The goal of using an orbital gradiometer mission to provide an accurate (1 to 2 mgal), high resolution (1 by 1 deg), global map of the earth's geopotential is currently being investigated. This investigation involves the simulation of the satellite ephemeris and the corresponding gradiometer measurements which can be used in the study of various techniques and methodologies that were proposed to recover the parameters used in modeling the geopotential. Also, the effects on the mission of various time varying forces acting on the spacecraft were included in the studies

Bettadpur, S.

Lundberg, J. B.

Schutz, B. E.

Tapley, B. D.

English

NASA Technical Reports Server

N9o- o5 8
STUDIES OF AN ORBITAL GRADIOMETER MISSION
B. E. Schutz, J. B. Lundberg, S. Bettadpur, and B. D. Tapley
Center for Space Research, The University of Texas at Austin
Abstract
The goal of using an orbital gradiometer mission to provide an accurale (1-2 regal), high
resolution (1 ° by 1°), global map of the earth's geopotential is currently being investigated. This
investigation involves the simulation of the satellite ephemeris and the corresponding
gradiometer measurements which can be used in the study of various Techniques and
methodologies that have been proposed to recover the parameters used in modeling the
geopotential. Also, the effects on the mission of various time varying forces acting on the
spacecraft have been included in the studies.
Introduction
The goal of these studies is to create an accurate ephemeris and set of 'perfect' gradiometer
measurements to study various techniques to recover the parameters of the geopotential model
and to study the effects of various force model, ephemeris, and measurement uncertainties on the
recovery of the geopotential parameters. This research effort began by assuming a Geopotential
Research Mission (GRM) scenario involving a dual satellite configuration in which the principal
measurements are the relative range-rates between the satellites. In the initial simulations, a
geopotential model complete through degree and order 180 was used (Schutz et al., 1987). The
GRM simulation was extended to include a geopotential model complete through degree and
order 360 (Schutz et al., 1988). The satellites in the GRM scenario were assumed to be in
'frozen', polar orbits with a mean altitude of 160 km and a repeat ground track period of 32
sidereal days. For these simulations the earth was assumed to have a constant angular velocity
and a static geopotential.
To create as accurate an ephemeris as resources would permit, the equations of motion
were solved numerically using Encke's formulation and a class II, fixed mesh multistep algorithm
of order 10. The simulations were carried out on the CRAY X-MP/24 computer at the University
of Texas using single precision arithmetic which represents floating point numbers using a 48 bit
mantissa and a 16 bit exponent. The first simulation ($8705) was carried out using an integration
stepsize of five seconds and required 5.6 hours of CPU time. The second simulation ($8703) was
carried out using an integration stepsize of four seconds and required 19.2 hours of CPU time.
The nearly four fold increase in CPU time reflects the increase in the size of the geopotential
model from degree and order 180 to 360. In both simulations the ground track closed to within
two kilometers after 32 sidereal days (White, 1987).
In addition to creating these simulations, special studies were can'ied out on the effects of
solid and ocean tides, luni-solar and planetary gravitational forces, and the mass distributions
caused by ocean eddies (McNamee, 1986) on the relative range-rate measurements. A study of
the disturbance compensation system (DISCOS) was also carried out to verify the proposed
control law and estimates of fuel consumption (Antreasian, 1988).
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SCHUTZ ET AL.: Studies of an Orbital Gradiometer Mission
Simulation of a Gradionleter Mission
The products of the GRM simulation have been used to simulate the orbital gradiometer
mission. The orbit for the gradiometer mission is assumed to have the same characteristics as
GRM, i.e, a polar, frozen orbit with a mean altitude of 160 km and a repeat ground track period of
32 sidereal days. The ephemeris computed for the lead satellite of simulation $8703 is taken to
be the true ephemeris for the gradiometer mission. The true gradiometer measurements are
simulated by the gravity gradient tensor which is computed using the true ephemeris and a
geopotential complete to degree and order 360. The measurement interval is taken to be the same
as the ephemeris interval, i.e., four seconds.
Analysis of a Simulated Gradiometer Mission
The simulated true ephemeris and gradiometer measurements can be used to study the orbit
determination requirements of the mission, i.e., the effects of orbit uncertainties on the solution
for the geopotential parameters, and to study various techniques for recovering the geopotential
parameters using gradiometer measuremenLs. The simulation also provides a common data set
for the evaluation of results from different researchers and a basis for comparison of gravity
gradient computations.
Various tracking systems may be considered for this mission including GPS, satellite laser
ranging, PRARE, DORIS, or other suitable systems. Since some form of global tracking will be
required for the mission, GPS will probably be part of the overall tracking system (Yunck et al.,
1986). Whichever tracking system is used, there are two possible approaches that can be used to
determine the orbit of the satellite and recover the geopotential parameters. The first, or
gradiometer, approach uses only the tracking system information to establish a nominal orbit for
the mission and, once the nominal orbit is computed, uses the gradiometer measurements to
recover the geopotential parameters. The second, or dynamical, approach involves using the
tracking data along with the gradiometer measurements to determine the orbit and recover the
geopotential coefticients simultaneously. The fundamental difference between these two
approaches is that the lirst approach is similar to creating a gravity surface by adjusting the
geopotential parameters while the second approach is dependent on the orbit perturbation
frequency spectrum including the amplified effects due to resonance.
To illustrate the gradiometer approach, a simulation was carried out in which the
observations were the gradiometer measurements computed along the true orbit using only the
geopotential coefficients from degree and order 10 through degree and order 15. The gradiometer
measurements were used with a least squares algorithm in attempt to recover the geopotential
coefficients for arc lengths of up to 60 hours. To represent the effect of orbit uncertainties on the
solution, the partial derivatives of the observations with respect to the coefficients were evaluated
along a nominal orbit which is randomly perturbed in the radial direction from the true orbit. The
accuracy of the solution for the coefficients is evaluated by computing the absolute value of the
normalized differences between the estimated coefficients and the true coefficients, i.e.,
8C=
CTRIJ E -- CESTIMATE D
C I'RUE
Figure 1 shows the results for the sohmon of CI4,12 which is typical of all the solutions. The
effects of radial orbit uncertainties (lc_) of 0 cm, 5 cm, 30 cm, and 50 cm are represented in
177
SCHUTZETAL.:StudiesofanOrbitalGradiometerMission
Figure1. Forthecaseof noorbiterror,ninetotensignificantfiguresof C14,12 are recovered. For
the case of 50 cm radial uncertainty, six to seven significant digits are recovered. This is a very
limited example since the solution involved a relatively small number of coefficients and a
relatively low degree and order coefficient.
To illustrate the dynamical approach, nominal orbits were determined using the entire 32
day, true ephemeris ($8703) as simulated tracking data and the Goddard Earth Model 10B
(GEM-10B) as the a priori gravity model while solving for selected resonance coefficients. The
selected resonance coefficients were the first two pairs (Cnm and Snm) of orders 33, 49, and 82.
The results of these solutions are given in Table 1. The results indicate that the resonance terms
will have amplified effects in the dynamical solution for the gravity field.
Future Research
Future research of the orbital gradiometer mission includes studies of the gradiometer or
dynamical approaches to determine the orbit of the satellite and recover the geopotential
parameters, studies of the effects of various error sources on the solutions including tidal errors,
nontidal ocean phenomena, and spacecraft attitude errors, and to investigate the adequacy of a six
month mission.
Acknowledgemenls
This research is sponsored under NASA Grant NAG5-528. The authors would like to
acknowledge the contributions and support for this project by the Center for High Performance
Computing at the University of Texas.
References
Antreasian, P. G., "An Orbit Simulation Study of a Geopotential Research Mission Including
Satellite-to-Satellite Tracking and Disturbance Compensation Mechanism," Thesis,
Department of Aerospace Engineering and Engineering Mechanics, The University of
Texas at Austin, May 1988.
McNamee, J. B., "The Perturbation of Low Altitude Satellite Motion Due to the Presence of
Ocean Eddies," Thesis, Department of Aerospace Engineering and Engineering Mechanics,
The University of Texas at Austin, December 1986.
Schutz, B. E., B. D. Tapley, J. B. Lundberg, and P. Halamek, "Simulation of a Geopotential
Mission for Gravity Studies," manuscripta geodaetica, Vol. 12, 1987, pp. 51-63.
Schutz, B. E., J. B. Lundberg, L. K. White, and P. G. Antreasian, "Developments in the
Simulation of a Geopotential Research Mission," AIAA/AAS Astrodynamics Conference,
Minneapolis, Minnesota, August 15-17, 1988, pp. 323-328.
White, L. K., "Simulation and Analysis of a Geopotential Research Mission," Center for Space
Research, Department of Aerospace Engineering and Engineering Mechanics, The
University of Texas at Austin, 1987.
Yunck, T. P., S. C. Wu, and J. T. Wu, "Strategies for Sub-decimeter Satellite Tracking with
GPS," IEEE Position Location and Navigation Symposium, November 1986, Las Vegas,
Nevada, pp. 122-128.
178
SCltUTZ ET Air.: Studies of an Orbital Gradiometer Mission
RMS (meters)
radial
transverse
normal
Table 1
Resonant Effects on the Orbit Residuals
with order 82' with orders 33,82" with orders 33,49,82"
64.9 60.4 58.1
619.5 180.3 148.6
31.8 21.7 33.8
* after small adjustments in position vector
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1 t 1 1 1
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'i /
/ o cM RADIALORBIt ERROR
li/ >_--_---K 5 CUS RADIAL ORBIT ERROR
I_------m---_ll 30 CMS RADIAL ORBIT ERROR
Z_--_--_ 50 CMS RADIAL ORBIT ERROR
10.00 20.00 30.00 40.00
TZME (HRS)
50.09 60. O0
Figure 1
Normalized errors in the solution for C14,12 as a function
of radial orbit uncertainty and orbital arc length
179
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