiation, light reflectedoff the Moon toward the spacecraft causes an acceleration.Only the sunlit hemisphere contributes of course, but due toblack body radiation, the whole Moon acts as a source for the

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http://imbrium.mit.edu


0 90 180 270 360

70

80

90

100

110

120

130

140

150

Angle from ascending node (deg)

Day

of Y

ear

2012

asc.node longitude

Figure 4. Summary of the solar illumination of GRAIL-A over the primary mission. The horizontal axis indicatesthe time after the spacecraft ascending passage through theequator, and the vertical axis shows the date (as day of year2012) of that crossing. The shade is darker when the space-craft is in darkness. The slanted lines show the longitude ofthe ascending node, which can be used to tell if the start ofeach horizontal line is on the dayside or nightside.

planetary thermal radiation pressure. Albeit mathematicallysimilar to their direct solar counterpart, both the short wave-length and long wavelength terms of the planetary radiationpressure result from an extended source. Following Knockeet al. [1988] and Lemoine [1992, Chapter 9], the visible sur-face of the Moon is discretized in a series of ringed sectorialelements, each treated as a point source. The model for thealbedo of the lunar surface is a spherical harmonic expansionto degree and order 15 from Floberghagen et al. [1999].

[32] The surface emissivity is assumed to depend only onlatitude and local time and is derived from simple tempera-ture model:

T = max(Tmax(cos')1/4, Tmin), (3)

where Tmax = 375ıK, Tmin = 100ıK, and ' is the angle tothe subsolar point.

[33] Figure 5 shows the planetary thermal emission fluxresulting from equation (3) using the Stefan-Boltzmann rela-tion. The peak thermal emission reaches 1121 W/m2. Thisvalue approaches that of the solar constant, the solar radia-tive flux at 1 AU of Fsun � 1372 W/m2, and shows thatfor a low-orbiting lunar satellite passing near the subsolarpoint, this source of radiation as a perturbation on the satel-lite trajectory cannot be ignored when considering precisionorbit computations. For use in GEODYN, we developed adegree and order 9 spherical harmonic fit to represent thelunar planetary thermal emission.

3.4. Ephemerides[34] We use the JPL DE421 planetary ephemerides

[Williams et al., 2008] for the positions of the Sun, plan-ets, and Moon in the Solar System Barycenter referenceframe, as well as for the orientation of the Moon. We use the

principal axis (PA) frame, which is best suited for orbit deter-mination and gravity field estimation. The LOLA topogra-phy, typically mapped in the mean Earth (ME) frame, wasredetermined in the PA frame to conduct the comparisonswith gravity, which are presented in section 5.

3.5. Measurement Modeling3.5.1. KBRR Data

[35] The range rate data used in our GRAIL gravity solu-tions are a Level 1B product of the GRAIL mission andare produced at the Jet Propulsion Laboratory (JPL) by thesame group that has been responsible for the production ofthe GRACE mission Level 1B data. The Level 1B inter-satellite range rate data are a highly derived product that isbased on the dual one-way Ka-band ranging between the twosatellites. The Level 1B range Ka-band range rate (KBRR)data are time tagged at Barycentric Dynamic Time (TDB)epochs, consistent with the reference frame of the solar sys-tem barycenter. The KBRR is derived as instantaneous rangerates (no finite counting interval) between the two centersof mass of the GRAIL satellites. Thus, we do not apply anylight time or tracking point calculations in order to processthe KBRR data in the GRAIL orbit solutions. The processingrequired to derive the GRAIL Level 1B data is even moreinvolved than the processing of GRACE Level 1B data. Thetime tagging is particularly difficult without the advantageof the tie to the GPS satellites that the GRACE missionenjoys. Details of the time tagging are given in Kruizingaet al. [2012] and Kruizinga et al. [2013]. The procedureis dependent on the quality of reconstructed orbits of theGRAIL satellites. As the gravity modeling improved dur-ing the GRAIL mission, trajectory accuracy improved andseveral versions of the KBRR data were released. Our pro-cessing is based on the latest available release (version 2).The hardware of the GRAIL satellites leads to a small ambi-guity in the time tag of the KBRR data. The ambiguity isstable at the millisecond level over periods of several days.As such, we estimate a KBRR timing bias per data arc. Dur-ing the primary mission, the KBRR data are available at 5 sintervals. Examination of KBRR residuals in orbit solutions

Local Solar Time (hrs)

Latit

ude

(deg

)

0 6 12 18

−80

−60

−40

−20

0

20

40

60

80

0

200

400

600

800

1000

1200

Figure 5. Lunar planetary thermal emission (in W/m2)computed using the surface temperature model ofequation (3), displayed in local solar time versus latitude.

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shows that like the GRACE KBRR data, the GRAIL KBRRdata are precise to better than 0.1 μm/s.3.5.2. Doppler Data

[36] Our processing of the two-way DSN Doppler is basedon Moyer [2003]. The Doppler data are time tagged at Coor-dinated Universal Time (UTC) epochs corresponding to thereceive time of the signal at the midpoint of the 10 s count-ing interval. The Doppler can be reconstructed by computingand differencing the two-way ranges received at the end andat the start of the count interval and dividing by the Dopplercount time. The two-way Doppler is observed in the Earthframe, but due to our reliance on the planetary ephemeris(i.e., DE421 [Williams et al., 2008]), the two-way rangesmust be modeled in the solar system barycentric frame. Inorder to model the data in the solar system barycentric frame,the light time solution begins by converting the UTC epochat receive time to a TDB time tag. This is described as anindirect effect of relativity [Moyer, 1981a, 1981b]. The lighttime solution for the satellite position yields time tags atTDB epochs. In this way, the trajectory of the satellite isreconstructed at TDB epochs and so the TDB time tags pro-vided with the Level 1B KBRR data are very convenient.The light time solution continues from the satellite timetag and the TDB epoch of the transmit time at the stationis reconstructed. At this point, the modeled two-way rangeis consistent with the solar system barycentric frame. Inother words, the elapsed time from transmit epoch to receiveepoch associated with the modeled two-way range is the dif-ference between two TDB time tags. The elapsed TDB timetag is converted to an elapsed Terrestrial Dynamical Time(TDT) time tag (consistent with the Earth frame observation)using the transformation described in Moyer [1981a, 1981b]and is considered to be a direct effect of relativity. The rela-tivistic range delay due to gravity also has direct and indirecteffects [Moyer, 2003]. The delay is taken into account wheniterating for time tags in the light time solutions (indirecteffect). The delay is also added to the elapsed time (directeffect).3.5.3. Media Corrections for Radiometric DSN Data

[37] We apply media corrections to the DSN radiometrictracking data. Because the two-way Doppler measurementswe used are in the S band, we consider the ionosphericdelay in addition to the tropospheric media effects. Thetropospheric refraction corrections are computed based onthe Hopfield zenith delay [Hopfield, 1971] with the Niellmapping function [Niell, 1996], using meteorological datacollected at each of the DSN sites. The ionospheric correc-tions, which can be significant at�2.2 GHz (up to �10 m inrange), are computed from total electron content (TEC) col-umn abundance maps reconstructed by JPL and delivered inIONEX format to the NASA Crustal Dynamics Data Infor-mation System [Noll, 2010]. The Global Ionospheric Model(GIM) files typically span 1 day with discrete maps every2 h. Following Schaer and Gurtner [1998], we use the “rota-tion and interpolation” method. For a given time t, we firstfind the two reconstructed GIM maps at times ti and ti+1 forwhich ti < t < ti+1. Each map is rotated from its epoch to tto compensate for Earth’s spin (e.g., from ti and t). A sim-ple linear interpolation is then performed to obtain a GIM att, and we compute the TEC value at the location where theray connecting the ground station to the spacecraft reachesthe reference altitude of the GIM maps (450 km). The range

0.060.080.100.120.140.160.180.200.220.24

DS

N fi

t [m

m/s

]

01 09 17 25 01 09 17 25 01 09 17 25March April May

month

No correctionIon correction

Figure 6. RMS of fit of the Doppler measurements ofGRAIL-A during the primary mission without (red) and with(green) ionospheric delay corrections.

delay due to the charged media is then calculated with thefollowing equation derived from Allnut [1989]:

�Rion =�

40.3f 2

��

TECcos(�z)

(4)

where �Rion is the ionospheric delay (meters), f is theradio frequency considered, TEC is the total electron con-tent derived from the GIM, and �z is the zenith angle to thespacecraft (seen from the ground station). For GRAIL, weprocessed DSN data down to an elevation of 15ı above thehorizon at the tracking stations.

[38] Figure 6 shows we obtain a marked and consis-tent improvement in the root-mean-square (RMS) of fitof the Doppler measurements when using the computedionospheric corrections. The residuals were computed withrespect to an interim degree 420 model, a predecessor of theGRGM420A solution.

4. Gravity Solution Development[39] This section describes the data processing strategies

for the orbit determination of the GRAIL satellites and themethods followed for derivation of the new GRAIL gravitymodels. The precision orbit determination (POD), describedin section 4.1 was performed on a cluster of computerworkstations at NASA GSFC. As described in section 4.2,we employed the Gauss-Newton approach for the solutionsup to degree 420 and a method based on “QR” factoriza-tion [Golub and van Loan, 1989] for the degree 540 and660 solutions. The supercomputers of the NASA Center forClimate Simulation (NCCS) at the NASA Goddard SpaceFlight Center were used to create and aggregate the partialderivatives of the adjusted parameters and to generate theactual geopotential solutions.

4.1. Solution Development Strategy[40] The satellite data are divided into arcs, continuous

spans of time over which the orbit is integrated and theparameters are estimated. In the case of GRAIL, the mod-eling of the KBRR data requires knowledge of the positionof both satellites, so that the two GRAIL spacecraft orbitsare integrated simultaneously. The arc length was initiallychosen to be 1 day in order to limit the build up of gravityerrors. As we included more GRAIL data into our solu-tions, and as we expanded the size of our models (see also

1683


section 5.1), we also extended the arcs, up to a maximumlength of 2.5 days. Arc boundaries were chosen to avoidintegrating through angular momentum desaturation (AMD)maneuvers, since we did not want any residual unmodeledaccelerations from the AMD events to influence the orbitdetermination solutions.

[41] Initially, we used one-way and two-way Doppler dataand KBRR data to determine the GRAIL satellite orbits. Theorbit determination was performed in three successive steps:first, starting with the two-way data Doppler; second, addingin the one-way Doppler data (downweighted heavily withrespect to the two-way data because of the different countintervals); and third, adding in the KBRR data. The con-verged orbits from steps 1 and 2 were used as inputs to theprocessing of steps 2 and 3. The rationale for processing thedata in this fashion was to obtain orbits that were as welldetermined as possible before adding in the KBRR data. Inthe early stages, this made it easier to identify spurious datapoints and prevent outliers of the different data types fromunduly affecting the orbit solutions. Later, as we updated thegravity models and our orbits improved, we first discardedthe one-way data for reasons cited earlier (see section 2.2)and then we processed the two-way Doppler data and theKBRR data simultaneously in a single step.

[42] When using multiple data types in an estimationproblem, the relative weighting of these data types is alwaysa challenge. While in the gravity inversions, we eventuallyestimated scale factors for the different statistical sets (seesection 5.4), we had to choose a priori sigmas for the track-ing data in the orbit determination. Here we were guided bythe actual levels of fit and by considerations of the expectedcontribution of each data type to the gravity estimation.While it is true that the a priori sigmas can effectively beadjusted, the a priori values do influence the determinedorbits, and thus indirectly the values and the fidelity ofthe partial derivatives of the measurements with respect tomodel parameters.

[43] We initially processed the GRAIL data with theSGM150J model [Goossens et al., 2011a]. DSN fits withrespect to this model were typically a few mm/s, whileKBRR fits were several hundreds of μm/s, while the weightsthat were used were 0.3 mm/s for DSN data and 1 μm/sfor KBRR data. We found that if weights closer to this apriori level of fit were used, the actual level of (post)fit ofboth types would increase, and the KBRR data especiallywould show stronger once-per-orbital-revolution variations.Considering the sensitivity of KBRR data with respect toradial orbit error [e.g., Rowlands et al., 2002], we pre-ferred to over-weight both data types when we performedthe orbit determination, in order to get the better postfit per-formance with the resultant gravity solutions. A summaryof weights used for various expansions will be given insection 5.1.

[44] The estimated parameters in our orbit determinationprocess are divided into arc parameters (having direct influ-ence on measurements only within a given orbital arc) andcommon parameters (those directly influencing all measure-ments). The common parameters that we included in ourestimation were the coefficients of the lunar gravity fieldmodel in expressed in spherical harmonics, the lunar grav-itational constant GM, and the tidal Love numbers thatdescribe the (time variable) elastic response of the Moon’s

gravitational field to the solid body tides raised by the Earthand Sun, including k20, k21, k22, and k30.

[45] Arc parameters included the state vector for bothsatellites at the initial epoch of each arc, a measurementbias on the KBRR data per arc, one frequency bias persatellite-station combination per arc when one-way datawere included, a timing bias on the KBRR data per arc,a scaling coefficient for solar radiation pressure CR persatellite per arc, and empirical accelerations. We includedthe empirical accelerations to accommodate force modelerror. The empirical accelerations were estimated in thelocal spacecraft frame, and included a constant, a sine,and a cosine term once-per-orbit revolution (OPR), alongtrack and cross track to the orbit. The acceleration termswere estimated for every revolution of each GRAIL satellitethroughout the arc. This means that per satellite, six empir-ical acceleration parameters were estimated roughly every2 h. Orbit error tends to manifest itself as a OPR signalthat changes slowly in phase and amplitude. We estimate theacceleration sets so frequently to ensure that orbit error doesnot contaminate the KBRR residuals. It is the continuoustracking coverage provided by the KBRR data that allows usestimate these many acceleration sets. However, to ensuresmooth transitions between acceleration periods, we applytime-correlation constraints on the accelerations. If there areN acceleration periods, then there are N � (N – 1)/2 distinctpairings of periods. For each of the six types of accelerationparameters, a constraint equation is written tying togethereach of the distinct parings of like parameters from the sep-arate periods. Each constraint equation is a hard constraintforcing a pair of parameters to be equal. The degree to whichthe constraint is implemented by the solution is reflected inthe residual of the constraint equation and is strongly influ-enced by the weight assigned to the constraint equation.Following Luthcke et al. [2003] and Rowlands et al. [2010],the weight assigned to a constraint equation is

Wij = S � exp�

1 –tijT

�, (5)

where Wij is the weight given to the constraint between twoparameters i and j (of the same satellite and the same direc-tion and the same parameter type), S = 1/� 2

A , �A is the sigmaassigned to control the amplitude of the estimated parame-ters, tij is the difference in time between the two time tagsof the two parameters i and j (each parameter is assigned themid-interval time stamp for the period over which the accel-eration is effective), and T is the correlation time. We chose Tto be one orbital revolution, and �A was set to 0.5�10–8 m/s2,which we chose to make sure that the estimated accelerationswere time coherent.

[46] As we expanded our model size beyond degree andorder 420, we also increased the frequency with which weestimated the accelerations from once per revolution to onceper quarter revolution, in order to remove some remain-ing once-per-revolution signal in the KBRR residuals andto improve the DSN fits. We thus increased the numberof accelerations by a factor of 4. Accordingly, we alsochanged the correlation time T, and the parameter �A wasset to 0.5 � 10–9 m/s2. We also found out that correlationsbetween certain accelerations and the solar radiation pres-sure coefficients CR were too large which prevented thedetermination of both parameters simultaneously. Thus, we

1684


fixed the CR to values determined from previous iterations.This greatly reduced the number of iterations for the indi-vidual orbital arcs. We will briefly discuss the impact ofempirical accelerations on the estimated gravity coefficientsin section 5.8.

[47] With this parametrization in terms of arc length andestimated parameters, and chosen data weights, we pro-cessed the GRAIL data in the following way: First withGEODYN, we estimated the arc parameters from the data(while leaving the common parameters fixed) via batch esti-mation using iterative least squares, as mentioned at the startof this Section. Second, with the converged sets of orbits,we used GEODYN to create files of the partial derivativesof the measurements with respect to all the adjusted param-eters (both arc and common). These partial derivative filesare then input to standalone software on the NCCS super-computers that we wrote to estimate the lunar gravity fieldmodel coefficients and other parameters (see section 4.2).

4.2. Method of Solution[48] The model relating the DSN and KBRR measure-

ments to the force and measurement model parameters hasbeen established in section 3. In this section we outlineprocedures for adjusting these model parameters in optimalways in order to recover the best description of the lunargravity field.4.2.1. Gauss-Newton Method

[49] Let the measurement data vector d be related to themodel vector a(x) such that

d = a(x) + �, (6)

where x is the vector of model parameters and � is the noisevector, which is assumed to be a realization of a zero-mean,Gaussian process of covariance C, denoted as � � N (0, C).We seek a solution Qx that minimizes a cost functional definedas the `2 norm | � |2 of the standardized � vector N� = L–1�,where C = LLT and L is the Cholesky square root ofC. Because a is a nonlinear function of x, we apply theGauss-Newton (GN) method [Seber and Wild, 2003] whichfirst linearizes the observation equations in equation (6)and then solves a progression of linear subproblems untilconvergence. The kth step of the GN method is given by

GN

8<:min�xk

ˇL–1 (�dk – Ak�xk)

ˇ22

xk+1 = xk +�xk

, (7)

where �dk � �d(xk) = d – a(xk) are the residuals of thedata vector d with respect to the model evaluated at xk, Ak �A(xk) is the Jacobian of the model vector evaluated at xk, and�xk are the adjustments to the current parameter vector xk.

[50] Typically, GN is ill conditioned and additional infor-mation is required to stabilize or constrain the solution. Thisis usually accomplished by introducing other quadratic termsinto the functional to be minimized such that a new problem,denoted CGN, is defined as

CGN

8<ˆ:

min�xk

ˇL–1 (�dk – Ak�xk)

ˇ22 +

�ˇF–1 (xk +�xk)

ˇ22

xk+1 = xk +�xk

, (8)

where F is the square root factor of an a priori covariancematrix P–1 = FFT that along with the Lagrange multiplier� [Bertsekas, 1999], specifies the deviation of the solutionfrom a preferred model, which for the case shown is the nullvector 0. We in fact use the Kaula rule [Kaula, 1966] in ourstudy to bound the Stokes coefficients toward zero, and soit has this form. For our Kaula rule, F is a diagonal matrixwith values of O/`2 corresponding to Stokes coefficients ofdegree `. The O values we use may be assigned a priori ormay be estimated as discussed in section 4.2.4.

[51] The solution to the kth step of CGN is found bysolving the normal equations:

f�xk = N –1k bk, (9)

where Nk = ATk C –1Ak + �P is the normal matrix and bk =

ATk C –1�dk – �Pxk is proportional to the gradient of the cost

functional at step k. Explicit solution of the normal equationsmay be numerically unstable because of the possibly largecondition number of Nk [Golub and van Loan, 1989], butthere are alternative solutions to CGN which are not as proneto these difficulties.4.2.2. Square Root Information Filter

[52] One alternative is based upon the “QR” factorization[Golub and van Loan, 1989] of an m � n matrix H, withm > n, such that

H =�

Qk Q?� � R

0

�, (10)

where the columns of the m � n matrix Qk are orthonormaland span the column space of H, the columns of the m�m–nmatrix Q? are the orthonormal complement to Qk, R is anupper triangular n�n matrix, and 0 is an m–n�n zero matrix.

[53] Consider that the `2 norm of a vector is invariant torotation by an orthogonal matrix Q such that | N�|22 = |Q N�|22. Ifwe define H as

H =�

L–1Ak L–1�dkp�F –1 –

p�F –1xk

�, (11)

then its QR factorization allows us to rewrite CGN as

CGN

8<:min�xk |r – R11�xk|22 + e2?

xk+1 = xk +�xk

, (12)

where the corresponding R of the factorization is given by

R =�

R11 r0T e?

�. (13)

It is immediately clear that the minimizing �xk is given byf�xk = R–111r and the weighted residual sum of squares is

e2?

. This is the basis of the “square root information filter”(SRIF [Bierman, 1977]) used in many estimation softwarepackages. The advantage of this method over the normalequations is that H TH is never explicitly formed, but ratherstable orthogonal transformations are performed directly onH whose condition number is the square root of the former.4.2.3. Processing Arc Parameters

[54] It should be noted that often a sizable fraction of theparameters comprising x in equation (6) are “arc” parame-ters; that is, they have direct influence on measurements onlywithin a given orbital arc. For the normal equations, these

1685


are usually handled by first forming the Schur complementmatrix [Demmel, 1997] of the arc-arc portion of Nk, solv-ing for the “common” parameters (those directly influencingall measurements), and then solving for the arc parame-ters through back substitution. This keeps the portions ofNk needed at any step of the inversion in equation (9) to amanageable size.

[55] For the SRIF, the formation of the Schur complementmatrix is carried out by considering the submatrix of H inequation (11) corresponding to the ith arc

Hi =�

Aki Bki �dki

, (14)

where Aki and Bki are the Jacobians of the arc and commonparameters, respectively, and where we have assumed pre-multiplication by L–1 has taken place. The Schur processbasically eliminates the projection of the column space of Aki

from Bki and �dki . Given the QR factorization Aki = QARA,this is accomplished by forming a new matrix:

H 0i =�I – QAQT

A� �

Bki �dki

, (15)

which is then used to assemble the matrix H in equation (11)from all arcs. This new H is solved by the SRIF for the com-mon parameters f�xkc . Back substitution for the ith set of arcparameters is carried out as

f�xki = R –1A QT

A��dki – Bki

f�xkc

�. (16)

4.2.4. Variance-Component Estimation[56] At this point we need to specify the Lagrange multi-

plier � which controls the trade-off between the first (misfit)and second (regularization) terms in the cost functional ofCGN. This can be done in a variety of ways including L-curve analysis [Hansen and O’Leary, 1993] and generalizedcross validation (GCV [Wahba, 1977]). However, we wishto broaden our scope by including unknown scale factors inC that represent variance components on noise covariancesof statistically independent data subsets. This is importantbecause the DSN, KBRR, and even Kaula constraint com-prise very different data types whose interplay may not beobvious a priori. If there are J statistical subsets, then thisimplies that C has a block-diagonal structure with blocksCj = � 2

j LjLTj , j = 1, : : : , J, where � 2

j is the variance com-ponent to be estimated. Clearly, we may treat Kaula as anadditional system where LJ+1 = F and � 2

J+1 = �–1.[57] Kusche [2003] has presented a method to solve the

CGN step for both �xk and the set of variance compo-nents

˚� 2

1 , : : : , � 2J+1, which is based upon equilibrating the

observed weighted residual sum of squares, which shouldhave a �2 distribution if the underlying data noise and Kaulasignal are Gaussian distributed, with its expected value foreach statistical set. If f�xk is the solution to CGN at step kfor a fixed set of variance components

˚� 2

1,OLD, : : : , � 2J+1,OLD

,

then define�2

j = rTj C –1

j rj, (17)

where

rj =

8<:�dkj – Akjf�xk , j � J,

xk + f�xk , j = J + 1, (18)

Table 2. Parameters for Several Landmark Modelsa

Data Weight

KBRR DSN Step SizeMax Degree Data N (μm/s) (mm/s) (s)

270 14 April 73,437 1 0.3 5.0360 8 May 130,317 1 0.3 2.0420 29 May 178,081 0.1 0.12 1.5540 29 May 292,677 0.1 0.12 1.5660 29 May 436,917 0.05b 0.12 1.1

aThe column N denotes the number of spherical harmonics coefficients.bA weight of 0.1 μm/s was used for some low periapse arcs.

and�dkj and Akj are the jth subset vector and matrix of�dkand Ak, respectively. The expected value of �2

j can then beshown to be

Eh�2

j

i=

�2

j,NEW

�2j,OLD

! �Nj – Tr

�Rj�

, (19)

where E [�] and Tr [�] are the expectation and trace oper-ators, respectively, Nj is the number of measurements inthe jth data subset (for Kaula this is the number of Stokescoefficients being constrained), and

Rj =

8<:N –1

k ATkj

C –1j Akj , j � J,

�N –1k P , j = J + 1

. (20)

Equating equation (17) with (19) then leads to the followingestimate for � 2

j,NEW:

�2j,NEW = �2

j,OLD

rT

j C –1j rj

Nj – Tr�Rj! . (21)

[58] For fixed �dk, Ak, Lj, F, and xk, the algorithm ofKusche [2003] solves a fixed-point problem by iteratingon equation (21) until convergence to obtain a final set ofestimates of

˚� 2

1 , : : : , � 2J+1

and subsequently �xk. Because�2

j = E��2

j

for each j at the converged solution, the for-mal residual statistics derived from the converged parametererror-covariance matrix N –1

k match the observed statistics.Thus, N –1

k is a calibrated error-covariance matrix. We expectthat this variance-component estimation (VCE) will alsoguard against overfitting of noise.

5. Solution Presentation and Evaluation[59] In this section we present the results of the pro-

cessing that were described in the previous sections. Wegive an overview of the solutions that were obtained, andwe describe these solutions in more detail, focusing on thepower spectra, the correlations with topography the a pos-teriori fits, and the independent tests with Lunar Prospectortracking data.

5.1. Summary of Primary Mission Models[60] We started the processing of the GRAIL data with

the SELENE + historical data-based model, SGM150J[Goossens et al., 2011a]. Over the course of the primary mis-sion, as more data were collected, we developed a successionof models, each a full iteration of the previous model. Themodel size was gradually expanded from degree 150 withthe a priori SGM150J to degree 660 with GRGM660PRIM.

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10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

RM

S

100 150 200 250 300 350 400

Degree

GRGM420AGRGM420A, unconstrained

0 50 0 50 100 150 200 250 300 350 400

Degree

SGM150JGRGM420AGRGM540AGRGM660PRIM

(a) (b)

10−10

10−9

10−8

400 500 600

(c)

power

error

Figure 7. Power spectra of the primary mission GRAIL gravity solutions, compared with SGM150J[Goossens et al., 2011a].

All the models used only GRAIL data. As we developedthese successive models, the changes included the amountof data processed, the expansion size of the model, the apriori data weights for the DSN and KBRR data, and theintegration step size. We list the important interim models inTable 2, where we show the maximum degree, which dataspan was used (indicating the last day of data included), thenumber of spherical harmonic coefficients that were esti-mated, the data weights, and integration step. We changeddata weights as data fits improved, although in the inversionscale, factors were determined (see section 5.4). Integrationtime steps (which are fixed in GEODYN) were changed asexpansions increased to ensure that a gravity field of certainresolution would be sampled adequately, based on the maxi-mum field size and the mean orbital velocity of 1.7 km/s. Wealso note that the number of spherical harmonics coefficientslisted does not include the arc parameters. Even though theyare not estimated explicitly (see section 4.2.3), they are takeninto account in the solution. For our ` = 420 model andfor higher expansions, this would amount to an extra 54,000parameters.

[61] We used the normal equation approach for our mod-els up to degree and order 420. For the next model size,` = 540, we tested the normal matrix approach against thesquare root information filter (SRIF) method and found neg-ligible differences between them, for a constrained model.Then, for all subsequent models to degree 540 and degree660, we used the SRIF method, on account of its use of stableorthogonal transformations, as mentioned in section 4.2.2.

[62] Our final primary mission models are expansionsof maximum degree and order 420, 540 and 660. Thesemodels contain all of the primary mission data of GRAIL,as indicated by “29 May” in Table 2. The motivation forexpanding the solutions to degree 540 and degree 660 isshown first by the history of periapse heights shown inFigures 1 and 2, where we showed the periapse altitudereached as low as 15 km. Second, a degree 420 model stillhad higher RMS of fit for the arcs at the beginning and atthe end of the mission as we discuss in section 5.5. Whileit is true the degree 540 and 660 models probably do nothave global support, these higher expansions are necessaryto exhaust the signal in the tracking data for the primary

mission. In addition, the solutions to degree 540 and 660mitigate aliasing of high degree signals evident in the degree420 solution. These models are denoted as GRGM420A,GRGM540A, and GRGM660PRIM, respectively. The coef-ficients of the model GRGM660PRIM are available fromthe NASA Planetary Data System at http://pds-geosciences.wustl.edu/grail/grail-l-lgrs-5-rdr-v1/grail_1001/ and in thesupporting information.

[63] We applied a Kaula constraint, O/`2, to all of thesemodels. The constraint was only applied for ` > 330. ForGRGM540a and GRGM660PRIM, O = 25� 10–4, while forGRGM420A we applied a slightly looser Kaula constraintwith O = 36 � 10–4. We also tested a degree 420 model withno Kaula constraint for comparison with GRGM420A. Wenote that VCE effectively adjusts the value of O, along withthe a priori data sigmas of the KBRR and DSN data in theprocess of creating a solution with errors that have been cali-brated (see sections 4.2.4 and 5.4). Finally, it should be notedthat GRGM420A, GRGM540A, and GRGM660PRIM aresuccessive models, as in subsequent GN iterations.

5.2. GRAIL Solution Power Spectra[64] We express for the RMS power of the geopotential

coefficients (or the coefficient sigmas), RMS`(U), at degree` according to following relation:

RMS`(U) =

"(2` + 1)–1

Xm=0

C 2`m + S 2

`m

#1/2

. (22)

[65] The RMS power for the primary mission GRAILgravity solutions (GRGM420A, GRGM540A, andGRGM660PRIM) is shown in Figure 7, where we showthe power in the different models, and the RMS power forthe coefficient standard deviations, which are calibrated byVCE as described in section 5.4. In Figure 7a, we show thepower spectra for the GRGM420A solution and a solution tothe same degree without application of the Kaula constraint.The “no-Kaula” solution is stable in power to about degree330. This was the basis for applying the Kaula constraintonly beyond degree 330 in all the primary mission GRAILgravity models. Figure 7 also shows that the calibratedpower curve for the coefficient standard deviations and the

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http://pds-geosciences.wustl.edu/grail/grail-l-lgrs-5-rdr-v1/grail_1001/



10−10

10−9

10−8

10−7

10−6

10−5

10−4

RM

S

0 50 100 150 200 250 300 350 400 450 500 550 600 650

Degree

nearsidefarside

Figure 8. Power spectra of the primary mission GRAILgravity solution, GRGM660PRIM, localized to the nearsideand the farside.

GRGM420A coefficients do not intersect, suggesting thatthere is more signal in the KBRR data than can be accom-modated in a model to only degree 420. In Figures 7b, itsinset, and 7c, we show the degree variances for the GRAILgravity models compared to SGM150J. At the low degrees(` < 150), the GRAIL models show improvements withrespect to SGM150J of up to more than 3 orders of magni-tude at spatial scales that sample major impact basins. TheRMS power curves for the coefficients and the coefficientstandard deviations intersect between degrees 450 and 470.

[66] Using the work of Wieczorek and Simons [2005], welocalized the RMS power spectra for GRGM660PRIM overthe nearside and farside by choosing spherical cap regionswith a cap radius of 90ı centered on either the nearsideor the farside, using a single taper with a concentrationfactor of 99.99% and a spherical harmonic bandwidth of4 (see Figure 8). Matsumoto et al. [2010] already pointedout with the Kaguya data that the farside gravity field to` = 70 has more power than the nearside gravity field

due to the much rougher topography. This dichotomy holdsto higher degrees (to ` = 400) with GRGM660PRIM, aswould be expected from the high correlations of gravity withtopography discussed in section 5.3.

5.3. Correlations With Topography[67] One of the key findings of the initial results of

the GRAIL mission as presented in Zuber et al. [2013b]was the high correlation between gravity and the grav-ity predicted from topography. Here we again evaluatethis correlation for our primary mission models. Ratherthan using LOLA topography coefficients (in the principalaxis system) directly, we use gravity induced by topog-raphy, computed following the scheme of Wieczorek andPhillips [1998] which takes finite amplitude topographyinto account, assuming an overall density of 2560 kg/m3

[Wieczorek et al., 2013], and using the topography expansionup to the 9th power.

[68] Global and localized correlations for various mod-els are shown in Figure 9. The GRAIL models areindicated by their expansion size, and they correspondto the models listed in Table 2 (note that L420 corre-sponds to GRGM420A, L540 to GRGM540A, and L660to GRGM660PRIM). For comparison, correlations forSGM150J are also included. Figure 9a shows the globalcorrelations and shows clearly that each increase in theexpansion (and addition of new data) led to improvedcorrelations with topography being achieved for higherand higher degrees. The difference in correlations betweenthe successive models GRGM420A, GRGM540A, andGRGM660PRIM is much smaller however, although thosefor GRGM660PRIM are slightly above the lower degreemodels. Considering the low altitudes achieved duringparts of the primary mission (cf. Figures 1 and espe-cially 2), we might have expected more improvementsfor GRGM660PRIM when compared to GRGM420A orGRGM540A. However, the low altitudes occur only in cer-tain locations which indicate there might not be globalsupport for a full 660 degree and order expansion. More-over, the inclusion of a Kaula rule for all these models means

Cor

rela

tion

Degree

0.0

0.2

0.4

0.6

0.8

1.0

100 150 200 250 300 350 400 450 500 550 600 650

SGM150J L270L360 L420L540 L660

Degree

0.0

0.2

0.4

0.6

0.8

1.0

0 50 0 50 100 150 200 250 300 350 400 450 500 550 600 650

nearside farsidesouthern nearside

(a) (b)

Figure 9. Correlations between gravity solutions and gravity computed from LOLA topography. (a)Global correlations for various models and (b) correlations for the nearside and farside separately forGRGM660PRIM; correlations for the southern nearside are included as well, for comparison with thoseover the farside.

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Table 3. Weights Determined by VCE for GRGM420Aa

Data Set Dates Sigma

DSN low 1–12 March, 5–18 May 0.107 mm/sDSN high 13 March to 4 May 0.106 mm/sKBRR 1 March to 18 May 0.0503 μm/sDSN May 19–29 May 0.118 mm/sKBRR May 19–29 May 0.181 μm/sKaula 0.628

aKaula was applied only above degree 330, with a rule of 36 � 10–4/`2

(which is then scaled by the weight in the table).

that they will be smoothed, especially at the higher end ofthe spectrum. Therefore, the main justifications for going tomodels larger than degree and order 420 are to improve fit tothe KBRR data, as will be shown in section 5.5, to mitigatethe aliasing of higher degree signals, as well as to prepare amodel for the processing of GRAIL extended mission data.

[69] Following Wieczorek and Simons [2005], we alsolocalized the correlations between gravity and topogra-phy over the nearside and farside. Figure 9b shows theselocalized correlations for GRGM660PRIM. The differencesbetween nearside and farside are profound, with consid-erably higher correlations for the farside. The correlationsfor the farside also reach stable values (over large rangesof degrees) at much lower degrees, because the farside islargely dominated by the highlands, whereas for the near-side the large mascon basins result in negative correlationsat lower degrees. This is further shown by the correlationslocalized over the southern nearside in Figure 9b. Here weapplied localization over a cap with a radius of 30ı, centeredon coordinates (0ıE, 45ıS), thus covering the southern high-lands on the nearside. For this area, the correlations are verymuch like those over the farside.

[70] Both global and local correlations as shown inFigure 9 start to drop steeply for higher degrees. At thedegree where this occurs, the models start to break downbecause of aliasing and leakage of information present inthe data but beyond the modeled resolution. For all mod-els, this occurs in a global sense at around ` = 330, butFigure 9b shows that locally there is some variation. Thebreakdown does not mean that the model degrades beyond acertain degree, as will be shown in section 5.5. It does mean,however, that the model will show more stripes in certainareas, which will impact geophysical analysis, which shouldconsequently be limited to degrees smaller than the full reso-lution of the model. The local variations in the breakpoint onthe other hand mean that geophysical signal is still presentbeyond ` = 330 in certain areas. This breakpoint in thespectrum will be further discussed in section 5.7.

5.4. Calibration of Solutions With VCE[71] As discussed in section 4.2.4, variance-component

estimation (VCE) was used to both determine the relativedata weights and calibrate the covariance of the solutions[Kusche, 2003].

[72] For GRGM420A, six statistical sets were used,divided by data type and altitude. The KBRR data were splitinto two sets, from 1 March to 18 May and from 19 to 29May. The second set corresponded to the period of low-altitude phase when periapse varied between 15 and 33 km(see Figure 1). The DSN data were split similarly, but the 1

March to 18 May set was further divided based on periapseheight. The periapse was highest from approximately 13March to 4 May (see Figure 1), so this span of the DSN datawas treated separately for VCE. The final weight for eachstatistical set as determined by VCE is shown in Table 3.

[73] For GRGM420A, VCE decreases the errors com-puted from the solution error covariance by upweightingthe early mission KBRR data from an a priori sigma of 0.1to 0.0503 μm/s. VCE naturally accommodates the KBRRdata in the second set (19–29 May) with the extremely lowperiapse coverage by increasing the data sigma to 0.181μm/s.

[74] The larger size of the 660 model meant that VCEcould only be performed on two sets: data (including bothKBRR and DSN data) and a Kaula constraint, due to thelimits on the computational resources. The scale factorsdetermined by VCE were 1.734 for the data and 1.210 forKaula, corresponding to weights of 0.0380 μm/s for KBRR,0.0911 mm/s for DSN, and 0.909 for Kaula (scaling a ruleof 25 � 10–4/`2).

[75] VCE calibrates the covariance and therefore alsoadjusts the error spectrum. In the case where there is onlyone statistical set, the calibration reduces to a single scalefactor and the spectrum is simply shifted up or down. Whenthere are multiple systems however, the shape of the spec-trum can change as well. Figure 10 shows the results of VCEon the error spectrum of GRGM420A.

5.5. Data Fit Results[76] Data fit is one figure of merit of how well the models

represent the geophysical signals present in the observations,since the models are estimated by minimizing the residuals(in an `2 norm sense; see section 4.2.1). However, the fit-ting of data is not the objective of our study, but rather thebest determination of geophysical parameters. To avoid fit-ting noise may entail slightly higher levels of fit—and this isthe reason we chose to apply variance-component estimation(VCE). The intrinsic noise level of the data determines theproper level of fit that should be obtained with an adequatemodel. We note that for DSN data at the S-band frequency,it is thought to be 0.1 mm/s, and that for KBRR data in theprimary mission, it is thought to be 0.05 �m/s.

[77] Figures 11a and 11b show the postfit residual statis-tics for the DSN and KBRR data for the final GRAIL gravity

Figure 10. Error spectrum for GRGM420A, comparingthe uncalibrated solution (where all systems were combinedwith a factor of 1) to the calibrated VCE solution.

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0.16

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DS

N fi

t [m

m/s

]

GRGM420AGRGM540AGRGM660PRIM

0.000.050.100.150.200.250.300.350.400.450.50

KB

RR

fit [

µm/s

]

01 08 15 22 29 01 08 15 22 2901 08 15 22March April May

month

(a)

(b)

Figure 11. Postfit residuals in a root-mean-square (RMS) sense per arc for (a) DSN data and (b) KBRRdata using the GRAIL-only gravity models GRGM420A, GRGM540A, and GRGM660PRIM.

models that we discuss in this paper. The orbit fits depend onthe quality and resolution of the gravity models as well as theparameterization chosen in the precision orbit determination.The parametrization for the data fit evaluations for all threemodels shown was the same, and the estimated parametersare described in section 4.1.

[78] For both data types, the fits are close to the expectednoise levels. For DSN data, the fits are close for all threemodels shown, but in general GRGM660PRIM shows a bet-ter fit, especially for the late May arcs. The real improvementin GRGM660PRIM however comes from the improvementsin the fits with respect to the KBRR data. GRGM660PRIMshows consistent improvements over the smaller (and ear-lier) models. For some arcs, especially in April whereGRGM540A and GRGM660PRIM have very similar fits,this may stem from an increased KBRR data weight (cf.Table 2). But for early March and late May, when the GRAILsatellites had lower periapse altitudes as can be seen inFigure 1, it is the larger expansion of GRGM660PRIM thatallows for the improved fits. And as Figure 11 clearly shows,GRGM660PRIM obtains a consistent fit close to 0.05 �m/sfor the entire primary mission.

5.6. Gravity Anomalies and Geoid[79] Figure 12 shows the free-air gravity anomalies from

GRGM660PRIM to degree 320. We also show the anoma-lies computed with SGM150J to degree 140 for comparison.The SGM150J model already brought into focus the signa-tures of a number of farside basins in the free-air anomalies,for example, Apollo (209ıE, 36ıS), Ingenii (163ıE, 34ıS),Mendeleev (141ıE, 6ıN), and Poincaré (162ıE, 57.5ıS).The Lunar Prospector-derived model, LP165P [Konopliv etal., 2001], had already partially resolved farside mascons insix farside basins, including Hertzsprung (231.5ıE, 1.5ıN),

Korolev (203ıE, 4.5ıS), and Freundlich-Sharonov (175ıE,18.5ıN). With GRGM660PRIM to degree 320, we now seethese features more clearly, in addition to seeing the entiresurface of the Moon in more detail; for example, minorcraters in Apollo and Freundlich-Sharonov are resolved; anda gravity high is shown to encircle the central mascon inHertzsprung. Given the high correlations of gravity withtopography (discussed in section 5.3), it is not surprisingto see complex craters such as Vening-Meinesz (87 km indiameter; 162.6ıE, 0.3ıS), and Wegener (88 km in diameter;246.7ıE, 45.2ıN) resolved in the gravity field.

[80] As an example of the detail which can be resolvedwith the GRGM660PRIM, we show in Figure 13 gravityanomalies to degree 660 compared with the lunar topogra-phy in the vicinity of Wegener. This crater is well resolvedin the new field, as are smaller craters such as Lacchini(252.5ıE, 41.7ıN) with a diameter of 58 km, visible tothe lower right of the same figure. Not only are thesecraters resolved but also their structural elements—cratercavity, rim, etc. The resolution of the gravity field is spa-tially variable because of the varying orbital altitude of theGRAIL spacecraft. Extending the field to the highest degreeand order needed to exhaust the signal in areas of low-est altitude coverage introduces minor artifacts in areas ofhigher-altitude coverage. For example, evaluating the field atits full resolution (degree 660) as shown in Figure 12 intro-duces striations in this region, because of noise in the higherdegree coefficients (` > 350). Nonetheless, in comparisonto the anomaly maps of the earlier gravity models such asLP150Q and SGM150J, the lateral striations that are coinci-dent with the orbit tracks of the Apollo subsatellites are nolonger present in the GRAIL fields.

[81] For GRGM660PRIM, the range of anomalies at ` =320 are –538.30 to 336.99 mGal with the minimum at

1690


(a)

(b)

−60˚ −60˚

−30˚ −30˚

0˚ 0˚

30˚ 30˚

60˚ 60˚

−60˚ −60˚

−30˚ −30˚

0˚ 0˚

30˚ 30˚

60˚ 60˚

−500 0 500 1000

mGal

Figure 12. Free-air gravity anomalies computed using a reference radius of 1738 km (a) for SGM150Jto degree 140 and (b) for GRGM660PRIM to degree 320. The units are mGal. The anomalies are shownwith a Mollweide projection, centered on 270ıE, with the lunar farside on the left and the nearside onthe right.

164.13ıE, 9.88ıN and the maximum at 225.63ıE, 19.63ıN.We have evaluated the differences in the gravity anomaliesbetween SGM150J to degree 140 and GRGM660PRIM alsoto degree 140. The RMS differences to ` = 140 between

SGM150J and GRGM660PRIM are 18.49 mGal for thenearside and 59.84 mGal for the farside.

[82] Figure 14 shows the lunar geoid (selenoid) fromGRGM660PRIM to degree 320, which we compute by

Wegener (88 km)

-120˚ -115˚ -110˚40˚

45˚

50˚

-8 -6 -4 -2 0 2 4 6km

GRGM660PRIM

-120˚ -115˚ -110˚-105˚ -105˚

-500 0 500 1000mGal

Figure 13. (right) Gravity anomalies to degree 660, evaluated with GRGM660PRIM for the regionaround the crater Wegener (246.7ıE, 45.2ıN). (left) Comparison with lunar topography derived fromLOLA over the same region [Smith et al., 2010].

1691


−60˚ −60˚

−30˚ −30˚

0˚ 0˚

30˚ 30˚

60˚ 60˚

−600 −500 −400 −300 −200 −100 0 100 200 300 400 500 600

meters

Figure 14. Lunar geoid (selenoid) computed using a reference radius of 1738 km for GRGM660PRIMto degree 320. The units are meters. The geoid is shown with a Mollweide projection, centered on 270ıE.The contour interval is 100 m.

removing the J2 term. The geoid minimum –535.5 m occursat 186.88ıE, 69.13ıS in South Pole Aitken, and the geoidmaximum of 568.3 m occurs at 201.38ıE, 5.13ıN in thehighland regions of the lunar farside. Even though the highfrequency parts of the gravity field are attenuated in thegeoid [cf. Heiskanen and Moritz, 1967], we still see thesignature of the major farside basins as well as the lunarmaria. The differences in the selenoid between SGM150Jand GRGM660PRIM (to degree 140) are 1.74 m RMS overthe nearside and 6.44 m over the lunar farside. The expan-sion from degree 140 to 320 with GRGM660PRIM changesthe geoid globally by 3.06 m RMS.

5.7. Bouguer Gravity[83] Bouguer gravity can be computed by taking into

account the gravitational effects of surface topography. Bysubtracting the topographical contribution from the esti-mated gravity field Bouguer gravity is obtained, whichreflects subsurface structures and is often used in studiesof compensation states [e.g., Watts, 2001] and crustal thick-ness [e.g., Wieczorek et al., 2013]. We computed Bouguer

gravity for our solutions by subtracting the topography-induced gravitational potential, with a density of 2560 kg/m3

and which was computed as explained in section 5.3.[84] Figure 15 shows a map of Bouguer anomalies. The

range of degrees used for this map was ` = 7–320. Thestarting degree of 7 was chosen such that the South PoleAitken basin’s strong positive Bouguer anomaly is filteredout, since it would obscure much of the structure within thebasin. When the Bouguer anomalies are compared to thefree-air anomalies from Figure 12, it is apparent that the pos-itive anomalies in the farside highlands disappear. Obviouslythis follows from the high correlations between gravity andtopography noted for this area. The cutoff degree at the highend of the spectrum was chosen to be 320 to avoid stripes inthe solution, as discussed in section 5.3.

[85] Figure 16 shows the amplitude of the Bouguer grav-ity coefficients (defined here as

qCB

2lm + SB

2lm, with CBlm and

SBlm being the Bouguer gravity coefficients obtained aftersubtraction of the topographic potential), as well as the RMSpower of the Bouguer spectrum (which is defined the sameas the RMS power spectrum for the gravity field itself as

-60˚

-30˚

0˚

30˚

60˚

-500 0 500 1000mGal

Figure 15. Bouguer anomalies on a reference sphere of radius 1738 km for GRGM660PRIM. Thedegree range used for plotting the anomalies is ` = 7–320. The map uses a Mollweide projection centeredon 270ıE.

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0

50

100

150

200

250

300

350

400

450

500

550

600

650

orde

r

0 100 200 300 400 500 600

degree

0 2 4 6 8 10

x10−9

10

10

10

10

10

10

10

10

10

Bou

guer

pow

er

0 50 100 150 200 250 300 350 400 450 500 550 600 650

degree

0.0

0.2

0.4

0.6

0.8

1.0

Cor

rela

tion

BouguerErrorsCorrelations

(a) (b)

Figure 16. Amplitude of the (a) Bouguer gravity coefficients and (b) power spectrum. The latter alsohas the global correlations with topography-predicted gravity.

discussed in equation (22)). For comparison, Figure 16balso includes the error spectrum and global correlations withtopography. The Bouguer amplitudes show large power forthe lower degrees (which are dominated by South Pole-Aitken and the mascon basins) and then decreasing powerfor the higher degrees. Between degree 200 and 400, theyare relatively smooth. Beyond degree 400, effects of sig-nal noise start to appear. At order 350, there is also somestrong anomalous signal that we interpret as a resonance sig-nal in our solutions. As mentioned in Floberghagen [2002],because of the slow rotation of the Moon, resonances areonly to be expected at high degrees. With an orbital period ofabout 113 min, the GRAIL satellites will make roughly 346revolutions while the Moon rotates once. This means thatresonances occur at order m � 346, which is close to wherethe signal starts to appear. By either adding in more data orby weighting the data differently (especially the DSN data),we expect to push out the resonance-induced noise signa-tures to higher degrees. For GRGM660PRIM however, thissignal in the solution determines the cutoff degree beyondwhich the model breaks down in terms of its global geo-physical use. This effect is also illustrated in Figure 16b. TheBouguer RMS spectrum starts to follow the solution errorspectrum around degree 320, which is also the point wherethe global correlations start to break down.

5.8. The Degree 2 Coefficients and the PotentialLove Numbers

[86] As mentioned in section 4.1, we also estimated tidalLove numbers and the lunar gravitational parameter GM.Together with the low-degree gravity field coefficients, theLove numbers can be used to constrain models of the deeplunar interior [e.g., Williams et al., 2001; Williams, 2007;Zhong et al., 2012]. In addition, determining the potentialdegree 2 Love number k2 with a precision of 1% is one of themeasurement requirements of the GRAIL mission [Zuberet al., 2013a]. Table 4 lists the results for degree 2 coeffi-cients and Love numbers, as well as GM and the degree 3Love number k30. We refer to (J. G. Williams et al., Lunarinterior properties from the GRAIL mission, manuscript inpreparation, 2013), hereinafter referred to as Williams et al.,manuscript in preparation, 2013) for a detailed discussion of

these numbers and their implications for models of the lunarinterior structure.

[87] Since our gravity field solutions are developed in thePA frame (see section 3), it is to be expected that C21, S21,and S22 terms would be close to zero because they are relatedto off-diagonal elements of the moment of inertia matrix.Table 4 shows that they are, but S21 is relatively much larger.

[88] A result for the degree 3 Love number k30 is alsoincluded in Table 4. The value of 0.00734 is relatively closeto the expected model value of 0.0095 (Williams et al.,manuscript in preparation, 2013). Our result did not use thecomplete primary mission data set, because adding in thelast arcs with the low periapse data after 25 May resulted inunreasonably high k30 values. Obviously, this indicates sen-sitivity issues with the data, and the result presented here istherefore highly preliminary. Unlike the degree 2 Love num-bers, k30 was not systematically estimated and updated in ourprogressive analysis of the GRAIL data, but rather, it wasonly estimated for GRGM660PRIM. We expect that contin-ued analysis of the primary mission data, combined with theextended mission data, will lead to a better determined valuefor k30.

[89] The uncertainties quoted in Table 4 are those result-ing from the estimation of variance components along with

Table 4. Solutions for Degree 2 Gravity Field Coefficients(Normalized), Potential Degree Love Numbers, and the LunarGravitational Parameter, GM (m3/s2)a

Parameter Value Remark

GM 4.90279981� 1012 ˙ 7740.0 –C20 –9.08829� 10–5 ˙ 1.53� 10–10 no permanent tideC21 8.50� 10–11 ˙ 6.17� 10–12 sameC22 3.46709� 10–5 ˙ 4.87� 10–11 sameS21 9.77� 10–10 ˙ 7.18� 10–12 sameS22 –2.41� 10–10 ˙ 9.27� 10–12 samek20 0.024615˙ 0.0000914 –k21 0.023915˙ 0.0000132 –k22 0.024852˙ 0.0000167 –k30 0.00734˙ 0.0015 data to May 25

aAll values are for GRGM660PRIM and use the whole of the primarymission data of the GRAIL model unless stated otherwise. The quoteduncertainties are the errors with VCE.

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(c)

−60˚ −60˚

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5 10 15 20 25 30 35 40 45 50 55 60

mGal

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

mGal

−60˚ −60˚

−30˚ −30˚

0˚ 0˚

30˚ 30˚

60˚ 60˚

15 20 25 30 35 40

mGal

Figure 17. Projected errors of the gravity anomalies computed from the error covariances of SGM150Jand GRGM660PRIM: (a) SGM150J to 140 � 140, (b) GRGM660PRIM to 320 � 320, and (c)GRGM660PRIM to 660 � 660. The units are mGal. The anomaly errors are shown with a Mollweideprojection, centered on 270ıE, with the lunar farside on the left and the nearside on the right, and areoverlain on a digital shaded relief map of the lunar topography [Smith et al., 2010].

the model parameters. However, Williams et al. (manuscriptin preparation, 2013) compare our solution to that gener-ated by JPL and find discrepancies well beyond the formalerrors for the lower degrees. For this reason, Williams etal. (manuscript in preparation, 2013) apply a factor of 25to the errors quoted here. Such large differences between

the two solutions are likely due to differences in the pro-cessing, such as data weight and parametrization. We alsoexpect that the frequent estimation of empirical accelerationparameters contributes to these differences. The accelera-tions have a period of one orbital revolution, which is alsowhere much of the signal from gravity and other forces is

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]

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Ove

rlap

diff

[m]

01 08 15 22

June 1999

(a)

(b)

Figure 18. (a) RMS of postfit residuals and (b) orbit over-lap differences in the radial direction for various gravityfield models applied to Lunar Prospector extended missiontracking data.

concentrated. While the estimation of these empirical accel-erations results in better determined orbits, these parameterswill also likely absorb signal related to geophysical parame-ters. However, when switching from accelerations estimatedonce-per-orbit to once-per-quarter-revolution (as mentionedin section 4.1), the values for the k2m parameters did notchange significantly. Together with the agreement with JPL’svalues at the 1% level [Konopliv et al., 2013; Williams et al.,manuscript in preparation, 2013], this gives us confidence inLove number results.

5.9. Analysis of the Error Covariance[90] We show in Figure 17 the projected errors in the

gravity anomalies for the a priori model SGM150J and thenew model GRGM660PRIM, calculated from the full errorcovariance matrices of both solutions. In Figure 17a weshow the errors for SGM150J to 140 � 140, in Figure 17bwith GRGM660PRIM to 320 � 320, and in Figure 17c withGRGM660PRIM to 660 � 660. The error maps carry theimprint of the tracking coverage of the satellites that con-tributed to the solutions convolved the intrinsic accuracyof the data. In the case of SGM150J, the low-inclinationband of lower errors on the nearside (5–10 mGal) is associ-ated with the Lunar Orbiter tracking data. As pointed out byMatsumoto et al. [2010], SELENE (Kaguya) contributedmore four-way Doppler data in the southern hemisphere ofthe farside compared to the northern hemisphere, so that theanomaly errors are smaller (around 40 mGal) in the southernhemisphere for SGM150J.

[91] For SGM150J, the global anomaly errors to 140 �140 are 29.00 mGal RMS, the nearside errors are 16.98mGal RMS, and the farside error is 37.35 mGal RMS. ForGRGM660PRIM, the global anomaly errors to 320 � 320are 0.77 mGal RMS, the nearside RMS is 0.67 mGal RMS,and the farside error is 0.85 mGal RMS. In contrast toGRGM660PRIM, the errors for SGM150J were not cali-brated with VCE—so these can be thought of as formalstatistics which are derived from the sigmas applied to theSELENE data in the original analysis of the tracking data[cf. Matsumoto et al., 2010; Goossens et al., 2011a]. For

GRGM660PRIM to 660�660 (Figure 17c), the RMS globalanomaly errors are 30.46 mGal, 29.71 mGal for the near-side, and 31.17 mGal for the farside. In general, the areaswith the largest projected anomaly errors (longitudes 45ıEto 135ıE and in the northern polar regions) correspond to theareas with higher minimum periapse altitudes for the GRAILspacecraft (see Figure 2).

5.10. Results With Lunar Prospector Tracking Data[92] The results related to the data fit as discussed in

section 5.5 were based on the same GRAIL data that wereused in the estimation of the models that were evaluated, andhence they are not independent. Other satellite tracking datanot included in the models can then be used as independentdata to evaluate the models. Here, we used data from LunarProspector (LP). We processed LP data using 2 day arcs andthen propagated the orbit for 12 h. The propagated part ofthe orbit was then compared to the orbit spanning the sametime, as determined from data from the next (adjacent) arc,in order to compute orbit overlap differences. Together withdata fit, orbit overlaps are a standard tool in gravity field andorbit determination evaluation.

[93] We focused especially on LP data in June 1999, sinceLP had its lowest periapse altitudes, between 14 and 25 kmabove lunar surface in that period. Determining how wellour GRAIL-only gravity models perform for these data is anindication of the overall quality of the models, in terms oftheir orbit determination precision. For the processing of LPdata, we compare our GRAIL models with both LP150Q andSGM150J, models that used LP data extensively. We usedonly the Doppler data and weighted these at 1 mm/s. Weestimated the initial state at epoch, a solar radiation pressurescaling coefficient, and measurement biases on the Dopplerdata (on average one per station per arc) because LP wasa spin-stabilized spacecraft and the spin induces a bias onthe data. For more information about LP, see Konopliv et al.[2001].

[94] The results using the independent LP data areshown in Figure 18. Both data fit and orbit overlapresults are shown. The data fit results in Figure 18a aremost telling, with GRGM660PRIM fitting below 1 mm/s,whereas both LP150Q and SGM150J fit higher at sev-eral mm/s. The average fits for GRGM660PRIM, SGM150,and LP150Q are 0.82, 3.5, and 4.52 mm/s, respectively.GRGM660PRIM also fits at a consistent level, whereasLP150Q and SGM150J show more variation with orbitalgeometry. Note that the data noise for LP was reported tobe around 0.2 mm/s [Konopliv et al., 2001] (at 10 s countintervals, which is also what we used). GRGM660PRIM isof course a model with higher resolution than LP150Q orSGM150J, but it should again be stressed that the LP data

Table 5. Average and Median Values for the Total Orbit Over-lap Differences (in a Root-Sum-Square Sense) for LP ExtendedMission Data Using Various Gravity Field Models

GRGM660PRIM

Model LP150Q SGM150J ` = 420 ` = 660

Average 32.89 m 33.98 m 15.21 m 20.62 mMedian 20.77 m 18.64 m 8.16 m 9.64 m

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were not used in the estimation of this model. In fact, ifGRGM660PRIM is to be used at the same resolution asLP50Q and SGM150J (` = 150), it would perform consider-ably worse, because its coefficients are not tuned to the LPdata as they are for the other models.

[95] The orbit overlap results in Figure 18b are for theradial direction. We list average and median values for thetotal overlaps (in a root-sum-square sense over the threeorthogonal directions) in Table 5. The overlap results areless dramatic than those for the data fit, but GRGM660PRIMis still an improvement over other models. Again, the dif-ferences are more consistent and less dependent on orbitalgeometry. Table 5 shows that the total overlap differences forGRGM660PRIM are about half of those of the other models,for both the average and median values. We show the medianto mitigate the effect of outliers, which can bias the overallorbit overlap results especially with SGM150J. Both Table 5and Figure 18b also include orbit differences for the casewhere GRGM660PRIM was used only to degree and order420, and those improve markedly over those using the fullexpansion. GRGM660PRIM shows tuning to the GRAILorbits in the same way as LP150Q and SGM150J did towardLP, especially for the higher degrees and orders, as discussedearlier in this section.

6. Summary and Conclusions[96] Using the KBRR data from GRAIL, we have devel-

oped solutions for the gravitational field of the Moon to420 � 420, 540 � 540, and 660 � 660 in spherical harmon-ics (GRGM420A, GRGM540A, and GRGM660PRIM).The spherical harmonic model, GRGM660PRIM is avail-able from the NASA Planetary Data System (PDS) (http://pds-geosciences.wustl.edu/grail/grail-l-lgrs-5-rdr-v1/grail_1001/) and in the supporting information. The NASA GSFCsolution, GRGM660PRIM is the counterpart to the degree660 GRAIL lunar gravity solution derived by the JPLgravity team [Konopliv et al., 2013]. The solutions used aGauss-Newton approach relying on a Cholesky decomposi-tion, or the square root information filter (SRIF) relying on“QR” factorization. Another innovation is the applicationof variance-component estimation (VCE) [Kusche, 2003] tocalibrate the errors of the resultant solutions.

[97] The solutions that were developed were much higherin degree than was envisaged initially, where solutions toonly degree 180 were promised [Zuber et al., 2013a], andwhere solutions to degree 360 were believed to be possi-ble based on simulations [Park et al., 2012; Asmar et al.,2013]. As a result, we had to rewrite our software and intro-duce new inversion strategies to develop these high degreesolutions. All the solutions were developed on the supercom-puters at the NASA Center for Climate Simulation (NCCS)at the NASA Goddard Space Flight Center—and withoutwhose computational resources the development of thesehigh degree solutions would not have been possible.

[98] The GRAIL spacecraft, unlike GRACE, did not carryan accelerometer. Thus, the precise orbit determination(POD) required the detailed modeling of the nonconser-vative forces, including both solar radiation pressure andplanetary radiation pressure. Since the Moon lacks a sub-stantial atmosphere, atmospheric drag was not an issue.We used a detailed 17-plate macromodel, combined with

detailed modeling of the planetary radiation pressure tomodel the nonconservative forces. Particular attention hadto be paid to the details of solar eclipse modeling wherewe used the actual solar array telemetry (verified by useof LOLA-derived topography) to model the solar radia-tion pressure levels during eclipse transits of the GRAILspacecraft. The GRAIL POD also relied on empirical accel-erations estimated frequently in a time-coherent fashion toremove residual model error.

[99] It is well known that the S-band DSN tracking datawill be sensitive to ionosphere and solar plasma effects. Tocorrect these data for the Earth ionosphere-induced delay, weused GPS ionosphere models (GIMS) [Schaer and Gurtner,1998; Noll, 2010]. We note that while the DSN furnishescalibration polynomials for correction of ionosphere delayon a per-spacecraft basis, the more general approach that wehave implemented has also allowed us to correct the S-bandtracking data to the Lunar Reconnaissance Orbiter (LRO),improving in particular the consistency of the LRO S-bandrange data [Mazarico et al., 2013].

[100] In section 5.8 we presented the GRAIL-only esti-mates for the degree 2 Love numbers, k20, k21, and k22, aswell as preliminary estimate for the degree 3 Love number,k30. Williams et al. (manuscript in preparation, 2013) discussthe interpretation of these new estimates. In this paper wehave not estimated the periodic harmonics of NC21 and NS21 thatrepresent the expected signature of core motion [Williams,2007]. The estimation of these periodic terms will be thesubject of future work.

[101] Efforts are already underway to incorporate datafrom the extended mission of GRAIL into solutions forthe lunar gravity field. During the extended mission, fromSeptember to December 2013, GRAIL orbited the Moon ata mean altitude of 23.5 km, vastly increasing the resolutionof the GRAIL tracking data.

[102] Acknowledgments. We acknowledge the GRAIL Level 1B datateam at the Jet Propulsion Laboratory (JPL) led by Gerhard Kruizinga fortheir production of the KBRR data. The high-degree GRAIL geopotentialsolutions were produced using the supercomputers at the NASA Center forClimate Simulation (NCCS) at the NASA Goddard Space Flight Center,and we acknowledge their support. Despina E. Pavlis (SGT Inc. and NASAGSFC) supported the upgrade of GEODYN to handle the DSN and KBRRtracking data from GRAIL. Our localized spherical harmonic analyses andassociated Legendre polynomial evaluation tests made use of the freelyavailable software archive SHTOOLS (http://shtools.ipgp.fr). The GenericMapping Tools (GMT) package was used to produce some of the figures inthis paper [Wessel and Smith, 1998]. This work was supported by the NASAGRAIL Mission under the auspices of the NASA Discovery Program.

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