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\title {\bf Precise Orbit Determination \\ for  Mars Global Surveyor
\\ During Hiatus and SPO}
\author {F. G. Lemoine\thanks{Code 926, Laboratory for
Terrestrial Physics, NASA Goddard Space Flight Center, Greenbelt, MD 20771; E-mail: flemoine@olympus.gsfc.nasa.gov},\\
D.D. Rowlands, G. A. Neumann\thanks{Also at Department
of Earth, Atmospheric, and Planetary Sciences, Massachussetts Institute of
Technology, Cambridge, MA 02139-4307.}, D. E. Smith \\
 Laboratory for Terrestrial Physics \\ NASA Goddard Space Flight
 Center \\  Greenbelt, Maryland  \and
D. E. Pavlis, D. S. Chinn, S. B. Luthcke \\ Raytheon ITSS, Greenbelt, Maryland}
\date{}
\begin{document}
\maketitle
\begin{abstract}
The Mars Global Surveyor (MGS) spacecraft arrived at Mars on September 12,
1997. The Mars Orbiting Laser Altimeter (MOLA)
collected data during the Hiatus Phase from October 13 to November 7, 1997,
and during the Science Phasing Orbit(SPO) in March to April 1998,
and June to July 1998.
The Doppler tracking data acquired from the Deep Space Network (DSN) were used
to compute orbits for the MOLA Science Team. After development of
tuned gravity models to 70x70 in spherical harmonics with the
MGS tracking data, the RMS radial orbit difference of  overlapping MGS arcs
during SPO had an overlap consistency of 30 to 35 m,
compared to 160 to 200 m with
the {\it a priori} GMM-1 solution. We also demonstrate that
the use of altimeter crossovers derived from MOLA altimetry
improved the MGS orbits in SPO-1 (March
27 - May 1, 1998) by 15 percent (2.8 m) in the radial component, and
60 percent (600 m) in total position.
\end{abstract}

\section*{{\bf INTRODUCTION}}
The Mars Global Surveyor spacecraft arrived at Mars on September 12, 1997.
The initial plans called for three to four months of aerobraking
to circularize the orbit by January 1998. Concerns with the
-Y panel solar array led to the suspension of aerobraking
in October 1997, and an alternate plan to reach the
mapping orbit by February of 1999$^1$.  Aerobraking resumed from
November 1997 to March 1998, at a reduced rate$^2$.
After the end of the first phase of aerobraking, the spacecraft
was located in an orbit with a period of 11 hours and 38 minutes, and
a mean periapse altitude of 171 km from March 27, 1998 to September
23, 1998. Interim science observations were collected by the MGS instruments
in the Science Phasing Orbit-1 (SPO-1; from March 27 to May 1, 1998)
and Science Phasing Orbit-2 (SPO-2; May 28 to Sept. 23, 1998), as well
as in Hiatus Phase (October - November 1997).


A primary goal for the Mars Orbiter Laser Altimeter (MOLA) is to derive
a global, geodetically referenced, 0.2$^\circ$ x 0.2$^\circ$ topographic
grid of Mars with a vertical accuracy of better than 30
meters$^3$.
Topographic mapping of the entire planet will occur after the MGS orbit
is circularized at the end of aerobraking.  Nevertheless, MOLA
collected data during the Hiatus, SPO-1, and the first two months of SPO-2.
A single pass of MOLA data was also obtained after Mars Orbit Insertion
(MOI), on September 15, 1997.   The orbit characteristics
for the insertion, Hiatus, and SPO orbits are summarized in Table 1.
All these orbits are highly eccentric, so that MOLA data were obtained
only for  20 to 30 minutes during each periapsis pass.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|ccccc|}
\multicolumn{5}{c}{Table 1: MGS Orbit Characteristics during MOLA Data Acquisition}\\
\hline\hline
\rule{0mm}{3mm} Mission & Dates & Orbital  & Periapsis  & Periapsis\\
           Phase        && Period (hr) &Ht. (km) & Latitude (deg)\\
\hline
Insertion & September 12-16, 1997 & 44.993 & 263 & 31$^\circ$ N\\
Hiatus    & Oct. 13 - Nov. 7, 1997 & 35.43 & 170 &  35$^\circ$ N\\
SPO-1     & March 27 - May 1, 1998 & 11.64 & 171& 61$^\circ$ - 71$^\circ$  N\\
SPO-2     & May 28 - July 30, 1998 & 11.64 & 171& 82$^\circ$ - 86.3$^\circ$ N \\
 & & &  &86.3$^\circ$ - 76$^\circ$ N\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

The instrument acquired data whenever the spacecraft was below about 785 km,
at a rate of 10 Hz.  Under the best of conditions, the MOLA data have
a precision of 37.5 cm, when the surface slope is less than 2.0$^\circ$ and when
the instrument is pointing at nadir$^{3,4}$.
Off-nadir data were collected during spacecraft roll-outs, and
specially targeted off nadir passes including the campaign to map the
northern Mars polar cap during SPO-2.  The argument of periapsis of the
MGS orbit precessed from 146$^\circ$ in October 1997 to 76$^\circ$ on
July 30, 1998, so that periapsis was located exclusively in the Northern
Hemisphere when MOLA measurements were made. A total of 2.6 million MOLA
ranges were obtained over 208 passes in Insertion, Hiatus, and SPO.

In order to properly analyze the MOLA data, the spacecraft orbit must be
known to an accuracy comparable to the performance of the altimeter.
Careful attention must be made in the reduction of the DSN tracking data
to both the measurement and spacecraft force modeling.  The principal
concerns in the force modelling include the Mars gravity field perturbations,
atmospheric drag, solar radiation pressure, and the periodic angular
momentum desaturations.  Detailed modeling of the nonconservative forces
requires that the spacecraft attitude and shape be properly modelled.
The main concerns of the measurement model include proper modelling of
the ramped two-way DSN Doppler data, and the one-way Doppler
data obtained from the MGS Ultrastable Oscillator (USO)$^5$.
The offsets of the antenna from the spacecraft center of mass must be
included, especially for the one-way data.  Each of these issues
is explored in this paper, after a brief review of the available data.

\section*{{\bf MGS TRACKING DATA AND STATION COMPLEMENT}}

During Hiatus, aerobraking, and SPO-1 and SPO-2,
the MGS spacecraft was tracked by the antennae of the Deep Space Network (DSN)
using two-way and three-way ramped Doppler and two-way ramped range data.
All data were obtained at X band.
The ramped Doppler observable is described
by Moyer$^{6,7}$,
and has been used on previous interplanetary space missions, such as
Magellan and Voyager.  MGS used unramped Doppler during cruise to Mars, but
switched to ramped Doppler at the start of aerobraking in September 1997
for operational reasons$^8$.

Since the High Gain Antenna (HGA) is fixed to the spacecraft
prior to the deployment of the antenna arm during mapping,
no direct tracking data can be obtained from the HGA
while the instrument panel is pointed
toward nadir during passage through periapsis, unless fortuitously
the HGA is pointed within 3-4$^\circ$ of Earth.
During SPO-1, the HGA was aimed from 20$^\circ$ to 50$^\circ$ off of
Earth point during the periapse pass, and in early SPO-2
the geometry was even more unfavorable$^8$.
Since no direct tracking was possible through periapsis using the HGA,
high resolution gravity mapping of the Northern Hemisphere
of Mars would have been compromised.

The orbits during SPO represented an unique opportunity, in that it would
be possible for the first time to obtain high resolution gravity information
over the northern  polar regions of Mars from a relatively
low altitude (170-300 km).  Earlier low-altitude tracking  data had
been obtained in the 1970's  from S Band tracking of the Viking Orbiters,
but mostly from the equatorial regions of the
planet$^{9,10,11}$.

The solution to the HGA Earth visibility problems during SPO-1 and
early SPO-2 was to use the spacecraft low gain antenna (LGA).
Special arrangements were made to collect one-way tracking data
from the MGS Ultrastable Oscillator (USO)$^5$.
One-way Doppler differs from two and three-way Doppler in that the
signal originates at the spacecraft.  For two-way and three-way Doppler,
the signal originates at an Earth tracking station, and is simply
transponded by the MGS spacecraft$^{5,12}$.
One-way Doppler rather than two-way Doppler were obtained to prevent
loss of valuable low-altitude data in the time it would take for the
Earth tracking station to lock up on the signal from MGS  in two-way mode.
During SPO-1, valid
one-way Doppler data were obtained starting three to eight minutes before
periapsis.  The spacecraft switched
to the HGA again after the end of the periapse pass and the return to
Array Normal Spin (ANS) attitude.  During periapse passes in SPO-2,
one-way data were obtained on nearly every other  orbit through July 8, 1998.
Two-way Doppler data over four periapse passes (389, 392, 396, and 398) in
SPO-2  were obtained from the low gain antenna (LGA)$^8$.
The geographical distribution of the one-way Doppler data acquired during
SPO-1 and SPO-2 is depicted in Figure 1.

Figure 1: Geographic location of Doppler data acquired from Mars Global Surveyor,
below 500 km altitude during SPO-1 and SPO-2 shown in a north pole
stereographic projection to 55$^\circ$ N  (a) Data acquired below 300 km.
(b) Data acquired between 300 and 500 km altitude.


Mars Global Surveyor was tracked primarily by the 34 m High Efficiency
Antennae (stations 15, 45, 65), and the newer 34 m  beam-waveguide (BWG)
antennae (stations 34, 54, 25)$^{13}$  of the Deep Space Network (DSN).
In addition to the one-way and two-way Doppler
data, some passes of three-way Doppler data were obtained.
Three-way Doppler differs from two-way Doppler in that
different antennae are used for transmission  and reception of the
Doppler signal$^{12}$.  The averaging times for the two-way
Doppler data was 60 seconds during Hiatus and SPO-1,
and 10 seconds during SPO-2.
For the one-way Doppler most of the data were averaged over 10 seconds.


\section*{{\bf FORCE AND MEASUREMENT MODELLING}}
\subsection*{{Representation of the Gravitational Potential}}
The Mars gravity potential is modeled in spherical harmonics using
the expression$^{14}$,

\begin{equation}
U = {GM \over r} \sum_{l=0}^{\infty} \biggl({a_e\over r}\biggr)^l
\sum_{m=0}^{l}
\bigl\lbrace
\bar C_{lm}  \cos m\lambda + \bar S_{lm}  \sin m\lambda \bigr\rbrace
\bar P_{lm}(\sin\phi)
\end{equation}
\noindent where the expansion is defined using the spherical coordinates
of radius, $r$, latitude, $\phi$, and longitude, $\lambda$; $\bar C_{lm}$ and
$\bar S_{lm}$ represent the normalized geopotential coefficients; $\bar P_{lm}$
are the normalized associated Legendre functions of degree $l$ and order $m$;
$a_e$ is the reference equatorial radius; $G$ is the  universal constant
of gravitation, and $M$ is the planet mass.
A solution for the Mars gravity field consists of estimates for
the coefficients $\bar C_{lm}$ and $\bar S_{lm}$ and the gravitational
constant $GM$.

\subsection*{{Definition of the Box-Wing Model}}
The nonconservative force model must take into account the complex geometry
and material properties of the spacecraft. To that end,
the spacecraft is treated as a combination of flat plates arranged in the
shape of a box with attached solar arrays and drag flaps
(ie. `Box-Wing' or `Macromodel'). The solar radiation
pressure, atmospheric drag, and Mars albedo and infrared emission forces
acting on each flat plate are computed and vectorially summed to compute
the total nonconservative force acceleration acting on the spacecraft.
The same approach has been used to model the nonconservative forces
acting on other spacecraft, for instance TOPEX/Poseidon$^{15}$,
and TDRSS$^{16}$.
Self shadowing of spacecraft elements is not considered.
A fourteen plate box-wing model for MGS was derived for use
during Hiatus and SPO from engineering drawings of the spacecraft.

The radiation pressure acting on a flat plate can be computed using the
following equation, assuming Lambertian diffusion$^{17}$,

\begin{equation}
\Gamma_S = - {GA \cos\theta \over Mc} \lbrack 2 (\delta/3 + \rho\cos\theta)
\vec n + (1-\rho)\vec s \rbrack]
\end{equation}

\noindent where $\Gamma_S$ is the acceleration due to the solar radiation
pressure on a flat plate, $G$ is the radiation flux from the source (the
Sun) in W/m$^2$, $A$ is the surface area of the flat plate, $M$ is the
satellite mass, $c$ is the speed of light, $\delta$ is the diffuse
reflectivity (defined as a percentage of the incoming radiation), and
$\rho$ is the specular reflectivity (also defined as a percentage of the
incoming radiation), $\theta$ is the angle between the surface normal
and the solar incidence, $\vec n$ is the surface unit normal vector, and
$\vec s$ is the unit vector from the source.


The albedo and infrared accelerations use a similar equation
to equation(2), however the flux magnitude and source vector differ.
The source is the vector from the Mars surface spot to the satellite.
The spot definition and location are defined by Knocke et al.$^{18}$.
The Mars albedo and thermal emission is modelled using zonal spherical
harmonic models of the Mars albedo and 3x3 spherical harmonic models of
the planetary thermal emission as a function of Mars areocentric
longitude, $L_s$$^{11}$.  In our implementation, we calculate
the solar reflected radiation and planet thermal emission separately
for 43 surface spots in three concentric rings and a central cap beneath
the subsatellite point. The total albedo and infrared acceleration
can be represented as,

\begin{equation}
\Gamma_A = - \sum_i^N \sum_j^{Jmax} {G_j A_i \cos\theta_{ij} \over Mc }
\lbrack 2(\delta_i/3 + \rho_i \cos\theta_{ij})\vec n+i + (1 - \rho_i)\vec s_j
\rbrack
\end{equation}

\noindent where the index $i$ refers to plate $i$, and the summation takes
place over $N$ plates, the index $j$ refers to the surface spot $j$, and the
summation occurs over $Jmax$ surface spots. If the angle $\theta_{ij}$,
between the normals to the surface spot and the plate is greater than
90$^\circ$, then the acceleration is not computed since the plate cannot
observe the surface spot.

The atmospheric drag is computed according to the equation,
\begin{equation}
\vec D = -{1 \over 2} {C_D \over m} \rho \vec v |\vec v| \sum_i^N A_i\cos\theta
\end{equation}
where $C_D$ is the drag coefficient, $\rho$ is the atmospheric density,
and $\vec v$ is the satellite velocity with respect to the atmosphere.
The angle $\theta$ is the angle between the plate normal unit vector and
the spacecraft velocity vector.

\subsection*{MGS Attitude during Hiatus and SPO}
The definition of a geometric model for the MGS spacecraft must
be accompanied by specification of the spacecraft attitude, so
that the forces perturbing spacecraft motion can be computed properly
and so the tracking data can be corrected for the motion of the antennae
with respect to the spacecraft center of mass.
The GEODYN orbit determination (OD) software can calculate spacecraft
orientation either using spacecraft quaternions or an internal attitude
model$^{19}$.  However, the MGS spacecraft quaternions  cannot
be relied on exclusively since gaps occur in the quaternion data stream.
Thus, it is important to understand the various spacecraft attitude
regimes, and develop an analytical attitude model to `fill-in' during
the quaternion gaps.


The origin of the MGS spacecraft body-fixed coordinate system is located on
the spacecraft-launch vehicle interface plane and
the center axis of the spacecraft (ie. through the bottom center of
the propulsion nozzle) (see Figure 2). The +Z axis
is aligned with the nadir instrument deck, and +X axis
points toward the stowed position of the  HGA.
The Y axis completes the orthogonal set.


Figure 2:  Nominal MGS spacecraft configuration during Array Normal Spin
(Figure from ref. 20).



During Hiatus and SPO, two attitude regimes predominated:  nadir pointing and
array normal spin (ANS).  Near periapsis the spacecraft instrument panel
(+Z face of the spacecraft) was usually  pointed at nadir to collect
science data.  Far from periapsis the spacecraft was in array normal
spin, with the high gain antenna (+X spacecraft body fixed axis)
pointed at the Earth (see Figure 2), rotating
once every 100 minutes about this axis.
In addition, for up to sixty minutes per orbit,
ANS was interrupted to turn the spacecraft
and warm the MOLA instrument.  On some periapse passes the instrument panel
was deliberately pointed off nadir to acquire science data.

During the periapse passes in SPO-2,
orbit geometry and low gain antenna viewing constraints
dictated that the spacecraft be yawed such that the MGS spacecraft +Y axis
was aligned with the velocity vector, so that the Earth could receive
the one-way data from the low gain antenna.
This yaw mode was implemented on every other revolution,
with alternate revolutions devoted by the Mars Orbiter Camera
(MOC) to targeted imaging.
In addition during 12 passes, the nadir instrument panel was deliberately
pointed as much as 49.2$^\circ$ off nadir so MOLA could acquire
measurements over the
center of the North polar cap\footnote{If the MOLA instrument had remained
nadir pointed, altimeter coverage in SPO would only have reached 86.3$^\circ$N,
because of the inclination of the MGS orbit.}.

In our macromodel, we modelled the solar arrays as swept forward 30$^\circ$
in the +X direction, consistent with the mission documentation for the array
normal spin attitude$^{20}$.
However, analysis of the spacecraft solar array quaternions shows that this
cant angle varied between 23.95$^\circ$ and 33.68$^\circ$ between
Insertion, Hiatus, and SPO.  In addition, starting later in SPO-2, the
attitude of the solar array was altered from nominal during periapsis passes.

It is evident that unlike mapping, the MGS spacecraft during
SPO-1 and SPO-2  experienced a multitude of attitude regimes.
An analytical or semi-empirical model that handled all contingencies
and special circumstances (Phobos flybys for instance) would
attain inordinate complexity. Therefore, as a first approximation,
the analytical attitude algorithm modelled only the
nadir attitude during the periapsis pass, and the ANS attitude
far from periapsis, with the presumption that
the model would come into play only during quaternion data gaps, and
that the spacecraft quaternions would handle off-nadir targeting passes,
rollouts (transitions from nadir to ANS attitude), and MOLA warming maneuvers.

The fidelity of the macromodel is illustrated in Table 2, which summarizes
the RMS of fit to the tracking data for 4.5 day arcs in SPO-1, using
a box-wing model, and a `cannonball.'  In the
absence of a macromodel, as far as the force model is concerned,
the spacecraft is effectively modelled
as a `cannonball' with a constant cross-sectional area.
The RMS of fit to the one-way Doppler data in these nine arcs improves by
0.64 mm/s or about 25 percent, with the macromodel.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|lcc|cc|cc|}
\multicolumn{7}{c}{Table 2: RMS of fit (in mm/s) for Arcs in SPO-1:
Comparison of Macromodel vs. Cannonball
\footnote{The orbit determination results in this table were all computed
with the MGM0890 (70x70) gravity model.}}\\
\hline\hline
\rule{0mm}{3mm} Arc  & \multicolumn{2}{c|}{No. of Doppler Obs.}
& \multicolumn{2}{c|}{Cannonball} & \multicolumn{2}{c|}{Macromodel}\\
   & 1-way & 2-way  & 1-way Dop. & 2-way Dop. & 1-way Dop. & 2-way Dop.\\
\hline
980327 & 387 & 2524 &  2.66 & 0.46 & 1.71 & 0.46\\
980403 & 900 & 3098 &  1.64 & 0.44 & 1.11 & 0.44\\
980406 & 630 & 2243 &  1.64 & 0.54 & 1.31 & 0.54\\
980410 & 1420 & 3813 & 2.62 & 0.61 & 1.40 & 0.61\\
980413 & 875 & 1427 &  1.38 & 0.47 & 1.27 & 0.47\\
980417 & 1154 & 2929 & 2.91 & 1.06 & 1.54 & 1.07\\
980420 & 232 & 2605 &  1.41 & 1.72 & 1.40 & 1.72\\
980424 & 961 & 2941 &  3.31 & 1.81 & 2.04 & 1.82\\
980427 & 206 & 2283 &  1.76 & 2.15 & 1.76 & 2.14\\
\rule{0mm}{3mm} {\bf Average} & & & 2.15 & 1.03 & 1.51 & 1.03\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

It is not just the fit to the tracking data, but the actual orbit
consistency which improves with the application of a macromodel.
Successive 4.5 day arcs that overlapped by 12 hrs
in SPO-1 (from 980327 to 980501) and SPO-2 (from 980531 to 980710)
were used to compare modelling the MGS orbits with a cannonball and
with a macromodel.  No other aspect of the modelling, in terms of adjusted
parameters or measurement model corrections, were changed in these tests.
Once-per-revolution along-track empirical acceleration parameters were
not adjusted in thes orbit tests.
As summarized in Table 3, the application of the macromodel improves
the orbit consistency by 15 percent (6 m)
in the radial direction, and about 10 percent (70 m) in total position.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|lcccc|}
\multicolumn{5}{c}{Table 3: Summary of Overlap Statistics in SPO-1 and SPO-2:}\\
\multicolumn{5}{c}{Cannonball vs. Macromodel}\\
\hline\hline
\rule{0mm}{3mm} & \multicolumn{4}{c|}{Average RMS for 19 overlaps (meters)}\\
  & Radial  & Cross-track & Along-track & Total\\
\hline
 Cannonball & 39.1 & 542.8 & 454.2 & 773.1 \\
 Macromodel & 33.2 & 505.8 & 399.9 & 701.8 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}




\subsection*{Measurement Offsets and Tracking Point Corrections}
The MGS spacecraft has five antennae: a High Gain Antenna (HGA) and
four low gain antennae (LGA's). Two of the LGA are reserved for transmission
and two for reception. The HGA is not deployed
until mapping, and is stowed next to the +X face of the spacecraft
(see Figure 2). The coordinates of the geometric centers of these
antennae are summarized in Table 4, and provided in a
MOLA instrument kernel$^{21}$. The coordinates refer to the
MGS spacecraft body-fixed frame. The center of gravity of the spacecraft
(or the offset of the center of mass from the origin of the spacecraft
body-fixed frame) must also be specified, and is also summarized in Table 4.


\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|lccc|}
\multicolumn{4}{c}{Table 4:MGS Antenna and CG Offset Corrections}\\
\hline \hline
\rule{0mm}{3mm} & \multicolumn{3}{c|}{Coordinates (meters)}\\
 & X  & Y  & Z \\
\hline
HGA (stowed)         & 0.999 & 0.000  & 1.680\\
LGA-transmit (front) & 0.999 & -0.716 & 2.069\\
LGA-transmit (back)  & 0.526 & -0.763 & 2.072\\
LGA-receive (+X)     & 0.806 &  0.134 & 0.446\\
LGA-receive (-X)     & -0.655 & 0.000 & 1.268\\
CG-Launch            & -0.0010 & -0.0007 & 1.1320\\
CG-Mapping           &  0.0298 & 0.0718 & 0.9970\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

The two transmit LGA's are attached to the TWTA (traveling wave
tube amplifier) box which is mounted on the HGA reflector.
Hence, the coordinates for the transmit LGA's apply only while
the HGA is stowed. The `front' transmit LGA was used for transmission
of one-way Doppler data during SPO-1 and SPO-2. The `rear' transmit
LGA was not used since its boresight is oriented about 160$^\circ$ from
the +X axis$^{22}$.

To underscore the importance of applying the measurement model offsets
for the one-way Doppler data, we show in Figure 3 the residuals for a short
seven hour arc centered on periapsis 238 (April 14, 1998). In the first
case the measurement offset corrections detailed in Table 4 were not
applied, and the RMS reaches -6 mm/s. Although the spurious signal in the
residuals occurs  during a spacecraft roll-out (transition from nadir-pointing
attitude to array normal spin) and at a relatively high altitude (approximately
2000 km), such a signal would  corrupt the one-way Doppler data
and their contribution to gravity field mapping.
The RMS for this pass (consisting of 207 one-way Doppler observations)
is 1.45 mm/s without the measurement model correction,
and 0.84 mm/s when the correction is applied.
Not every periapsis pass has this dramatic a signature, but
nevertheless the one-way Doppler data show substantial improvement
when these corrections are applied. In contrast, the application of
the measurement offset corrections  change minimally the  RMS of fit to the
two-way Doppler data.
This is logical since nearly all the two-way Doppler
data were obtained on the HGA while the spacecraft was in array normal
spin (hence Earth pointing). In this orientation, the measurement offset
correction for the two-way Doppler data would not be discernible.

Figure 3:  Effect of the measurement offset correction on one-way Doppler
residuals for periapse pass 238 (April 14, 1998).

\subsection*{Other modelling issues}

The MGS spacecraft must periodically fire its thrusters to  desaturate the
reaction wheels, which absorb angular momentum from disturbance torques
acting on the spacecraft.
These thruster firings can impart some velocity impulse
($\Delta V$) to the spacecraft, effectively altering the spacecraft orbit.
Nine angular momentum desaturation maneuvers  were documented
in Hiatus (October 13 to November 7, 1997), four in SPO-1 (March 27 to
May 1, 1998), and sixteen in SPO-2 (May 31 - Sept. 23, 1998).
In our orbit determination, a three-axis (radial, along-track,
cross-track) constant acceleration is estimated over the duration of
these maneuvers.


The DE403 planetary ephemerides were used as the source of ephemeris
information for the measurement model, and for the calculation of
the third body perturbations$^{23}$.
Solid tides raised by the Sun on Mars were also included using a $k_2$
Love number of 0.05 from Balmino et al.$^{9}$.
The Stewart atmosphere model was used to compute the atmospheric density$^{24}$.
The third body perturbations due
to Phobos and Deimos were applied with an ephemeris from
B. Jacobson$^{25}$ and GM values of 720,000 m$^3/$s$^2$
for Phobos$^{26}$, and 120,000 m$^3$/s$^2$
for Deimos$^{27}$\footnote{MGS made several close (less than 1000 km)
flybys of Phobos during SPO-1 and SPO-2, coming as close as 260 km on
Sept. 12, 1998, when MOLA measurements were obtained from the satellite. The
geometry of the MGS orbits during Hiatus and SPO did not favor `close'
approaches to Deimos, and the closest approach distance was 3000 km.}.
The orientation of Mars is computed using the 1991 system of IAU
constants$^{28}$.
The DSN provides weather data in the form of temperature, barometric pressure,
and relative humidity for each of the Deep Space Complexes at Goldstone,
Madrid, and Canberra$^{29}$.
The weather data are provided to the MGS project
at half-hour intervals.  The weather data are interpolated to the appropriate
transmit and receive times of each measurement, and an Earth troposphere
refraction correction is applied to the tracking data using a Hopfield
model$^{30}$.

The arcs in Hiatus, SPO-1, and SPO-2 were four to five days in length.
The adjusted arc parameters included the spacecraft position and velocity;
drag coefficients (once per orbit in Hiatus, and once every twelve hours
in SPO); and a solar radiation reflectivity coefficient.
Three axis accelerations were estimated over the duration of any angular
momentum desaturation maneuvers.
Pass-by-pass biases were estimated for the range data.
A frequency offset ($f_b$), as well as the first and second derivatives
of this frequency offset with time($\dot{f_b}$ and $\ddot f_b$)
were adjusted in each arc which included one-way Doppler data.


\section*{ORBIT DETERMINATION RESULTS}
\subsection*{Gravity Model Determination}
\subsubsection*{{\underline {Data Used}}}
The quality of the orbits computed for MGS during Hiatus and SPO is
intimately associated with the Mars gravity model used to reduce the tracking
data, and which satellites contributed to that gravity model's development.
The first analysis of the SPO-1 and Hiatus data relied on GMM-1$^{10}$,
a 50x50 model developed from a reanalysis of the Mariner 9 and Viking Orbiter
tracking data. However, it was obvious that the
determination of improved orbits for analysis of MOLA data, required that the
MGS tracking data be incorporated into Mars gravity solutions.
So, a sequence of models were developed that incorporated
more MGS tracking data, and gradually refined the weights of both the new
and the historical (Mariner 9 and Viking) tracking data.
At several stages, the MGS tracking data used up until that point were
reanalyzed with the newer MGS derived gravity model. New normal equations
were created for the MGS data, and these `updated' normal equations were
included in the subsequent gravity model solutions.  This process of
iteration was essential, since the knowledge of the Mars gravity field in
the northern polar regions was quite coarse prior to the start of the
gravity mapping campaigns in SPO-1 and SPO-2.
We report on the performance  of five landmark models,
to show the evolution in orbit
performance: (1) GMM-1, the {\it a priori}, 50x50, solution from Mariner 9 and
Viking Orbiter tracking$^{10}$; (2) MGM0827E, a 70x70 solution incorporating
only MGS data from SPO-1 in addition to the historical data; (3) MGM0883,
a 70x70 solution that included the historical data, as well as most of
the available MGS data in SPO-1 and SPO-2; (4) MGM0890, a 70x70 solution
that added the remaining MGS data from SPO-2; (5) MGM0890a, a 80x80
solution identical in data content to MGM0890.
The measurement offset correction was not applied in the orbit analysis
prior to the derivation of the MGM0890 models.  After the end of SPO-2,
all the one-way Doppler passes were rescrutinized, and the data reedited
after applying the measurement offsets discussed earlier.
In addition, the one-way Doppler data were assigned an {\it a priori} sigma
of 1.0 Hz (3.57 mm/s) in the orbit determination (OD)
analysis for MGM0827E and MGM0883, but
these data were upweighted and assigned an {\it a priori} sigma of
0.2 Hz (0.71 mm/s) in the orbit determination (OD) analysis
for the MGM0890 series of
solutions.  The two-way and three-way Doppler data were assigned
an {\it a priori} sigma of 0.1 Hz (0.36 mm/s) in all the OD analyses.

As a matter of interest, it is possible to derive a Mars gravity solution
solely from the MGS tracking data in Hiatus and SPO,
however the historical data are
still needed to stabilize the solution at long wavelengths, and provide
higher resolution information in regions not sampled at low altitude
by MGS.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|lccccc|}
\multicolumn{6}{c}{Table 5: Data Summary for Mars Gravity Solutions}\\
\hline \hline
\rule{0mm}{3mm}   & \multicolumn{4}{c|}{Number of Observations} & Comment\\
Model & Mariner 9 & VO-1 & VO-2 & MGS & \\
\hline
 GMM-1 (50x50) & 49,878 & 95,370 & 87,075 & 0  & {\it Smith et al.} [1993].\\
 MGM0827E (70x70)
 & 69,992 & 109,844 & 74,373 & 61,270 & Add MGS data to 98/05/01.\\
 MGM0883 (70x70)
 & 69,992 & 109,176 & 76,005 & 219,988 & Add MGS data to 98/09/01.\\
 MGM0890 (70x70)  &69,992 & 109,176 & 79,545 &
 257,165 & Add MGS data to 98/09/17.\\
 MGM0890a (80x80) &69,992 & 109,176 & 79,545 &
 257,165 & Same data used as MGM0890.\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}
The MGM0827E and later models incorporated some changes in the historical
(Mariner 9 and Viking Orbiter) data.  All the newer models incorporated
Mariner 9 data from April - July 1972, not included in GMM-1. In addition,
Viking-1 Orbiter tracking from June and July 1976, as well as S Band
range data to the Viking Orbiters were added in MGM0827E and subsequent
solutions (see ref. $11$ for a complete description of the historical data).


\subsubsection*{\underline {Method of Solution}}

The Mars gravity solutions were computed using least squares
collocation$^{31,32}$
which minimizes the sum Q of signal and noise according to the relation,

\begin{equation}
Q = \sum_{l,m} {{\bar C_{lm}^2 +  \bar S_{lm}^2} \over \sigma_l^2} + \sum_t
\sum_{obs_j} {r_{ik}^2 \over \sigma_k^2} f_k
\end{equation}

\noindent where the signal is given by normalized spherical harmonic
coefficients
$\bar C_{lm}$ and $\bar S_{lm}$; $\sigma_l$ = 13.0 x 10$^{-5}/l^2$ is the
RMS of the coefficients of degree $l$ from the {\it a priori} power law
based on the 18x18 solution of Balmino et al. (see ref. $9$);
$r_{ik}$ is the observation residual for the $i_{th}$ observation of satellite
tracking data group $k$, and $\sigma_k$ is the RMS of the observation residuals
for data group $k$, and $f_k$ is a downweighting factor to compensate
for unmodelled error effects for each data group.  Without some type
of constraint in addition to the observation equations,
the least squares solution would yield
spherical harmonic coefficients with excessive power in the high degree terms.
We apply a spectral constraint using a power law in the derivation
of our gravity solutions, however other types of constraints are possible,
for instance the spatial and spectral technique described by Konopliv
and Sjogren in their derivation of the Mars50c gravity
solution$^{33}$.

The method of Lerch$^{34}$ using subset solutions was applied to
adjust the {\it a priori} weights of each data set.  The essence of this
method is that the aggregate changes in the spherical harmonic coefficients
caused by the addition of an independent set of data should be commensurate
with the aggregate change in the standard deviations of the spherical harmonic
coefficients (See also
ref. $35$ for a detailed discussion of how the
method was used to calibrate the relative weights of tracking data in
a satellite-only solution for the Earth).
The MGS data were grouped into four sets for purposes of
calibrating the data: Hiatus, SPO-1, SPO-2 through June 30, 1998, and
SPO-2 from July - Sept. 1998. The historical (Mariner 9 and Viking Orbiter)
data were grouped according to satellite periapsis height and mission
phase to determine their calibrations.

\subsubsection*{\underline {Orbit Performance}}
The orbit performance of these different gravity models can be evaluated by
computing the orbit position discrepancy between overlapping arcs.
The set of overlaps included nine 4.5 day arcs in SPO-1 form March 27-May 1,1998
and 23 4.5 day arcs in SPO2 from May 31-September 8, 1998. All arcs
overlapped by only 12 hours, or just over one revolution of MGS
about Mars.  The orbits were compared every 20 seconds in the overlap
interval.  The overlap between the arcs starting on July 10 and July 14
was eliminated since the radial component was anomalously high (800-3000 m)
for all the gravity models tested.  The orbit overlaps are reported in Table 6.
The radial orbit consistency was improved from 160 to 200 meters with the
{\it a priori} GMM-1 to 30 - 36 meters with the most recent model including
MGS data (MGM0890).
For the SPO-2 orbits, the MGM0827E model (including MGS data from SPO-1)
is already an improvement over the {\it a priori} GMM-1 model. Not surprisingly,
though, the MGM0827E model has superior performance for the SPO-1 orbits
(with an average RMS radial overlap of 39 m) than the SPO-2 orbits
(with an average RMS radial overlap of 98 m).
The improvement of the ensemble of SPO-1 and SPO-2 orbits required that
all available MGS tracking be included  in the gravity model solutions.
The difference in orbit overlap quality  between the  MGM0890 (70x70)
and MGM0890a (80x80) model is negligible. The RMS of fit to the tracking
data also shows little change between the two models (see Table 7) implying
that the extension of the field to 80x80 does not benefit the
orbit determination for MGS.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|lcccc|}
\multicolumn{5}{c}{Table 6: Orbit Overlap Statistics for Mars Gravity Models}\\
\hline \hline
\multicolumn{5}{|c|}{Average RMS Overlaps for 9 Arcs in SPO-1 (m)}\\
 Model & Radial & Cross-track & Along-track & Total\\
 \hline
 GMM-1 & 206.3 & 4951.9 & 2816.6 & 5741.8\\
 MGM0827E & 38.7 & 1819.6 & 1084.4 & 2132.2\\
 MGM0883  & 36.5 & 1315.1 & 707.1 & 1503.1\\
 MGM0890  & 35.9 &  934.1 & 424.0 & 1034.1\\
 MGM0890a & 35.1 & 923.5 & 418.2 & 1023.2\\
\hline
\multicolumn{5}{|c|}{{Average RMS Overlaps for 23 Arcs in SPO-2 (m)}}\\
 Model & Radial & Cross-track & Along-track & Total\\
 \hline
 GMM-1 & 164.1 & 1080.7 & 3641.3 & 3827.9\\
 MGM0827E & 98.1 & 839.9 & 2833.0 & 2980.6\\
 MGM0883  & 32.3 &  220.3 & 775.5 &  814.4\\
 MGM0890  & 29.1 &  116.5 & 491.2 &  510.5\\
 MGM0890a & 28.6 & 115.1 & 485.8 &  504.8\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

The RMS of fit to the tracking data has been dramatically reduced
(for example, from 39 mm/s with GMM-1 to 1.5 mm/s
with MGM0890 for the one-way data)
confirming that the signal in the tracking data has been
substantially accommodated by the new gravity models.
Nevertheless, the RMS of fit to the two-way and three-way
Doppler data remains in excess of
the nominal noise floor for the X Band Doppler data of 0.1 mm/s.
In the vicinity of periapsis, systematic signals remain in the
one-way Doppler residuals as well - some of them of high frequency
as seen for example in Figure 4. We depict the  two-way and three-way
Doppler residuals
from a 4.5 day arc starting on March 27, 1998 using GMM-1, and
MGM0890. The one-way Doppler residuals are shown two periapse passes:
208, and 209.  Whether, the high frequency signal depicted in the one-way
Doppler residuals with MGM0890 is due to a surface gravity anomaly
or whether it is due to other unmodelled effects remains to be established.
Further improvements in the analysis of the Hiatus and SPO
tracking data are still possible.


\begin{table}
\begin{minipage}[tbhp]{\textwidth}
\begin{center}
\begin{tabular}{|lccc|}
\multicolumn{4}{c}{Table 7: RMS of Fit for Mars Gravity Models}\\
\hline \hline
\rule{0mm}{3mm} & \multicolumn{3}{c|}{RMS of Fit  for 9 Arcs in SPO-1}\\
  & 1-way Doppler & 2-way Doppler & 2-way Range \\
 Model & (mm/s) & (mm/s) & (m)\\
\hline
GMM-1 & 39.94 & 9.12 & 111.2\\
MGM0827E & 4.88 & 1.28 & 29.7\\
MGM0883  & 2.64 & 1.06 & 29.0\\
MGM0890  & 1.51 & 1.03 & 29.0\\
MGM0890a & 1.52 & 1.03 & 29.0\\
\hline
\rule{0mm}{3mm} & \multicolumn{3}{c|}{RMS of Fit  for 27 Arcs in SPO-2}\\
  & 1-way Doppler\footnote{Only nine arcs in SPO-2 had 1-way Doppler data,
  so for this data type, the average is only computed over those arcs.}
  & 2-way Doppler & 2-way Range \\
 Model & (mm/s) & (mm/s) & (m)\\
 \hline
GMM-1 & 14.06 & 5.15 &  67.9\\
MGM0827E & 5.42 & 2.82 & 36.7\\
MGM0883  & 2.27 & 0.77 & 28.3\\
MGM0890  & 2.16 & 0.63 & 28.0\\
MGM0890a & 2.31 & 0.62 & 28.0\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}


Figure 4: Doppler residuals for the 4.5 day arc starting on March 27, 1998 with
the {\it a priori} GMM-1 (50x50) gravity model, and the MGM0890 (70x70),
MGS-derived gravity model. The two and three-way Doppler residuals are
shown for the entire arc. The one-way Doppler residuals
from two periapsis passes in this arc are shown: periapsis 208 on March 30,
1998, and periapsis 209 on March 31, 1998.

\subsubsection*{\underline{Evaluation of MGS Orbits Using Altimetry Crossovers}}
The orbits released to the MOLA science team were evaluated using
altimeter crossovers\footnote{These orbits represent the gdn-ql06 series
of orbits for MGM0883, and the gdn-ql09 series of orbits for MGM0890.}.
Strictly speaking this evaluation cannot be viewed solely
as a test of the MGM0883 and MGM0890
gravity models, as other changes in the data processing  have occurred
in the creation of the two sets of MGS ephemerides during SPO.
This evaluation is interesting, since it tests the orbit quality where
MOLA  data are available (near periapsis, below 785 km), whereas the
orbit overlaps test the orbit consistency around the entire orbit.

Crossovers were computed internal to
both SPO-1 and SPO-2, and an observation timing bias of
115.2 ms was applied. The roll and pitch offsets of the laser altimeter
from the spacecraft body-fixed frame were set to values estimated
in a SPO-1 tracking data and crossover orbit determination run
(see the next section). The yaw offset of the MOLA altimeter from the
spacecraft frame came from the MOLA instrument kernel$^{21}$.
The L1 (median absolute deviation normalized to unity for a gaussian)
and L2 (root mean square deviation) norms
were evaluated for the set of SPO-1 and SPO-2
crossovers, and are summarized in Table 8.

The orbits show slight improvement in SPO-1 (about 0.6 m or five percent
as measured by the L1 norm), but a substantial
improvement in SPO-2 (about 26 m or 67 percent, as measured by the L1 norm).
The L2 norm is so much larger than L1 because
of the presence of large tails in the crossover distribution.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|lcc|}
\multicolumn{3}{c}{Table 8: Orbit Evaluation Using Altimeter Crossovers}\\
\hline \hline
\multicolumn{3}{|c|}{SPO-1}\\
   & MGM0883  & MGM0890 \\
\hline
L2 & 63.9     & 61.6 \\
L1 & 10.1     & 9.5 \\
N\footnote{N = number of crossovers.}  & 1188 & 1189\\
\hline
\multicolumn{3}{|c|}{SPO-2}\\
   & MGM0883  & MGM0890 \\
\hline
L2 & 141.9     & 70.9 \\
L1 & 38.3     & 13.9 \\
N  & 5879 & 5875\\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}


\subsection*{Use of Altimetry Crossovers In OD for SPO-1}

We experimented with using the MOLA altimetry directly to improve
orbit determination for MGS.  The motivation arises from the fact
that an altimeter crossover observation provides a powerful constraint
that ties together two separate orbit tracks.
Altimetry in the form of crossovers has often been used to provide a
constraint to orbit solutions (for example to Earth orbiting satellites
such as GEOSAT, ERS-1 and ERS-2). When altimetry observations are used
to form crossover constraint equations, less knowledge about the
planet surface is required than when the alimeter data  are used
directly to form observation equations. See ref. $36$ for a
detailed discussion of the use of altimeter crossover constraint
equations in orbit determination.

Conditions during Hiatus and SPO were quite different from those under
which crossover constraints have usually been applied. Previously,
crossovers have usually been used (1) over relatively flat
surfaces such as oceans and ice sheets, (2) with radar altimeters that
have a large footprint, (3) with satellites in a circular orbit,
and (4) with satellites whose trajectories have already been well
determined  from conventional tracking (excluding altimetry).

Under conditions (1-3) above, crossover constraints are relatively
insensitive to the horizontal (non-radial) components of orbit error.
Because of this relative insensitivity to horizontal
orbit error and also because the satellite trajectories have already
been well determined, crossover constraints have always been formulated
in terms of surface height discrepancies at predetermined latitudes
and longitudes.  For MGS, the altimeter data is obtained in part at least
over sloping topography, the laser altimeter has a narrow footprint (on the
order of 100 m rather than many km), and the orbit is highly eccentric.
In addition, the MGS orbit still has large orbit error even
with the improvements described in the previous sections.
Because of these differences, we are sensitive to horizontal (non-radial)
orbit error, and therefore, we have formulated our crossover constraint
equations differently. The crossover constraint
formulation is described in ref. $37$
so we will only summarize the main differences with the standard approach:

We define the crossover discrepancy to be the minimum distance
between two curves of geolocated altimeter bounce points
that have been traced out on the planet surface. The
two curves correspond to the ascending and descending
passes that surround a conventional crossover. The
planet fixed coordinates of points on the curves are
determined (geolocated) using orbit and attitude information, as well
as the altimeter ranges.  Our technique to minimize the crossover discrepancies
does not assume that a crossover will occur in exactly
the same location after the orbit parameters adjust in an OD solution.
In fact, the constraint equations are written in a way that allow
topography along the curves to be considered. The use
of topography can influence the adjustment of orbit and attitude parameters
so that the geolocated curves are relocated.
Conversely, since the attitude influences the geolocation of altimeter
bounce points, we can solve for attitude parameters using the the
altimeter crossovers.

The short passes of altimeter data (20-30 minutes for a revolution
of 11 hours and 38 minutes) available during SPO cause a complication
in the use of the crossovers.  The number of crossovers internal to a single arc
(four or six days in length) is limited and it becomes impractical
to rely solely on these `intra-arc' crossovers. Therefore, we
combined  six-day arcs  spanning all of SPO-1  into a
single simultaneous multi-arc solution using ground tracking and altimeter
crossovers. Bringing in `inter-arc' crossovers in this fashion makes it
possible to use hundreds of crossovers.

Two types of solutions were obtained using the altimeter crossovers in
SPO-1. In the first solution, 279 off-nadir crossovers (where either the
ascending or descending pass pointed more than 2$^\circ$ off-nadir) were
used to solve for attitude parameters, while the orbit was held fixed
to the orbit solution determined solely from the tracking data.
The 535 most nadir pointing crossovers were used simultaneously with the
ground tracking observations in an orbit solution.  In this
second solution, the attitude parameters were held fixed to the the
values obtained from the first (attitude parameter) solution.


Three parameters were estimated in the `attitude only' crossover solution.
The first is a time tag bias for the telemetered quaternions that describe
the spacecraft orientation. The other two are constant roll and pitch
offset parameters that describe the orientation of the MOLA instrument in
the spacecraft body frame.  The sense of these parameters is that a
a vector in the spacecraft body fixed frame is rotated to
the MOLA instrument frame by successively applying rotations in roll,
pitch, and yaw. The preflight values of the parameters were 0.021$^\circ$
for roll, -0.005$^\circ$ for pitch and 0.059$^\circ$ for yaw$^{21}$.
In flight, these parameters can also help compensate for offsets in the
the telemetered quaternions that describe the spacecraft
body orientation. Using the 279 SPO-1 off-nadir crossovers in
the `attitude-only' solution, we find a timing bias in the MGS
spacecraft telemetered quaternions of  1.160 seconds,
a roll offset of -0.0028$^\circ$, and a pitch offset of -0.0086$^\circ$.
In ref. $37$ it is demonstrated
that the use of these adjusted attitude parameters
improve the pointing of the MOLA instrument, at least during SPO-1.


To evaluate the effect of  the 535 nearly nadir altimeter crossovers
on the  determination of the MGS orbit, we compute the RMS orbit discrepancies
for each of the six-day arcs (which all overlap by 12 hours) after
obtaining the multi-arc solution.
These nadir crossover orbit solutions also included an  estimate for
a MOLA altimeter timing bias and we  obtained a value of 113.6 $\pm$ 1.0 msecs.
The crossover solution is compared
to a solution that relied solely on the DSN tracking data (Doppler and range).
(see Table 9). Note that the radial orbit overlaps for the `tracking-data-only'
solutions are superior (18 m compared to 30 m)
then those cited in the gravity model tests (compare with Table 6)
since empirical once-per revolution parameters were adjusted frequently
in the crossover arcs.

The crossovers improve the radial orbit consistency in SPO-1
by 15 percent (2.8 m), and 60 percent (600 m)  in total position.
Although the radial component is only slightly improved, the
along-track and cross-track components of the orbit are improved substantially.
This multiarc crossover solution strategy can also be applied in
SPO-2 and Hiatus, although the Hiatus period has no internal
crossovers. Hiatus arcs must be adjusted together with SPO arcs in
order to allow the crossovers to improve the Hiatus orbits.
Once mapping commences from the low-altitude, near-circular,
and polar orbit, sufficient altimeter data should be available to
rely solely on `intra-arc' crossovers.

\begin{table}
\begin{minipage}[t]{\textwidth}
\begin{center}
\begin{tabular}{|lcccc|}
\multicolumn{5}{c}{Table 9: RMS Orbit Overlaps Using Altimeter Crossovers}\\
\hline \hline
\rule{0mm}{3mm} & \multicolumn{4}{c|}{RMS Orbit Overlaps (m)}\\
Solution type  & Radial & Cross-track & Along-track & Total\\
\hline
No crossovers  & 18.5  & 907   & 505  & 1045\\
Include crossovers & 15.8& 347   & 201  & 402  \\
\hline
 \hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}


\section*{SUMMARY}
The Doppler tracking data from Mars Global Surveyor in Hiatus and the
Science Phasing Orbits have been incorporated into updated
models of the gravity field of Mars. These solutions, to 70x70
in spherical harmonics, have been used to compute refined orbits for
analysis of the MOLA altimeter data.  The RMS radial orbit differences
of overlapping arcs have been reduced from 160 to 200 m with the
{\it a priori} GMM-1 model to 30 to 35 meters with the latest MGS derived
solution, MGM0890.

For the first time we have used laser altimetry about another
planet in the form of crossovers as a tool to strengthen
determination of a spacecraft orbit.
The use of crossovers during SPO-1 slightly improved the orbits in
the radial direction, but improved them most especially
in the along-track and cross-track directions.

Further improvements in the orbits of MGS during Hiatus and SPO
will result from improving the macromodel to account for the articulation
of the solar arrays, and the incorporation of the altimeter crossovers
into the Mars geopotential solutions.


\section*{ACKNOWLEDGEMENTS}

We acknowledge Theodore Moyer  of JPL  for
providing information regarding the modelling of
the interplanetary one-way Doppler data;
Phil Tracadas  of MIT and Michael Griffin of Lockheed Martin
provided information to help define
the MGS macromodel. Susan Fricke of Raytheon
ITSS  provided computational support with GEODYN, and John McCarthy of
Raytheon ITSS supported the gravity field development with SOLVE.
We thank the NASA Center for the Computational Sciences (NCCS)
at GSFC for hosting our gravity model solution development on
their Cray J90 supercomputers.
This work was supported by the Mars Global Surveyor Project.


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\end{enumerate}

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