JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 98, NO. Ell,
PAGES 20,871-20,889, NOVEMBER
25, 1993
An Improved Gravity Model for Mars'
Goddard
Mars
Model
1
D. E. SMITH, 1 F. J. LERCH,1 R. S. NEREM, 1 M. T. ZUBER,1,2
G. B. PATEL,3 S. K. FRICKE,4 AND F. G. LEMOINE5,6
Doppler tracking data of three orbiting spacecraft have been reanalyzed to develop a new
gravitational field model for the planet Mars, Goddard Mars Model 1 (GMM-1). This model employs
nearly all available data, consistingof approximately 1100 days of S band tracking data collected by
NASA's Deep Space Network from the Mariner 9 and Viking 1 and Viking 2 spacecraft, in seven
different orbits, between 1971 and 1979. GMM-1 is complete to spherical harmonic degree and order
50, which correspondsto a half-wavelength spatial resolution of 200-300 km where the data permit.
GMM-1 represents satellite orbits with considerably better accuracy than previous Mars gravity
models and showsgreater resolutionof identifiablegeologicalstructures.The notable improvement in
GMM-1 over previous models is a consequenceof several factors: improved computational capabilities, the use of optimum weighting and least squarescollocation solution techniques which stabilized
the behavior of the solution at high degree and order, and the use of longer satellite arcs than employed
in previous solutions that were made possible by improved force and measurement models. The
inclusion of X band tracking data from the 379-km altitude, near-polar orbiting Mars Observer
spacecraftshouldprovide a significantimprovement over GMM-1, particularly at high latitudes where
current data poorly resolve the gravitational signature of the planet.
1.
INTRODUCTION
Knowledge of the gravitational field, in combination with
surface topography, provides one of the principal means of
inferring the internal structure of a planetary body. By
removing the gravitational signal of the topography, the
distribution of internal density anomalies associated with
thermal or compositional differences can be addressed.
Gravity can also be used to understand the mechanisms of
compensation of surf'acetopography, providing information
on the mechanical properties and state of stress of the
lithosphere.
The earliest global gravitational field models of Mars were
derived from Doppler tracking data of the Mariner 9 spacecraft [Lotell et al., 1972, 1973; Born, 1974; Jordan and
Lotell, 1975; Reasenberg et al., 1975; Sjogren et al., 1975].
These models provided estimates of low-degree spherical
harmonic gravity coefficientsthat yielded information on the
oblateness
and rotation
vector
orientation
of Mars.
Later
Balmino et al. [1982] resulted in what was then the highest
resolution Martian gravitational model to date: an 18th
degree and order field with half-wavelength resolution of
approximately 600 km. That field, which is characterized by
a spatial resolution comparable to what was then the highest
resolution (16 x 16) topographic model [Bills and Ferrari,
1978], was utilized in analyses of the state of stress of the
Martian lithosphere and the isostatic compensation of surface topography [Sleep and Phillips, 1979, 1985; Banerdt et
al., 1982, 1992; Willemann and Turcotte, 1982; Esposito et
al., 1992]. However, the resolution and quality of the current
gravity and topographic fields (particularly the latter) are
such that the origin and evolution of even the most prominent physiographic features on Mars, namely, the hemispheric dichotomy and Tharsis rise, are not well understood.
The resolution of the Balmino et al. [1982] gravity field
was limited not by data density, but rather by the computational
resources
available
at the time.
Because
this restric-
tion is no longer a limitation, we have reanalyzed the Viking
models that incorporated data from Mariner 9 and the Viking
1 and 2 orbiters [Gapcynski et al., 1977; Christensen and and Mariner data sets and have derived a new gravitational
field, designated the Goddard Mars Model 1 (GMM-1). The
Balmino, 1979; Christensen and Williams, 1979] resolved
higher degree gravity coefficients, showingthe higher power objectives of this study were (1) to develop the best possible
in the Martian gravity field compared to Earth's, and the a priori gravitational field model for orbit determination of
strong correlation of long wavelength gravity with topogra- the Mars Observer spacecraft in support of the radio science
phy. The subsequentinclusionof additional Doppler data by and Mars Observer laser altimeter investigations, (2) to
validate analysis techniques to be implemented in Mars
•Laboratoryfor TerrestrialPhysics,NASA GoddardSpace Observer gravity modeling studies, and (3) to improve scientific interpretations of geophysical and geological data
Flight Center, Greenbelt, Maryland.
2Alsoat Department
of Earth and PlanetarySciences,
Johns collected in previous missions to Mars.
Hopkins University, Baltimore, Maryland.
GMM-1 is complete to spherical harmonic degree and
3Hughes
STX Corporation,
Lanham,Maryland.
order 50. The corresponding half-wavelength resolution,
4RMSTechnologies,
Inc., Landover,Maryland.
5Colorado
Centerfor Astrodynamics
Research,
Universityof which occurs where the coefficients attain 100% error, is
Colorado, Boulder.
200-300 km where the data permit. In contrast to previous
6Now at Laboratoryfor TerrestrialPhysics,NASA Goddard models, GMM-1 was solved to as high a degree and order as
Space Flight Center, Greenbelt, Maryland, and Astronomy Pronecessary to nearly exhaust the attenuated gravitational
gram, University of Maryland, College Park.
signal contained in the tracking data. This was possible
Copyright 1993 by the American Geophysical Union.
mainly due to the use of optimum weighting and least
squares collocation solution techniques [Lerch et al., 1979],
Paper number 93JE01839.
0148-0227/93/93 JE-01839505.00
which stabilized the behavior of the solution at high degree
20,871
20,872
SMITH ET AL.' GODDARDMARS MODEL 1
TRACKING
DATA
•
APRIORI
MODELS:
MEASUREMENT,
SOLIDMARS
FIXED TIDES,
RELATIVITY,
/'
•._•,..
THIRD
BODY
EFFECTS
GM, DRAG, SOLAR RADIATION,
COMPUTE
ORBITS
f
ADJUSTED
IGRAVITY,
STATE
VECTORS
!
SELECTNEW
I
WEIGHTS
'
i
i
G EN ERATE
NORMAL
EQUATIONS
COMBINE
NORMAL
I
EQUATIONSAND
/ _ _ GRAVITY FIELD
COMPUTE
SOLUTION
!
I
_
SOLUTION
GMM-
1
RESIDUAL
ANALYSIS
Fig. 1.
and order
where
correlation
ORBIT ERROR PREDICTIONS
(DUETO GRAVITYFIELDERRORS)
Flow chart of the procedure used to derive GMM-1.
and data sensitivities
become
problematic. As discussedbelow, the extension of the model
to high degree and order significantly reduced errors resulting from spectralleakage comingfrom the omitted portion of
the gravitational field beyond the limits of the recovered
model. GMM-1 has a higher spatial resolution than preliminary versions of this model [Smith et al., 1990a; Zuber et
al., 1991] and, in addition, is fully calibrated to give a
realistic
error estimate
from the solution
1.1.
ing gravitational model may be input back into GEODYN for
residual analyses.
1.2.
Representation of the Gravitational Potential
The gravitational potential at spacecraftaltitude, V M, is
represented in spherical harmonic form as
GM
covariance.
In the following sections we discuss the development of
GMM-1 and make a detailed comparison of the field with the
previous model of Balmino et al. [1982]. We also include an
error analysis and a discussionof the implications of GMM-1
for Martian geophysics and for navigation and precision
orbit determination in support of the upcoming Mars Observer
ERROR COVARIANCE
Mission.
General Approach of Gravitational Field Recovery
Figure 1 shows a flow chart of the procedure used in the
recovery of the gravitational field model. The data were
processed using the GEODYN/SOLVE orbit determination/
estimation programs [Putney, 1977]. These programs have
previously been used in the derivation of a series of gravitational Goddard Earth Models (GEM) [e.g., Lerch et al.,
1979;Marsh et al., 1988, 1990] and have been adaptedfor the
analysis of planetary tracking data [Smith et al., 1990b;
Nerem et al., 1993]. GEODYN provides orbit determination
and geodetic parameter estimation capabilities, and numerically integrates the spacecraftCartesian state and the force
model partial derivatives employing a high-order Cowell
predictor-corrector method. The force modeling includes a
spherical harmonic representation of the gravity field, as
well as a point mass representation for the Sun, Earth,
Moon, and the other planets. Atmospheric drag, solar radiation pressure, measurementand timing biases, and tracking
station coordinates can also be estimated. The least squares
normal equationsformed within GEODYN may be output to
a file for inclusion in error analysesand parameter estimations. The SOLVE program then selectively combines the
normal equationsformed by GEODYN to generate solutions
for the gravity field and other model parameters. The result-
VM =
•
+ - •
r
•
/=2 m=0
film(Sin
c•)(Clm
COS
mA
q- S!m sin mA)
(1)
where r is the radial distance from the center of mass of Mars
to the spacecraft, •b and A are the areocentric latitude and
longitude of the spacecraft, r M is the equatorial radius of
Mars, GM is the product of the universal constant of
gravitation and the mass of Mars, P!m are the normalized
associatedLegendrefunctionsof degreeI and order m, C!m
and S !m are the normalized spherical harmonic coefficients
which were estimated from the tracking observations to
determine the gravitational model, and N is the maximum
degree representing the size (or resolution) of the field. The
gravitationalforce due to Mars which acts on the spacecraft
correspondsto the gradient of the potential, V M.
2.
REFERENCE SYSTEMS FOR DEEP SPACE NETWORK
TRACKING OF MARS SPACECRAFT
Spacecraft orbiting Mars are tracked from Earth through
the NASA deep space network (DSN) tracking stations at
Goldstone (California), Canberra (Australia) and Madrid
(Spain). The adopted planetary ephemeris for Earth, Mars,
and the other planets was the JPL DE-96 system [Standish et
al., 1976].
The inertial coordinate system for the Earth is defined by
the direction
of the Earth' s rotation
axis and the location
of
the vernal equinox. The orientation of the Mars rotation axis
is specified by the right ascension and declination of the
Martian pole, as given in Davies et al. [1986, 1992]. The z
axis of the Mars inertial coordinate system is the instantaneous Mars rotation axis. The direction of the x axis (the
SMITH ET AL ßGODDARDMARS MODEL 1
TABLE
1. Planetary and Astronomical Constants
Parameter
Value
GravitationalconstantGM
42828.28
Equatorial radius of Mars
3394.2
Spinrateof Mars
Solid tide amplitude k2
Flattening of Mars
Unit
km3 s-2
km
350.891983
degd-1
0.05
1/191.1372
Speedof light
2.99792458
x 108
Astronomicalunit
m s-1
1.49597870660x 10TM m
International Astronomical Union (IAU) vector) is defined
to be the intersection of the instantaneousMars equator with
the mean Earth equator of the appropriate ephemerisepoch.
For this analysis, 1950.0 was chosen as the epoch, and thus
the IAU
vector was defined with reference
to the Earth mean
equator and equinox of 1950.0(EME50). The prime meridian
of Mars is defined by Davies et al. [1989]. At the beginning
of this analysis, tests were performed using a reference date
and planetary ephemeris of 2000.0, but results showed that
the 1971 Mariner 9 data and 1976-1978 Viking data were
better
satisfied with a reference
date of 1950.0.
Various reference system constants that were used are
given in Table 1.
3.
DATA SUMMARY AND ORBITAL SENSITIVITY
OF GRAVITY
3.1.
Satellite
Orbital
Characteristics
The Mariner 9 (M9), Viking Orbiter 1 (VOl), and Viking
Orbiter 2 (VO2) spacecraft were in highly eccentric orbits
with periods of approximately 1 day for VOl and VO2 and
one-half day for M9. The satellite orbit characteristicsare
summarized in Table 2. The orbital periods are nearly
commensurate with the rotational period of Mars (24.623
hours) which produce dominant resonant perturbations
[Kaula, 1966] for all orders rn of the Viking spacecraft
(24-hour period) and for the even orders of M9 (12-hour
period). The resonant periods range mostly from about 1 to
50 days for shallow resonant terms and also include deep
(very long) resonant periods. The beat period, or fundamental resonant period, identifies the shift (or "walk") in successive ground tracks and is useful in mapping the orbital
coverage over Mars (a plus sign represents an eastward
"walk" and a negative sign for a westward "walk"). The
beat period changes after each maneuver of the VOl and
VO2 [Snyder, 1979]. For example, an orbit maneuver by
VO2 on March 2, 1977, produced a very slow walk to
synchronize with the Viking Lander (VL2). Subsequently,
on April 18, 1977, another maneuver resulted in a walk
around the planet in 13 revolutions, producing a beat period
of 12 days. Strong resonant perturbations of long period
were produced on VOl, commencingon December 2, 1978,
to provide a slow walk around the planet. Mission events
such as leakages, attitude control jetting, and other phenomena that cause variations in the orbital periods are described
by Snyder [1977, 1979].
The 300-km periapsis altitude orbits of VOl and VO2
provide the strongestcontributionof data to the solutionfor
the higher degree terms, particularly, the VO 1 low orbit with
a range of 180øfor the argument of periapsis (co)as compared
to 25ø for the VO2 low orbit. The observing period for the
20,873
VO 1 low orbit shown in Table 2 covers almost 2 years from
March 12, 1977, to January 27, 1979, and the periapsis point
varies in latitude from + 39ø to -39 ø during this period. The
VO 1 low orbit provides about a 9ø ground track "walk" per
revolution for the 1.5-year period from July 1, 1977 to
December 2, 1978. This correspondsto a "near repeat" of
the ground track for a 39-day period. After the 39-day
near-repeat period, the orbital ground track shifts by 1.8ø
from the previous repeat track, which correspondsto a deep
orbital resonance with a period of about 200 days. This
produces, over a 200-day coverage, a global grid (_+39ø
latitude) with approximately 1.8ø ground track separations
and provides for a high-resolutionrecovery of the gravity
field.
3.2.
S Band Doppler Tracking Data Used in the Solution
3.2.1. Data summary. The data set consisted of 265
orbital arcs, representingover 1100 days of S band Doppler
tracking data from the M9, VOl, and VO2 collected by the
DSN between 1971 and 1978. These data, grouped by
satelliteperiapsisaltitude and inclination, are summarizedin
Table 3. In total, over 230,000 observations were included in
the GMM-1
3.2.2.
solution.
Data
characteristics.
The
data consist
of two-
way S band (2.2 GHz) Doppler measurementscompressed
to 60 s (1-min data points). Data far removed from periapsis
(approximately greater than 12,000-km altitude) were compressed over 10-min intervals.
All observations were collected by three DSN sites located at Goldstone, Madrid,
and Canberra. They were
processed in the differenced-range Doppler formulation,
taking into account relativistic bending due to the Sun
[Moyer, 1971]. Observationsnear satellite periapsisare most
valuable for determining the gravity field, and periapsis is
generally observableby at least one of the DSN sites except
when occulted by Mars. The data distribution and coverage
per satellite orbit are discussedin section 3.4.
The signal is significantly degraded in precision during
solar conjunction due to the solar plasma effects when its
path comes within 5ø of the Sun from Mars. This occurred
for a period of about 1.5 months beginning November 7,
1976, for seven VOl and VO2 arcs. The data were downweighted in the solution for this period.
3.3.
Spectral Sensitivity of Gravity Signal
The spectral sensitivity of the gravity field is analyzed for
the M9, VO 1, and VO2 orbits. Sensitivity for the high-degree
terms (>30) is the main area of interest, and these are
comparedwith a thresholdlevel correspondingto the precision of the DSN signal. The signal, when compressed to
1-mindatapoints,hasa precision
of 1 mms-•, andapproximately0.3 mm s-• for 10-mindatapoints.A sensitivity
study for the above Mars orbiters has been made by Rosborough and Lemoine [1991] and Lemoine [1992] for terms
through degree 20. Analysis for the high-degree terms is
discussedin detail by Lerch et al. [1993]. A brief summaryis
given here.
The orbit perturbations were studied using linear perturbation theory and through numerical integration by GEODYN. The gravity signalfor sensitivity analysis employed a
formof Kaula'srule, 13 x 10-5//2, for termsof degreel,
20,874
SMITHETAL.:GODDARD
MARSMODELI
TABLE 2. SatelliteOrbit Characteristics
IncludingBeat Periodsand GroundTrack Walks
Periapsis
Periapsis
Altitude,
km
Epoch*
IncliOrbit
Nodal Beat
Walkper
nation, Eccen- Period, Argument, Rate Rate Period, Revolution,
deg
tricity hours
deg
deg/d deg/d days
deg
Mariner
1500
Nov. 16, 1971
9
64
0.62
11.81
-24
-0.02
-0.18
18.3
64
0.60
11.98
-24
-0.01
-0.16
19.5
New orbit
Trim maneuver
End of data
Dec. 31, 1971
April 19, 1972
Comments
Viking 1
1500
June21, 1976
Sept.13,1976
Sept.24, 1976
Jan.22, 1977
March12, 1977
March24, 1977
July1, 1977
Dec. 2, 1978
300
38
0.75
39
0.80
24.63
21.87
24.63
22.99
21.92
23.50
23.97
24.85
Jan. 27, 1979
47
56
60
79
99
102
120
264
0.17 -0.13
0.27 -0.21
>1000
8
> 1000
14
8
21
38
129
<1
44
<1
25
43
17
10$
3
10
> 1000
13
15
35
<1
29
25
270
Synchronous
9 rev/walk
Over Lander
Near Phobos
Low periapsis
New walk
Dual station
Near synchronous
End of data
Viking 2
1400
Aug.7, 1976
Aug.28, 1976
55
0.76
27.31
24.63
72
73
0.05 -0.09
1500
800
Oct. 2, 1976
Dec. 21, 1976
75
80
0.80
0.80
26.79
26.48
68
62
750
March5, 1977
24.73
55
215
2
700
April 18, 1977
Sept.26, 1977
22.72
24.29
51
34
12
78
29
5
600
300
Oct. 9, 1977
Oct. 25, 1977
July 25, 1978
24.20
23.98
33
32
2
60
39
6
9
-0.05
-0.08
-0.04
-0.03
-0.10
-0.04
New
orbit
Synchronous
New inclination
Low periapsis/
inclination change
Slow walk
13 rev/walk
Near Deimos
Near Deimos
Low periapsis
End of data
*Start epochof the orbit parameterscited.
?Walk for two 12-hour revolutions for Mariner 9 orbits.
$Secondary
walk is about2øover approximately
a 200-dayduration.
to Table 2, for the VOl orbit from July 1, 1977, through
December2, 1978, the near repeat of the groundtrace (to
paredwith the noiseof the DSN Dopplerdata. Both the within 1.8ø) after 38 revolutionsproducesa perturbationat
analyticaland numericalstudiesconfirmthe importanceof order 38 with a period of about 200 days.
The analyticalvelocityspectrumby degreeis presentedin
the resonanceperturbationsin determiningthe satellite
Table 4 for both M9 and the 300-, 800-, and 1500-km VOl
sensitivityto the Mars gravity field.
The resonanceson the Viking spacecraftfall into three andVO2 orbits.The analyticalvelocity spectrumis obtained
classes: (1) resonancesat low orders (characterizedby by computingthe Kepler element perturbationsusing
periodsof up to 40 days),(2) long-period
resonances
(periods Kaula's [1966] theory, and then mappingthese to velocity
greater than 50 days) at specifichigher orders, and (3) space.Sincewe are interestedin the satellitesensitivityto
intermediateresonancesat the other orders.The long-period the gravityfield over the periodsof the arc lengthsof data
resonancesresultfrom a nearrepeatof the groundtraceafter used in the GMM-1 solution, perturbationswith periods
an integernumberof spacecraftrevolutions.Thus,referring greater than 40 days were excluded.In addition,those
which was obtainedby Balmino et al. [1982]for the power
spectrumof Mars. The velocity perturbationswere com-
TABLE 3. Summaryof DSN TrackingData Used in GMM-I
Arcs
Satellite
Viking 1
Viking I set 2
Viking 2
Mariner
Total
9
Average
Average
Input rms
Altitude,
Inclination,
Arc Length,
Residuals,
km
deg
Total
days
cm/s
1500
300
1400
1500
38.2
39.1
55.4
75.1
80.1
80.2
64.4
29
95
12
11
54
37
32
4.2
4.8
3.8
4.7
3.8
4.2
4.3
0.446
2.844
0.256
0.507
0.210
6.282
0.212
778
300
1500
270
Average
Observations
1082
673
990
952
655
793
1559
Total
Observations
Total
Days
31,393
63,977
11,878
10,467
35,375
29,355
49,878
122
425
46
52
204
155
138
232,323
1142
SMITH ET AL..' GODDARDMARS MODEL I
TABLE 4.
20,875
Theoretical Harmonic Analysis of Velocity Perturbations by Degree
Altitude, km
VOl 300
VO2 300
VO2 800
VOl 1500
M9 1500
Epoch
Jan. 15, 1978
Dec. 17, 1977
Apdl 19, 197
Feb. 5, 1977
Apdl 10, 1992
144.452
39.761
16.519
7.771
4.248
2.498
1.617
1.197
1.118
1.523
2.686
1.195
1.350
0.211
0.048
0.012
0.004
0.000
0.000
0.000
0.000
0.000
0.000
0.000
64.324
36.417
13.981
5.659
2.270
1.051
0.579
0.304
0.158
0.093
0.056
0.031
0.019
0.010
0.002
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
828.128
100.049
55.852
24.812
12.463
6.003
2.669
1.135
0.490
0.260
0.167
0.105
0.061
0.002
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
2
3
4
5
6
7
8
9
10
11
12
13
14
18
22
26
30
34
36
38
40
42
46
50
566.175
353.495
207.188
107.358
66.205
40.783
28.610
18.054
13.176
9.049
6.808
4.711
3.858
1.390
0.581
0.317
0.260
0.370
0.507
0.468
0.373
0.241
0.265
0.322
653.411
218.135
106.629
59.610
34.485
21.119
13.923
9.347
6.677
4.736
3.555
2.649
2.053
0.780
0.381
0.249
0.110
0.047
0.034
0.028
0.029
0.036
0.038
0.027
A powerruleof 13 x 10-5 l 2 is used.DataarefromMarine9 andViking1 and2 sampled
orbits.
Periods >8 days wave their amplitudes multiplied by 8/period. Periods >40 days have been
excluded. Values are in units of centimeters per second.
perturbationswith periods between 8 and 40 days have been
prorated to 8 days by the factor 8/period. The VOl 300-km
orbithasa sensitivity
in excessof 1 mms-• (theaccuracy
of
the S band data) out to degree 50. In contrast, the VO2
300-km orbit is sensitiveonly to terms out to approximately
degree 30. As the periapsisaltitude is raised, the sensitivity
in degree is diminished. The limit is degree 18 for the VO2
800-km orbit, and degree 11 for the VO 1 1500-km orbit. The
Iooo
M9 orbit has stronger perturbations than the VOl 1500-km
orbit by virtue of its closeraveragedistanceto Mars, with its
twice per day revolution and smaller orbital eccentricity.
The sensitivity was also evaluated through numerical
integrationusingthe GEODYN programfor the VO 1 300-km
orbit. The spectral rms velocity perturbation by order is
shown in Figure 2 for different arc lengths. The results show
Gravity Perturbations
sensitivity
greaterthan 1 mm s-• for the highdegreeand
by Order for V•']qh•
Arc Lengths Using Power Rule 13x.10-'/[l_' _-_
.
Vtktni-I
300 •
Periapsis Orbit (epoch 75-01-15)
•
lo0
0.001
0
10
20
30
40
50
60
order
Fig. 2.
Spectral sensitivity of gravity signal (by order). GEODYN, numerically integrated perturbations' HAP,
harmonic analysis of perturbations from analytical theory; rss, root sum square.
20,876
SMITHETAL.' GODDARD
MARSMODEL1
(•0=269')
(•0=175')
Velocity
RSS'
Perturbations
392339
124383
28804
8027
2909
1297
675
400
268
229
203
187
179
186
199
VKGI:
2 ***
6 ******
10
14
18
22
26
30
34
36
38
40
42
46
50
* RSS taken
Velocity
In .001 cm/sec
DEG
RSS'
78-01-15
Arg. of periapsis
8 day arc length
469******
54'****'156
6 52***843
28
3 10138592237
8
1
2
7161289
76
1
1
2 20137143
0
0
1
2 33104
0
0
0
1 13 69
0
0
0
0
4 39
0
0
0
0
1 20
0
0
0
0
0
9
0
0
0
0
0
1
0
0
0
0
0
0
2
6 10
over
all
14
18
orders,
22
3
29
77
86
79
63
43
14
3
2
13
29
48
62
67
49
22
1
2
1
9
2
1
22
7
1
0
41 19
5
1
0
73 59 29
9
1
69 84 67 34 11
26
30
34
not
just
36
581650
72711
= 175 deg
38
40
42
0
0
46
0
50
sampled orders.
22705
9766
4959
2771
1631
984
604
483
378
316
244
162
110
Perturbations
In .001 cm/sec
DEG
VKGI:
2 ***
6 ******
10
14
18
22
26
30
34
36
38
40
42
46
50
***494336
******627
******274
762***554
148420411
120 47192
176108
38
133175148
157131
40
33 63 81
112106
60
67 69 52
32 37 35
ORD:
2
78-12-20
Arg. of periapsis
8 day arc length
6 10
* RSS taken over all
3
86
37
62
91
65
45
24
46
6
18
19
1
6
11
2
13
5
15
6
10
2
3
14
18
0
1
2
1
2
1
2
2
2
1
22
orders,
0
0
1
0
0
0
0
0
0
26
0
0
0
0
0
0
0
0
30
not just
0
0
0
0
0
0
0
34
- 269 deg
0
0
0
0
0
0
36
0
0
0
0
0
38
0
0
0
0
40
0
0
0
42
0
0
46
0
50
sampled orders.
Fig. 3. Signalsensitivityby degreeandorderfor VOl low orbit.
order termsfor arc lengthsgreaterthan 3 days. For arcsof 3
days the limit in sensitivity is approximatelyorder 30,
whereas for the 1-day arcs the limit in sensitivityis approximately order 20. The increasein sensitivityresultsfrom the
samplingof the medium-periodresonanceperturbations.
Although these results suggestit would be beneficial to
processthe VOl 300-kmdata in batchesof 8-16 days, this
was not possiblebecauseof insufficienttracking coverage
and errors in the nonconservative
force models.
separationbetweengroundtracksfor the orbitaldata setsis
indicatedin the plates by the term "walk." Also the data
spanis givenalongwith the walk to depictthe extentof the
coverageoverall dataof thistype asoriginallygivenin Table
2. In theseplateswe can seethe extent of periapsiscoverage
of the VOl low orbit (---39ø latitude), which is well complemented in the northern hemisphereby the VO2 low orbit.
Theseplatesshowreasonablygoodglobaldatacoveragefor
the VOl low orbit, VO2 low orbit, VO2 800-km orbit, and
alsofor M9. The globalcoverageprovidesfor good separa-
For the highly eccentric Viking orbits the sensitivityof
sphericalharmonic coefficientsdependsnot only on the bility of the lower degreeterms of the gravity field and
periapsisaltitude,but alsoon the locationof the argumentof possibly out through degree 30, consideringthe strong
periapsis.The GEODYN spectrumof rms velocity pertur- sensitivities to these terms. Plate 2 shows the combined
bationssampledby degreeand order are givenfor an 8-day coverageof the low orbits of VOl and VO2 from observaarc for two VOl
300-km orbits in 1978. In the first arc,
beginningJanuary 15, 1978, periapsisis located near the
equator (to --- 175ø). In the second test arc, beginning
December 20, 1978, the periapsisis located near 39øS(to •269ø).When tois near 180ø, the orbit tendsto be sensitiveto
termsof highdegreeand highorder, whereaswhen tois near
270ø, the orbit is sensitiveto terms of high degreeand low
order (see Figure 3).
tions with altitudes less than 5000 km and coverage by
2-
1.0 --
Another importantcharacteristicof theseeccentricorbits
is that significantsensitivityto the high-degreeterms exists
over a broad rangeof altitudes.As a demonstration,for the
VOl arc described in Figures 2 and 3 (epoch January 15,
1978),the perturbationsfor termsof order25 (degrees31-50)
are shown over a revolution in Figure 4. For these terms,
significantsensitivityis apparentup to an altitudeof 10,000
km, covering half of an orbit revolution.
In summary, the high degree sensitivity of the Viking
Orbiter tracking data to the gravity field of Mars is deter-
2 -
0.1 --
minedby the periapsisaltitude,locationof the argumentof
periapsis,and the length of the arcs used to processthe
tracking data.
40000
3.4.
Distribution of Observational Coverage
20000
0
20000
40000
altitude (kin)
The groundtrack for each of the Doppler observationsis
Fig. 4. Velocityperturbationfor VOl low orbit due to 25thplotted in Plates l a and lb for a completeset of ground orderharmoniccoefficientsfor degrees31-50 (curvea) and degrees
tracks coveringall major data setsusedin the solution.The 41-50 (curve b).
, V 0
II mm •
• Id
F,.
Z
0
ß
II
•
I
I
I
I
I
I i I
ß
A
?
II
!
SMITHET AL.: GODDARDMARSMODEL 1
20,879
o
o
o
v
ß +
'.'
!
i
I
20,880
SMITH ET AL.: GODDARDMARS MODEL 1
different
levels
of altitude
over
300 km.
This low-altitude
data coverage of the observation points along the ground
tracks for the VO 1 low orbit is seento be complementedby
the VO2 low orbit, particularly in the northern hemisphere.
However, the lack of complete data coverage near periapsis
indicates that separability will not be complete for the
high-degreeterms (30-50), as noted above in the argumentof
periapsis coverage for VO2.
Nevertheless, the result from Plate 2 indicates that there is
great sensitivity to the higher degree terms for altitudes less
than 2000 km for the low-altitude Viking orbits. Hence we
may expect that the ground track coveragefor the combined
VOl and VO2 low orbits, particularly, for the observed
coverage with altitude less than 500 km over a wide area, will
provide for good resolution of localized geophysicalfeatures
in the vicinity of these ground tracks.
4.
4.1.
MODELING
Physical Model
Because the Viking and Mariner data do not provide
uniform spatial coverageof Mars, the applicationof a priori
constraintswas critical to the development of a high degree
and order solution. The Viking and Mariner data were
initially processedusing the gravity model of Balmino et al.
[1982]. However, in the final iteration to produce GMM-1, an
intermediate solution, MGM-635, was used as the a priori
4.2.
Method of Solution
4.2.1. Least squares with a priori constraints. The
methodof solutionis a modifiedleast squaresprocess[Lerch
et al., 1979; Schwartz, 1976, 1978] which minimizes the sum
Q of signal and noise as follows:
+ E •fk
Q=
EC;m
+2•;m
•t ri2•
l,m
irl
irk
(2)
where the signal is given by Elm, Slm, which are the
normalized spherical harmonics comprising the solution
coefficients.
Theparameter
irl = 13 x 10-5/I2 isthermsof
the coefficients of degree l (a priori power rule) and is
introduced to permit solutions to degree and order 50. This
expression, which is based upon Kaula's [1966] rule, has
been obtained by Balrnino et al. [1982] and representsthe
power in that gravity model. The noise given by r ik is the
observationresidual (observed-computed)for the ith observation of satellitetracking data set type k, irk is the rms of
observationresidualsof data type k (generally significantly
greater than the a priori data precision), and fk is a downweightingfactor to compensatefor unmodeled error effects
for each data type k (ideally, fk = 1 for pure noise).
The optimum weighting method estimates the combined
weights directly, namely,
wk =
model.
The gravitational effects of the Martian solid body tide
obsi
2
irk
(3)
were includedin the satelliteforce model, and a value of k2
= 0.05 was adopted [Christensenand Balmino, 1979]. The
effects of atmospheric drag were incorporated into the
When minimizing Q above, using the least squaresmethod,
the normal matrix equation and error covariance are ob-
satellite force model using a spherical model for the satellite
body and the atmosphericdensity model developedby Culp
and Stewart [1984]. The coefficientof drag CD was adjusted
once per data arc, except for the VO 1 and VO2 low-periapsis
orbital arcs, where CD was adjustedonce per day. Solar
radiation forces were calculatedusinga sphericalmodel for
the spacecraft body and adjusting a reflectivity coefficient,
CR, once per data arc.
The solar flux at Mars at a given time was scaledfrom the
N• = R
Earth
value
for
that
date
to the
actual
distance
of the
tained as follows:
(4)
where • is the solution, N is the normal matrix, R is the
vector of residuals, and
V = N-1
N = E wkNk
(5)
is the approximate form for the variance-covariance error
matrix which must be calibrated by adjusting the weights.
Nk is the contribution for each satellite data set k to the
normals, where k = 0 corresponds to the normal equations
for the satellitea priori coefficientconstraintsfor which No
is the matrix of Kaula constraintsand the weight w0 is fixed
at unity for the constraints.
The process of minimizing both signal (by application of
the Kaula power rule constraints) plus noise in (2) is also
spacecraftfrom the Sun. One range rate bias was estimated
for each tracking station per arc. The measurementswere
corrected for tropospheric effects using the Hopfield [1971]
model. The tracking data records did not contain meteorological data or troposphericcorrections;thus the corrections
were computed assumingstandard temperature, humidity, known as collocation [Moritz, 1978]. The constraints bias the
and pressure, scaled to reflect the station height above sea coefficientstoward zero, where they are poorly observed.
level.
With the conventional least squares approach (noise-only
Third-body gravitational perturbationson the spacecraft minimization) there is a problem of separability due to the
were computed from the point massgravitationalforces due strong correlation between many of the high-degreecoeffito the Sun, the Earth-Moon system, the other planets, and cients. The absence of collocation (w0 = 0 in (5) for
Phobos, one of Mars' natural satellites. In addition, GEO- GMM-1) results in excessivelylarge power in the adjustment
DYN was modified to read an ephemerisfor Phobosand to of the potential coefficientsas in Figure 5. Hence we see the
add the point mass gravitational acceleration due to Phobos benefit of the constraints which permit resolution of the
to the total spacecraftacceleration. The ephemerisof Pho- high-degree terms wherever the data permit and provide
bos was prepared at Goddard Space Flight Center (GSFC) control of aliasing in the solution. Biasing of the coefficients
by processingoptical measurementsobtainedfrom the M9, is unimportant in those areas where data density is high.
VOl, and VO2 images of Phobos [Duxbury and Callahan,
4.2.2. Data weighting and error calibration. The
1988, 1989].
weightingtechniqueand error calibration [Lerch et al., 1988;
SMITH ET AL.' GODDARDMARS MODEL 1
20,881
(10) represent extremes in estimation: complete independence and complete dependence.
•
GtfIK-1
without
Belmino
Constraint
Truncated
to
35x35
lOOO
5.
5.1.
•
loo
RESULTS OF THE GMM-1
SOLUTION
Description of Solution
GMM-1 is a 50 x 50 spherical harmonic gravity model.
There are a total of 5250 estimatedparameters:2597 gravity
coefficientsplus GM, and approximately 2650 orbit and
spacecraft parameters. The GMM-1 gravity coefficients
throughdegreeand order 18 are shown in Table 5. The full
lO
set of coefficients can be obtained from Smith et al. [ 1993] or
fo........
Fig. 5.
'io........
Harmonic
•'o'.......
Degree
directly from the authors. Calibrated accuracy estimatesof
¾o
........
the coefficients
Gravity model uncertainties.
5.2.
have also been obtained for the model.
Gravity Model Tests Using Orbital
Observation
Residuals
Orbital arcs have been selectedfrom the seven major data
Lerch, 1991] of the solution (2)-(5) is based upon subset
solutions.The subsetsolution(Ck) is formed by deletinga
major data set k from the completesolution(C). The weight
wk is adjustedas in (8) below by requiringthat
sets summarized
in Table
3 and used to test the orbital
accuracy of the model by fitting the DSN Doppler data.
Observationresidualshave been computed from our 50 x 50
model and compared with the prior best available 18 x 18
gravity model of Balmino et al. [1982]. Table 6 is a compi-
lation of orbit tests for 14 arcs. Each arc is fit using the
Balmino et al. field and GMM-1. For all 14 test arcs, the rms
where K• is an error calibrationfactor which ideally should residual fits are significantly smaller, sometimes 5-10 times
equal to unity'
smaller, when computedusing GMM-1 than when using the
(6)
Balmino
l acll --
(c-
}1/2
2
(7)
5]
et al. field. Table 7 summarizes
the results of orbit
predictiontestsfor a subsetof the arcsin Table 6. The orbits
obtainedin the orbit accuracytests are projected forward in
time 2 or 3 days. Then rms residualfits are comparedfor the
data in the predicted time periods. Again, the fits are
significantly smaller when computed using GMM-1 than
when usingthe Balmino et al. field. The improvement in the
andwhererr• andrr2 arethevariance
of thesubset
andthe fits is not entirely due to the increasedresolution of GMM-1.
complete solutions,respectively. The sum in (7) is over all
the coefficients,and the scale factor K k is needed for the
errors, since the error covariance in (5) is only an approxi-
An 18 x 18 version of GMM-1 outperformed the 18 x 18
model of Balmino et al. in all cases except the 300-km VOl
and VO2 orbits, for which the performance was comparable.
mation [Lerch, 1991].
The new weights w•: shouldbe adjustedso that each K•
convergesto 1 for all k, and the new weights are computed
from
Wk
w•,=
2
K•
(8)
5.3.
Analysis of the Gravity Coefficients
Figure 6 is a plot of the degreevariance of the coefficients
(power) and error variance of GMM-1 per degree. Also
plotted are the power of the 18 x 18 gravity field [Balmino et
al., 1982]and a powerrule (13 x 10-5//2) takenfrom
Balmino et al. [1982], which is the basis of the constraint
(10)
matrix usedin GMM-1. The plot showsthat for degreesless
than 15, the power spectrumof GMM-1 and the Balmino et
al. field are about the same. However, above degree 15,
GMM-1 drops below, while the Balmino et al. field rises
above the power spectrumof Balmino's rule. The upward
turn of the Balmino et al. field is undoubtedly due to aliasing.
Aliasingadverselyaffectsthe performanceof a gravity field
with respect to orbit fits from independentdata, orbit predictions, and other geophysical information which are derived from the gravity coefficients.The truncation level of
GMM-1 at degree 50 is high enough that the high-degree
gravity signal is not significantly aliased into the lower
as indicated by (7). This meansthat in our case the covariance between the square of the difference of the two estimates of the coefficients is equal to the difference of the
variancesof the subsetand complete solutions.Thus (9) and
The power spectrumof GMM-1 dropsbelow the valuesof
the power rule for high degrees.Above degree22, the errors
of the coefficientsare larger than the coefficientsthemselves.
This drop-off in the power spectrumoccursbecausethe drag
The processis iterated by forming a new completesolution
and subsetsolutionsfrom the new weights, and this process
may continue until the weights converge.
In a case where two solutionsare based upon independent
data, then (in the above notation) for a single coefficient
parameter the two estimatesgive
E(C- C)2= •2 + rr2
(9)
whereas in (2) the data for the subset solution is wholly
embeddedin the complete solution in which case
E(C- C)2= •2_ or2
degree terms.
20,882
SMITH ET AL.: GODDARDMARS MODEL 1
TABLE 5. GMM-1Normalized
CoefiScients
Through
DegreeandOrder18in Unitsof 10-lø
Index
Index
n
m
Value
2
8
14
20
26
32
38
44
50
0
0
0
0
0
0
0
0
0
-8759770
1075
1627
932
-417
-303
- 103
-2
15
2
3
4
5
5
6
6
7
7
8
8
9
9
9
10
10
10
11
11
11
12
12
12
13
13
13
13
14
14
14
14
15
15
15
15
16
16
16
16
17
17
17
17
17
18
18
18
18
18
1
1
1
1
5
1
5
1
5
1
5
1
5
9
1
5
9
1
5
9
1
5
9
1
5
9
13
1
5
9
13
1
5
9
13
1
5
9
13
1
5
9
13
17
1
5
9
13
17
36
37473
42612
6140
-44655
18281
16382
13088
-1266
-178
-29578
2743
-20839
-11064
11266
2582
-15370
-11144
14309
-3755
-11655
6147
6846
-1541
15
9134
3215
3403
-1812
3125
9560
4884
-10076
-2844
-780
-1952
-4603
-3031
-47
-1891
5100
-2758
-2304
1294
928
2770
-659
-2052
-2537
Index
Index
n
rn
Value
n
rn
3
9
15
21
27
33
39
45
0
0
0
0
0
0
0
0
-119062
-1924
3462
-1254
-572
-57
44
32
4
10
16
22
28
34
40
46
0
0
0
0
0
0
0
0
GMM-1
65
252926
37090
20365
37260
-14489
16726
-1926
-12343
5158
-17748
-439
-14007
-6448
-4987
-12538
-15510
3564
11617
-3912
-4769
3962
3922
5749
-1555
9571
9523
4961
-8041
8702
21239
-1130
-3830
-1616
7182
-1748
3784
-9907
-5937
-6095
3698
-1126
-5787
3664
-1651
-2105
2574
-2866
-10316
2
3
4
5
2
2
2
2
6
6
7
7
8
8
9
9
2
6
2
6
2
6
2
6
10
10
10
11
11
11
12
12
12
13
13
13
2
6
10
2
6
10
2
6
10
2
6
10
14
14
14
14
15
15
15
15
16
16
16
16
17
17
17
17
2
6
10
14
2
6
10
14
2
6
10
14
2
6
10
14
18
18
18
18
18
2
6
10
14
18
Value
n
Index
Index
rn
Value
n
rn
Value
n
m
Value
0
0
0
0
0
0
0
0
-17635
-4064
1112
395
-73
-129
-50
-5
6
12
18
24
30
36
42
48
0
0
0
0
0
0
0
0
13340
1038
-4292
-927
-23
129
66
13
7
13
19
25
31
37
43
49
0
0
0
0
0
0
0
0
13025
-7098
-367
603
289
72
-10
-17
GMM-1 Zonals
51491
5
7098
11
2885
17
1274
23
611
29
107
35
-34
41
-36
47
Tesserals and Sectorials
- 846829
490611
-159844
- 10546
-41571
83160
-89776
-13689
3
4
5
3
3
3
351325
64742
33602
255554
- 1728
3147
4
5
4
4
2973
-45993
8244
16668
6
3
8376
2989
6
4
9447
27894
8686
-128554
-35011
26941
29707
-5593
7
3
11115
-3998
7
4
26621
-4873
-5563
15193
-9586
12058
8637
-19244
6245
-18195
1543
4988
7
8
8
9
9
7
3
7
3
7
4224
-14284
-5040
-8436
-4347
-17858
-14281
16982
-9009
8088
8
8
9
9
4
8
4
8
10334
-3135
9281
12605
741
-2319
15600
-1645
475
6192
-9821
2122
10
10
3
7
-7646
2874
3178
-5952
10
10
4
8
-18250
4783
-214
7701
3
7
8530
-7097
11
11
4
8
-10855
-9343
-4991
7947
11
3
7
11
-8925
7927
-513
-14660
1595
7044
1764
-5332
8870
-15713
3878
5169
-8478
12
12
12
13
13
13
4
8
12
4
8
12
-5403
-16783
-11
5277
-1440
-15838
1344
-6155
-1882
9094
2831
-3279
-3370
8017
-2557
-1785
3840
-29
-3514
5315
-469
-196
403
-11720
-228
19515
8259
-16113
12839
2873
-7846
-7620
11
11
11
12
12
12
13
13
13
8078
-507
-2433
-5192
-797
-13363
14
14
14
3
7
11
5749
-6641
-8531
1038
1808
1998
14
14
14
4
8
12
713
8070
-2997
-9173
1451
-4818
15
15
15
3
7
11
-3327
8499
-7197
-2122
3714
6605
15
15
15
4
8
12
-6995
8817
9103
-8219
7175
8887
1212
-2598
-1913
-127
-267
11
3
7
-2883
1901
-5631
-8491
-5447
4891
-4057
15
15
-4849
-4860
3250
-5903
-74
1094
5287
-3407
-3784
-1966
862
3735
-9887
3650
-3206
3532
-4601
16
16
16
16
17
17
17
17
3
7
11
15
3
7
11
15
-2155
4516
-780
-4941
-821
2686
4904
-5352
3452
2623
-6243
1661
2035
-1516
2276
2974
16
16
16
16
17
17
17
17
4
8
12
16
4
8
12
16
-5471
2082
3041
1691
2372
-5777
58
10094
1725
-280
5799
1144
1746
-4332
593
5379
-2322
1470
4920
-4524
3176
-4761
663
-135
18
18
18
18
3
7
11
15
823
-3103
2969
2471
-2615
1644
-2634
8800
18
18
18
18
4
8
12
16
5812
1927
1647
6505
1093
-288
-1097
3824
4125
-11150
2455
3926
parameters (once per day values) are absorbingpart of the
gravity signal. Also, the high degree terms are highly correlated, and hence the effect of the power rule constraintin the
solution is quite strong, which further explains the small
power for these terms. However, because the a priori
constraint does not have a major effect on the solution, the
terms do contain information on the short wavelength gravity field in the vicinity of the spacecraftperiapses.While the
power in the field falls below that predicted by the power
rule at high degrees,the field is a better representationof the
true gravitational signature of the planet at those wavelengths than would be the case if the field were solved to
SMITH ET AL.' GODDARDMARS MODEL 1
TABLE 6.
20,883
Orbit Accuracy Tests: rms of Orbital Fits
Arc
Arc
Satellite
Inclination,
Arc
deg
Epoch
No. of
Observations
Balmino
GMM-1
Length
18 x 18,
50 x 50,
days
cm/s
cm/s
1
M9
64
Jan. 15, 1972
1896
4
0.456
0.090
2
3
VOl 1500km
VO2 1500km
39
55
Aug. 22, 1976
Sept. 17, 1976
1326
1511
6
6
0.687
0.387
0.097
0.196
4
5
VO2 1500 km
VO2 800 km
75
80
Oct. 26, 1976
Jan. 2, 1977
1350
682
6
4
0.649
0.434
0.340
0.143
0.522
0.173
6
7
8
VOl 300 km
VOl 300 km
VOl 300 km
39
39
39
Nov. 22, 1977
Feb. 10, 1978
June 4, 1978
568
754
538
9
9
2
5.07
6.44
2.42
1.04
1.22
0.74
9
10
VOl 300 km
VOl 300 km
39
39
Aug. 11, 1978
Sept. 4, 1978
387
1025
2
8
1.43
8.58
0.09
1.24
4.79
0.87
VO2
VO2
VO2
VO2
80
80
80
80
Nov. 17, 1977
Dec. 17, 1977
May 16, 1978
May 26, 1978
1114
688
791
705
6
2
8
4
1.52
0.73
7.67
0.60
1.02
0.11
1.78
Average
Average
11
12
13
14
300 km
300 km
300 km
300 km
Average
0.15
0.77
2.63
For arcs 1-5 the arc parametersadjustedare position,velocity,Cr, and stationbiases.For arcs6-14 the arc parametersadjustedare
position,velocity, Cr, and Ca per arc. Arc 13 comesin as two separatearcsin GMM-1.
lower degreeand order and all of the high degreeand order
coefficients
were constrained
error of 1 cm s-1 while the VOl 300-kmdata groupis
assigned
anerrorof 0.71cms-1 (wt = 1/(0.71)
2 = 2). The
to zero.
Figure 7 shows the coefficient differences between
GMM- 1 and the 18 x 18 gravity field of Balmino et al. While
the rms differencesper degree are about the size of Balmino's rule for terms above degree 10, the differencesbetween
larger errors (indicatingdownweighting)for the data setsof
VO 1 at 1500 km and VO2 at 1500 km (55øinclination) are due
to the synchronous(repetitive)nature of the orbits (over the
Viking Landers)as shownin Table 2 and in Plates 1a and 1b
particular(C•m, S--•m)
pairsevenfor lowerdegreetermsare for the data distribution. The calibration factors (k) given in
seento be quite large. In fact, the rms differencesfor lower
degreeterms are over an order of magnitudegreaterthan the
error estimates of GMM-1 as given in Figure 6. The coefficient discrepancybetween these models reflects the large
differences seen in the orbital residuals for the two models,
as shown in Table 6.
5.4.
Table 8 indicatethat the model is reasonablywell calibrated,
where a factor of k = 1 indicates perfect calibration.
5.5.
Error Analysis
The calibrated
error covariance
matrix from GMM-1
was
used to project the orbital errors in satellite position and
velocity. Table 9 showsthe projectederrors for M9, VO 1 at
300 km, VO2 at 300 km, and Mars Observer (MO) for a 6-day
arc length. The results for the MO orbit are of special
interest, since they indicate the expected error in MO's
positionif the GMM-1 model were used in the orbit determination. Figures 8 and 9 give the error spectrumby degree
Calibration of Gravity Model Errors
The calibration of the gravity model error estimates is
based upon the method described in section 4.2.2 and is
developedin greater detail by Lerch [1991]. In the application of this method, weights of basic observationsets from
different orbits are adjustedbased on subsetsolutions.The and order for the radial and along-track position compodata are separatedinto sevengroups(see Table 7), yielding nents,respectively,of Mars Observer(cross-trackerrors are
sevensubsetsolutionsfor the weight adjustment.In Table 7 similar to the radial errors). Note the largest error is shown
eachgroupis assignedan a priori data weight which is based for resonant order 25, indicating that a field complete to at
on our experience in computingprevious gravity solutions. least degree30 is required to reasonablymodel these coefFor example, the VOl 1500-km data group is assignedan ficientsbasedupon the error spectrumby degree.The radial
TABLE
7.
Orbit Prediction Tests' rms of Prediction Fits
Predicted
Arc
1
2
3
4
5
Satellite
M9
VOl
VO2
VO2
VO2
1500 km
1500 km
1500 km
800 km
VOl
VO2
300 km
300 km
Inclination,
deg
Arc Epoch
No. of
Observations
64
Jan. 13, 1972
39
55
Aug. 22, 1976
Sept. 17, 1976
1308
840
720
75
80
Oct. 26, 1976
Jan. 2, 1977
697
367
39
80
Sept. 4, 1978
Nov. 17, 1977
353
597
Average
6
7
Period,
days
Balmino
GMM-1
18 x 18,
50 x 50,
cm/s
cm/s
5.61
4.20
1.62
8.04
21.40
8.17
105.6
102.9
0.36
2.58
0.68
2.38
0.73
1.34
13.2
30.1
20,884
SMITH ET ^L.' GODD^}•DM^}•S MODEL 1
Coefficient
andErrors
X 108
100000
10000
1000
•
Rule
100
Balmino's
18 x 18
10
Sigma's GMM- 1
GMM-
0
5
10
15
20
25
30
35
1
40
45
50
degree
Fig. 6. The rms of Mars gravitymodelCoefficients
and standarddeviationsper degree.
cases the estimate of GM is largely uncoupled from the
remainingcoefficientsof the Mars gravity field.
orbit error is expected to be the largest contributorto the
uncertaintyof the topographyof Mars derivedfrom the laser
altimeter [Zuber et al., 1992].
6.
5.6.
Recovery of GM
GEOPHYSICAL IMPLICATIONS
along with the other coefficientsof the Mars gravity field.
Plate 3 shows free air gravity anomaliescomputed from
GMM-1 complete to degree and order 50, and Plate 4
displaysaccompanyinggravity anomaly errors computed
The value of GM determined
from the error covariance matrix. As illustrated in Plates 1
In the GMM-1 solution, the GM of Mars was adjusted
in the solution was 42828.36 -+
0.05 km3 s-2. Lemoine[1992]analyzeda smallerset of
and 2, the satellitesused in this study are characterizedby a
complicateddistributionof low-altitude data. The shortest
wavelengthsresolved (200- to 300-km half-wavelength)occur within the latitudinal band of -+40ø, correspondingto the
(42828.32-+0.13km3 s-2 [Null, 1969]),Mariner6 (42828.22 data coveragefrom the VOl low orbit. This region also
-+ 1.83 km3 s-2 [Andersonet al., 1970]),and Mariner9 includesthe periapsiscoverage(0o-30ø latitude) of the VO2
(42828.35-+0.55km3 s-2 [O'Neillet al., 1973])arein close low orbit as seenin Plate 2. That plate also showsthat above
agreementwith the GMM-1 value. The Mariners 4, 6, and 9 40øN latitude there is still strong coverage of periapsis,
values are especially interesting, since they were derived extendingto 63øNlatitude, particularlyfor the VO2 800-km
from tracking of spacecraft from flybys of Mars. In these altitude orbit. This coverage is reflected in the gravity
Viking and Mariner 9 Doppler data as well as Viking Orbiter
rangedata and determineda value of GM of 42828.40-+ 0.03
km 3 s-2. The estimates of the Mars GM from Mariner 4
8
Difference
X 10
Balmtrio
Rule
3250
1444
812
52O
361
265
203
160
130
107
90
77
66
58
51
45
40
rms degree
51
32
61
71
93
80
90
76
77
78
93
58
62
48
57
49
44
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
76
31
2
115
96
96
39
54
116
165
116
58
142
48
88
118
69
28
27
35
61
44
65
48
60
132
37
64
44
2
27
101
225
113
133
123
87
64
65
64
64
42
21
43
49
100
94
22
12
40
74
38
44
47
60
55
48
26
55
102
78
83
28
81
75
26
13
8
20
21
57
120
78
84
50
58
45
22
101
42
45
15
22
62
52
169
77
66
38
61
59
20
41
71
38
101
130
54
62
69
25
28
51
114
14
99
171
59
42
14
85
54
45
10
9
29
90
45
59
46
88
67
14
19
54
37
32
88
58
3
76
23
25
40
37
24
84
26
21
67
61
25
12
68
3
4
5
6
7
8
9
10
11
12
13
68
84
57
41
56
49
17
4
34
7
106
24
42
69
61
50
8
0
56
49
67
6
62
90
27
27
25
71
59
73
49
65
30
43
0
1
2
66
49
31
80
34
57
19
11
71
77
19
44
36
69
14
15
16
17
18
52
35
62
40
69
order
70
80
54
99
Fig. 7.
56
58
63
69
Coefficientdifferencesfor Balminominus GMM-1.
20,885
SMITH ET AL..' GODDARD MARS MODEL 1
TABLE 8.
TABLE 9.
Calibration and Data Weights of GMM-1
Projected Gravity Orbit Error From GMM-1
Covariances
Subset Solution
Data Set Removed
A Priori
Satellite
GMM-
GMM-1
Arc
1
Calibration
Length,
days
Inclination,
deg
Weights,
cm/s
Weights,
cm/s
Factors
k*
55
75
80
80
39
39
1.0
1.0
0.71
0.71
1.0
0.71
1.0
4.1
1.5
0.72
1.0
3.5
0.8
2.0
1.2
0.81
0.81
1.16
0.96
1.05
0.99
Satellite
Along
Track
Radial
Cross
Track
Total
Orbit Position Error, rn
VO2
VO2
VO2
VO2
VOl
VOl
M9
1500 km
1500 km
800 km
300 km
1500 km
300 km
Mars Observer
Mariner 9
VOl 300 km
VO2 300 km
6
6
6
6
67
2
26
26
757
4
69
83
90
2
31
12
765
5
80
87
Orbit Velocity Error, cm/s
Here,k = [E(c - •)2/•(•.2 _ o.2)]1/2,
w' = w/k2, o.o= 1/wl/2,
o.b= 1/(w')'/2.
Mars Observer
Mariner 9
VO 1 300 km
VO2 300 km
6
6
6
6
66.3
0.07
1.9
2.0
8.0
0.03
0.8
0.9
8.0
0.02
0.2
0.1
67.1
0.08
2.1
2.3
*For complete convergence, k = 1.
Long-period terms excluded.
anomaly error map, which shows more longitudinal structure and better resolution than in the corresponding southern
hemisphere beyond the region of 40øS latitude. Plate 4 also
shows some artificial "striping," which is a consequence of
the distribution of orbit tracks, along which the gravity
anomaly errors are smallest.
In Plate 3, the free air anomalies are overlain by contours
of topography. The topographic field is a spherical harmonic
expansion, also complete to degree and order 50, of the Mars
Digital Elevation Model (DEM [Wu et al., 1986]). Our
spherical harmonic topographic model is plotted with respect to the GMM-1 geoid, so that all topographic values are
relative to an equipotential surface. Thus the spherical
harmonic model and DEM have similar hypsometric distributions; however, the reference levels differ. It is worth
noting that the range of topographic highs and lows is
reduced by 2 km as a result of referencing the topography to
the geoid rather than to an ellipsoid.
Plate 3 shows that, as in previous studies, the gravity
anomalies correlate well with principal features of Martian
topography, including volcanic shields, impact basins, and
the Vailes Marineris. Most major features exhibit anomalies
with considerably higher magnitudesthan in previous models. GMM-1 also showsgravity anomaliesin associationwith
some observed structures that were not previously detected.
For example, GMM-1 resolves all three Tharsis Montes,
while the model of Balmino et al. [1982] fails to resolve the
central volcano in the line, Pavonis Mons. GMM-1 also
shows considerably more detail associated with the Vailes
Marineris, and for several of the major impact basinsinclud-
ing Isidis and Argyre. However, it is important to interpret
short-wavelength features resolved in the model with cau-
Total
RMS
tion, as the individual coefficients associated with the highest
degree and order terms are 100% in error.
One of the most prominent physiographic features on
Mars is the hemispheric dichotomy, which is characterized
by a 2-km elevation difference between the northern and
southern hemispheres of Mars. Plate 3 indicates that the
dichotomy does not have a distinct gravitational signature
associated
with it. This observation
indicates
GMM-1
fail
to correlate
with
observed
surface
Asforthegravity
anomaly
representatic•n
ofthefield,the
geoid from GMM-1 as shown in Plate 5 exhibits a higher
dynamic range of power (2300 m versus 1950 m) than the
model of Balmino et al. [1982]. The distribution of geoid
errors shows a similar pattern to the gravity anomaly errors.
7.
SUMMARY
Reanalysis of Doppler tracking data from the M9, VOl,
and VO2 spacecraft has led to the derivation of a 50th degree
= 67m
RMS per Order
4O
a2 15
20
'•
-- 15
5
5
0
0
0
5
10 15 20
25 30 35
Harmonicdegree
Fig. 8.
40 45 50
features.
These include a 300-mGal negative anomaly on the western
edge of Tharsis (200øE longitude, 30øN latitude) and a
200-mGal positive anomaly in Utopia (105øE longitude, 50øN
latitude). Both of these areas are in the northern hemisphere
and may have been resurfaced. The latter feature was also
present in the field of Balmino et al. [1982], but the anomaly
was smaller in magnitude.
RMS per Degree
30
that the dichot-
omy boundary is isostatically compensated at the resolvable
wavelengths of GMM-1, perhaps due to a change in crustal
thickness across the boundary, as suggested in previous
studies [Phillips and Saunders, 1975; Lambeck, 1979; Phillips and Lambeck, 1980; Phillips, 1988].
It is significant that several prominent anomalies in
I
0
,,,,
I
,,.,
5
[
,.,,
10
i
,,,,
I
,,,,
15 20
I
,,,!
I
,,,,
25 30
I
,,,,
I
,,,,
35 40
I
,,,,
I
i
45 50
Harmonicorder
Projected radial position error on Mars Observer from GMM-1 gravity covariances.
20,886
SMITH ET AL..' GODDARDMARS MODEL 1
Total
RMS
= 757m
RMS per Degree
RMS per Order
600
•
'•700
,•300
o
0
................
0
5
0
10
15
•
25
:50
:35
40
45
50
0
5
10
Harmonicdegree
Fig. 9.
15
•
25
:50
:35
40
45
•3
Harmonicorder
Projected along-track position error on Mars Observer from GMM-I gravity covariances.
and order gravitational model for Mars. The model has a
maximum (half-wavelength) spatial resolution of 300 km
where the data permit, which represents a factor of 2
improvement over that attained by the previous field of
Balmino et al. [1982], which utilized essentially the same
data. Probable reasons for the improvement include increasedcomputationalcapabilities, the application of collocation and optimum data weighting techniques in the least
squares inversion for the field, and the use of longer arcs
(days versus hours) than used previously for Viking lowaltitude data made possibleby improved force and measurement models.
Error analyses based on the observation data, derived
power spectrum, and comparison with topography demonstrate that this field representsthe orbits with considerably
better accuracy and shows greater resolution of identifiable
Mar s F ree
Ai r
geological structures than previous models. The model also
shows a greater dynamic range of power in both the gravity
anomaly field and the geoid. The inclusion of X band
tracking data from the 379-km altitude, near-circular, polar
orbit of Mars Observer will allow significantimprovement of
the Martian gravitational field [Smith et al., 1990b; Esposito
et al., 1990; Tyler et al., 1992], the greatest refinement
occurring at high latitudes far removed from the M9, VOl,
and VO2 periapsis latitudes. That gravitational field, in
combination with topography data from the Mars Observer
laser altimeter [Zuber et al., 1992], will allow detailed
analyses of Mars' internal structure, state of lithospheric
stress, and mechanismsof isostatic compensationof surface
topography.
Note added in proof.
Grav i t y Anoma I i es
Since the loss of the Mars Ob-
Compu ted
F rom
GMM- 1
,!
Contour
60 --
interval
--------,.• .•..__..._•
•
: Anomol ies
= 50 mgals
/•
,
Topogrophy
,• .---..
,.
-.-'-'..
=
1 km
•
20
•2
(
0
v
C•'
'
-••
•7•0
20
0
•
'
eO
•0
•00
120
•&O
•0
•
•
•
,•
'
•80
200
ß
220
2 0
•
260
-.•
'
280
o
-
500
,
320
340
Longlfude
Plate 3. Free air gravity anomalies computed from GMM-1 (colors). The contour interval is 50 mGals. The
anomaliesare plotted relative to a reference ellipsoid with an inverse flattening of 191.1372. The line contours in the plot
represent a 50 x 50 spherical harmonic expansion of the Mars DEM [Wu et al., 1986], with topographic values
referenced to the GMM-1 geoid. The contour interval is 1 km.
360
SMITH ET AL.: GODDARDMARS MODEL 1
Mars
Oravi
Jy
Anomaly
Contour
Errors
Inferval
=
2.0
20,887
from
GMM-1
rngals
60
50:
40
-
$0
20-
0-
-20
-
-,30
-
-$0
-
-60
-
;.
0
2o
•o
•o
so
•oo
,20
14o
16o
,80
Longi
48
50
52
54
56
Plate 4.
Contour
•-'•
•o
ß"-'",,-"
'.".'•,.'N
-ø
o•
82
64
Geo i d
Interval
:--'"-'--/
. .)(\
-•o-
60
66
68
70
72
220
Surf
:
74
ace
Hetght$
76
78
80
82
84
2so
86
88
280
90
'
',•
,
-- 50
m.
,
3oo
s2o
3•o
360
92
g4
9E
98
100
,
1 km
c'
•
2-,••., .
oo -, '."-L.«,• ; -,
,.,__.----:'•
• -.-'v.•,-----, ....
.•,o.o.-•o•,
-..to-
,,
--
,.., ,'----'%k.
",.--
•
'• ,-
..,';,• ••..,,.•,
GMM-1
Topography
,,
-..o---. ß ,
(-• -,
From
Compu t,ed
(Z}
/-'"-•..-/2
'" •,"".,%,"•
•
(_ c,2..../o
- •o-
240
Gravity anomaly errors from GMM-1. The contour interval is 2 mGals.
Mar s
so
58
200
rude
/
_.••
' •
•
o
,. --.•',.•.
ß
k,k'
.,'
c
'
,-•.
c_._.• ;-..& ,
'"
_ ø
-50 -
-60
-
•'•• ...
-65
o
•o
•o
•o
so
•oo
l•o
•o
J8o
•o
=oo
2•o
•o
•o
•o
•oo
a•o
•4o
k ong i • ude
.::
.
Plate 5.
Geoid anomaliesfrom GMM-1. The contour interval is 50 m. Also shown are contours of topography,
produced as in Plate 3, with a contour interval of 1 km.
•o
20,888
SMITH ET AL.: GODDARDMARS MODEL 1
server spacecraftin August 1993, planning has been initiated
for a recovery mission with a probable launch in November
1996 and arrival at Mars in September 1997. While the details
of the mission are still being defined at the time of this
writing, the base line favors a spacecraft(or multiple spacecraft) with X band tracking capability and an orbit as close as
possible to the original Mars Observer orbit. If this occurs,
the conclusions regarding orbit and gravity field determination based on GMM-1 can be expected to apply for the Mars
Observer Recovery Mission.
Esposito, P. B., S. Demcak, and D. Roth, Gravity field determination for Mars Observer, AIAA Rep., 621-626, 1990.
Esposito, P. B., W. B. Banerdt, G. F. Lindal, W. L. Sjogren, M. A.
Slade, B. G. Bills, D. E. Smith, and G. Balmino, Gravity and
topography,in Mars, edited by H. H. Kieffer et al., pp. 209-248,
University of Arizona Press, Tucson, 1992.
Gapcynski, J.P., R. H. Tolson, and W. H. Michael, Jr., Mars
gravity field: Combined Viking and Mariner 9 results, J. Geophys.
Res., 82, 4325-4327, 1977.
Hopfield, H. S., Tropospheric effect on electromagneticallymeasuredrange: Prediction from surface weather data, Radio Sci., 6,
357-367, 1971.
Jordan, J. F., and J. Lorell, Mariner 9: An instrument of dynamical
science, Icarus, 25, 146-165, 1975.
Kaula, W. M., Theory of Satellite Geodesy, 124 pp., Blaisdell,
Acknowledgments. We are grateful to George Balmino for proWaltham, Mass., 1966.
viding the DSN tracking data used in the solution and for suggestions and analysesassociatedwith the developmentof our solution, Lambeck, K., Comments on the gravity and topographyof Mars, J.
including comparisonsof gravity models with his JEEP software.
Geophys. Res., 84, 6241-6247, 1979.
Thisstudyhasalsobenefited
significantly
fromcarefulreviewsby Lemoine, F. G., Mars: The dynamics of orbiting satellites and
Robert Reasenbergand Nicole Rappaport, from suggestionscongravity model development, Ph.D. thesis, 413 pp., Univ. of Col.,
cerning the development of the model from Bill Sjogren, Pat
Boulder, 1992.
Esposito, and Ed Christensen, and from discussionswith Bruce Lerch, F. J., Optimum data weighting and error calibration for
Bills, Herb Frey, and Walter Kiefer. Finally, we thank Dave
estimation of gravitational parameters, Bull. Geod., 65, 44-52,
Rowlands, Despina Pavlis, Scott Luthcke, and John McCarthy for
1991.
their analysis with the GEODYN interplanetary orbit program, and Lerch, F. J., S. M. Klosko, R. E. Laubscher, and C. A. Wagner,
Joe Chan, Doug Chinn, Huseyin Iz, Rich Ullman, and Sheila
Gravity model improvement using GEOS 3 (GEM 9 and 10), J.
Kapoor for analytical and computer supportin the preparation of the
Geophys. Res., 84, 3897-3915, 1979.
solution and material of this document.
Lerch, F. J., J. G. Marsh, S. M. Klosko, E. C. Pavlis, G. B. Patel,
D. S. Chinn, and C. A. Wagner, An improved error assessment
for the GEM-T1 gravitational model, NASA Tech. Memo.,
REFERENCES
100713, 1988.
Anderson,J. D., L. Efron, and S. K. Wong, Martian massand Lerch, F. J., D. E. Smith, J. C. Chan, D. S. Chinn, and G. B. Patel,
Earth-Moon mass from coherent S-band tracking of Mariners 6
High degree gravitational sensitivity from Mars orbiters for the
and 7, Science, 167, 227-229, 1970.
GMM-1 gravity model, NASA Tech. Memo., in press, 1993.
Balmino, G., B. Moynot, and N. Val/•s, Gravity field model of Mars
Lorell, J., G. H. Born, E. J. Christensen, J. F. Jordan, P. A. Laing,
in spherical harmonics up to degree and order eighteen, J.
W. A. Martin, W. L. Sjogren, I. I. Shapiro, R. D. Reasenberg,
Geophys. Res., 87, 9735-9746, 1982.
and G. L. Slater, Mariner 9 celestial mechanics experiment,
Science, 175, 317-320, 1972.
Banerdt, W. B., R. J. Phillips, N.H. Sleep, and R. S. Saunders,
Thick shell tectonics on one-plate planets: Applications to Mars,
Lorell, J., G. H. Born, E. J. Christensen, P. B. Esposito, J. F.
J. Geophys. Res., 87, 9723-9733, 1982.
Jordan, P. A. Laing, W. L. Sjogren, S. K. Wong, R. D. ReasenBanerdt, W. B., M.P. Golombek, and K. L. Tanaka, Stress and
berg, I. I. Shapiro, and G. L. Slater, Gravity field of Mars from
tectonics on Mars, in Mars, edited by H. H. Kieffer et al., pp.
Mariner 9 tracking data, Icarus, 18, 304-316, 1973.
249-297, University of Arizona Press, Tucson, 1992.
Marsh, J. G., et al., A new gravitational model for the Earth from
Bills, B. G., and A. J. Ferrari, Mars topography harmonics and
satellite tracking data: GEM-T1, J. Geophys. Res., 93(B6), 61696215, 1988.
geophysicalimplications, J. Geophys. Res., 83, 3497-3508, 1978.
Born, G. H., Mars physical parametersas determinedfrom Mariner
Marsh, J. G., et al., The GEM-T2 gravitational model, J. Geophys.
Res., 95, 22,043-22,071, 1990.
9 observations of the natural satellites and Doppler tracking, J.
Geophys. Res., 79, 4837-4844, 1974.
Moritz, H., Least squarescollocation, Rev. Geophys., 16, 421-430,
1978.
Christensen,E. J., and G. Balmino, Developmentand analysisof a
twelfth degree and order gravity model for Mars, J. Geophys. Moyer, T. D., Mathematical formulation of the double precision
Res., 84, 7943-7953, 1979.
orbit determinationprogram (DPODP), JPL Tech. Rep. 32-1527,
Christensen, E. J., and B. G. Williams, Mars gravity field derived
Jet Propul. Lab., Pasadena, Calif., 1971.
from Viking-1 and Viking-2: The navigation result, J. Guidance Nerem, R. S., B. G. Bills, and J. B. McNamee, A high-resolution
Contr., 3, 179-183, 1979.
gravity model for Venus: GVM-1, Geophys. Res. Lett., 20,
599-602, 1993.
Culp, R. D., and A. I. Stewart, Time-dependent model of the
Martian atmosphere for use in orbit lifetime and sustenance Null, G. W., A solution for the mass and dynamical oblateness of
studies, J. Astronaut. Sci., 32, 329-341, 1984.
Mars using Mariner IV Doppler data, Bull. Am. Astron. Soc., 1,
Davies, M. E., V. K. Abalakin, J. H. Lieske, P. K. Seidelmann,
356, 1969.
A. T. Sinclair, A.M. Sinzi, B. A. Smith, and Y. S. Tjuflin, O'Neill, W. J., J. F. Jordan, J. W. Zielenbach, S. K. Wong, R. T.
Reports of IAU Working Group on cartographiccoordinatesand
Mitchell, W. A. Webb, and P. E. Koskela, Mariner 9 navigation,
rotational elements of the planets and satellites, Celestial Mech.,
JPL Tech. Rep. 32-1586, Jet Propul. Lab., Pasadena, Calif., 1973.
29, 309-321, 1986.
Phillips, R. J., The geophysicalsignatureof the martian dichotomy,
Davies, M. E., V. K. Abalakin, M. Bursa, G. E. Hunt, J. H. Lieske,
Eos Trans. AGU, 69, 389, 1988.
B. Morando, R. H. Rapp, P. K. Seidelmann, A. T. Sinclair, and Phillips, R. J., and K. Lambeck, Gravity fields of the terrestrial
Y. S. Tjuflin, Report of the IAU/IAG/COSPAR working groupon
planets: Long wavelength anomalies and tectonics, Rev. Geocartographicand rotational elements of the planets and satellites:
phys., 18, 27-86, 1980.
1988, Celest. Mech. Dyn. Astron., 46, 187-204, 1989.
Phillips, R. J., and R. S. Saunders, The isostatic state of martian
Davies, M. E., R. M. Batson, and S.S. C. Wu, Geodesyand
topography, J. Geophys. Res., 80, 2893-2898, 1975.
cartography,in Mars, edited by H. H. Kieffer et al., pp. 321-342, Puthey, B. H., General theory for dynamic satellite geodesy, The
University of Arizona Press, Tucson, 1992.
National Geodetic Satellite Program, NASA Spec. Publ., SP-365,
319-334, 1977.
Duxbury, T. C., and J. D. Callahan, Phobos and Deimos astronometric observationsfrom Viking, Astron. Astrophys., 201, 169- Reasenberg,R. D., I. I. Shapiro, and R. D. White, The gravity field
176, 1988.
of Mars, Geophys. Res. Lett., 2, 89-92, 1975.
Duxbury, T. C., and J. D. Callahan, Phobosand Deimos astrono- Rosborough,G. W., and F. G. Lemoine, Sensitivity studiesof Mars
metric observations from Mariner 9, Astron. Astrophys., 216,
orbiters for Mars gravity recovery, J. Astron. Sci., 39, 327-358,
284-293, 1989.
1991.
SMITH ET AL.: GODDARDMARS MODEL 1
Schwartz, K. P., Least-squares collocation for large systems, Bull.
Geod. $ci., 35,309-324, 1976.
Schwartz, K. P., On the application of least-squares collocation
models to physical geodesy, in Approximation Methods in Geodesy, edited by H. Moritz and H. Sunkel, pp. 89-116, H. Wichmann-Verlag, Karlsrfihe, Germany, 1978.
Sjogren, W. L., J. Lorell, L. Wong, and W. Downs, Mars gravity
field based on a short arc technique, J. Geophys. Res., 80,
2899-2908, 1975.
Sleep, N.H., and R. J. Phillips, An isostatic model for the Tharsis
province, Mars, Geophys. Res. Lett., 6, 803-806, 1979.
Sleep, N.H., and R. J. Phillips, Gravity and lithospheric stresson
the terrestrial planets with reference to the Tharsis region of
Mars, J. Geophys. Res., 90, 4469-4489, 1985.
Smith, D. E., F. J. Lerch, R. S. Nerem, G. B. Patel, and S. K.
Fricke, Developing an improved higher resolution gravity field
model for Mars, Eos Trans. AGU, 71, 1427, 1990a.
Smith, D. E., F. J. Lerch, J. C. Chan, D. S. Chinn, H. B. Iz, A.
Mallama, and G. B. Patel, Mars gravity field error analysis from
simulated tracking of Mars Observer, J. Geophys. Res., 95,
14,155-14,167, 1990b.
Smith, D. E., F. J. Lerch, R. S. Nerem, M. T. Zuber, G. B. Patel,
and S. K. Fricke, An improved gravitational model for Mars:
Goddard Mars Model 1 (GMM-1), NASA Tech. Memo., 104584,
50 pp., 1993.
Snyder, C. W., The missions of the Viking orbiters, J. Geophys.
20,889
Tyler, G. L., G. Balmino, D. Hinson, W. L. Sjogren, D. E. Smith,
R. T. Woo, S. Asmar, M. J. Connally, and R. A. Simpson, Radio
scienceinvestigationswith Mars Observer, J. Geophys. Res., 97,
7759-7779, 1992.
Willemann, R. J., and D. L. Turcotte, The role of membrane stress
in the support of the Tharsis rise, J. Geophys. Res., 87, 97939801, 1982.
Wu, S.S. C., R. Jordan, and F. J. Schaefer, Mars global topographic map, 1:15,000,000 scale, NASA Tech. Memo., 88383,
614-617, 1986.
Zuber, M. T., D. E. Smith, F. J. Lerch, R. S. Nerem, G. B. Patel,
and S. K. Fricke, A 40th degree and order gravitational field
model for Mars (abstract), Lunar Planet. $ci., XXII, 1581-1582,
1991.
Zuber, M. T., D. E. Smith, S.C. Solomon, D. O. Muhleman, J. W.
Head, J. B. Garvin, H. V. Frey, J. B. Abshire, and J. L. Bufton,
The Mars Observer laser altimeter investigation, J. Geophys.
Res., 97, 7781-7797, 1992.
S. K. Fricke, RMS Technologies, Inc., Landover, MD 20785.
F. G. Lemoine, F. J. Lerch, R. S. Nerem, and D. E. Smith,
Laboratory for Terrestrial Physics, NASA Goddard Space Flight
Center, Greenbelt, MD 20771.
G. B. Patel, Hughes STX Corporation, Lanham, MD 20706.
M. T. Zuber, Department of Earth and Planetary Sciences, Johns
Hopkins University, Baltimore, MD 21218.
Res., 82, 3971-3983, 1977.
Snyder, C. W., The extended mission of Viking, J. Geophys. Res.,
84, 7917-7933, 1979.
Standish, E. M., Jr., M. S. W. Keesey, and X. X. Newhall, JPL
development ephemeris no. 96, JPL Tech. Rep. 32-1603, Jet
Propul. Lab., Pasadena, Calif., 1976.
(Received March 11, 1993;
revised July 6, 1993;
accepted July 7, 1993.)