JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 99, NO. B2, PAGES 2841-2851, FEBRUARY
10, 1994
Global gravity field recovery from the ARISTOTELES
satellite
mission
P. N. A.M.
Visser, K. F. Wakker, and B. A. C. Ambrosius
Faculty of Aerospace Engineering, Delft University of Technology, Delft, Netherlands
Abstract. One of the primary objectives of the future ARISTOTELES satellite
mission is to map Earth's gravity field with high resolution and accuracy. In order to
achieve this objective, the ARISTOTELES satellite will be equipped with a gravity
gradiometer and a Global Positioning System (GPS) receiver. Global gravity field error
analyseshave been performed for several combinations of gradiometer and GPS
observations. These analyses indicated that the bandwidth limitation of the gradiometer
prevents a stable high-accuracy, high-resolutiongravity solution if no additional
information is available. However, with the addition of high-accuracy GPS
observations, a stable gravity field solution can be obtained. A combination of the
measurementsacquired by the high-quality GPS receiver and the bandwidth-limited
gradiometer on board ARISTOTELES will yield a global gravity field model with a
resolution of less than 100 km and with an accuracy of better than 5 mGal for gravity
anomalies and 10 cm for geoid undulations.
Introduction
With the coming of the space age, technologieshave been
developed that enabled a systematic observation of the
entire Earth. In order to understand the complex nature of
the dynamic Earth, a multidisciplinary approach is required
in which the mapping of Earth's gravity and magnetic fields
with high resolution and accuracy is a prerequisite [Lainbeck, 1990]. High-resolution models of the gravity and
magnetic fields of Earth will help in modeling and understanding its structure and the driving forces behind plate
tectonics, mantle convection, lithospheric motions, etc.
[NASA, 1987]. In addition, a high-resolution gravity field
model will help to establish a physically meaningful reference surface for the oceans, in this case the geoid. With the
extensive altimeter data sets of past missions like GEOS 3,
Seasat, and Geosat, current missions, TOPEX and ERS 1,
and future altimeter missions, ocean variations with respect
to this surface can be studied on different geometrical and
temporal scales. In addition, by combining altimetry and
gravity, ocean currents can be deduced and possibly longterm effects like global sea level change can be studied.
A mission dedicated to the high-resolution mapping of
Earth's gravity and magnetic fields is the European Space
Agency's (ESA) ARISTOTELES
satellite mission. This
mission has been under study in Europe since the early
1980s.
The
name
of
the
mission
is an
abbreviation
of
"applications and research involving space techniques to
observe the earth's fields from low earth orbiting satellites,"
and the name already indicates that the aim of the mission is
to provide global models of Earth's gravity and magnetic
fields with high spatial resolution and accuracy [European
Space Agency (ESA), 1991; Schuyer et al., 1992]. In 1987,
ARISToTELES
became an ESA potential mission, which
has led to detailed satellite and mission design studies by the
European industry. In addition, almost all of the European
experts in the recovery of gravity and magnetic fields from
space have taken part in extensive data processing simulations to verify the feasibility of the concept. Although the
ARISTOTELES mission has been abandoned right now and
a possible realization delayed, this does not mean that this
mission should not be studied, since the ARISTOTELES
concept still has the promise of answering many open
questions in scientific fields addressed above.
According to the original plans, the launch of the satellite
into a 400-km-altitude
orbit with an inclination
of 96 ø would
have occurred in 1997, close to the next solar activity
minimum. Following launch, the first 2 months of the mission at 400 km altitude will be dedicated to checking out the
spacecraft subsystems and to the calibration of the instruments. Thereafter, the orbit altitude will be decreased to 200
km to measure both short-wavelength gravity and magnetic
anomalies with high sensitivity for a period of about half a
year. This altitude is a compromise, since the amplitude of
the effects of higher harmonics of both fields decreases
rapidly with increasing altitude, while the rate of orbital
decay increases dramatically with decreasing altitude. The
last 15 days at 200 km altitude will be preceded by a change
of the inclination to about 92ø to improve the coverage over
the poles. The last phase of the mission, at an altitude of 480
km, will last at least 3 years to measure the long-term
changes (secular variations) in the geomagnetic field. Because of budgetary problems of ESA the launch will probably not occur before 2004.
In this paper only the recovery of gravity field information
at the altitude of 200 km will be addressed. For the gravity
part of the mission the primary objective is to determine a
global high-accuracygravity field model with a resolution of
about 100 km, which is equivalent to a spherical harmonic
expansion complete to degree and order 180. For this
Paper number 93JB02969.
resolution,
theaccuracy
strivedfor is 5 mGal(1 Gal -- 10-2
m/s2)for gravityanomalies
and10cmfor geoidundulations.
0148-0227/94/93 JB-02969505.00
It is emphasized that although current gravity field models
Copyright 1994 by the American Geophysical Union.
2841
2842
VISSER
ET AL.:
ARISTOTELES
GLOBAL
exist complete to degree and order 360 [e.g., Rapp and Cruz,
1986; Rapp and Pavlis, 1990] these models are not very
accurate at the shorter wavelengths. At least four areas in
Earth sciencesand applicationswill greatly benefit from the
vast improvement in our knowledge of the gravity field that
will result from the ARISTOTELES mission [NASA, 1987]:
(1) geodynamics,(2) oceanography,(3) climate and sea level
change studies, and (4) geodesy, orbit mechanics, and navigation. In particular, the ability to accurately model the
shorter-wavelengthgeoid undulations will be of major importance, as it will facilitate for the first time the detection of
oceanic features with wavelengths shorter than the dynamic
sea surface topography that can be recovered nowadays
from satellite altimetry [Denker and Rapp, 1990;Engelis and
Knudsen, 1989; Marsh et al., 1989b; Nerem et al., 1988;
GRAVITY
FIELD
RECOVERY
sphericpropagationdelay can be recovered [Gurtner, 1985].
Apart from the C/A code and P code signals,the carrier itself
may also be used for ranging. Although the system was not
designed for this application, it was already early in the
development of the system realized that the highly stable
oscillators on board of the GPS satellites would allow very
precise range measurementson this signal. The precision of
these measurementscan be as high as 5 mm.
The most important applications of the on-board GPS
receiver could be the support of a precise orbit determination of ARISTOTELES
and the contribution
of additional
information about the low- and medium-degree part of the
gravity field of Earth. This could help to improve the
accuracy of current (low-degree) sphericalharmonic expansionsof the gravity field to a higher level, so that low-degree
Visser, 1992].
errors will not obscure the modeling of the higher-degree
The primary gravity measurement device to be imple- field from the gradiometer measurements. In addition, the
mented on board ARISTOTELES satellite will be a gravity GPS measurementswill be capable of delivering the missing
gradiometer, which senses Earth's gravity tensor at the
low-frequency gravity field information unobservableby the
satellite altitude. This gradiometer will consistof electrostatband-limited gradiometer.
ically suspended proof masses, which are separated by a
The capability of GPS to achieve this complementary
distance of the order of 1 m. The differences between the
objective has been shown extensively in several studies,
accelerations of these proof masseswill be measured. These
especially the studies related to the Gravity Probe B and
differential measurementswill have an accuracy of the order
TOPEX/Poseidon mission [Smith et al., 1988; Wu and
of 0.01EU (1 EU = 10-9 S-2) [Consortium
for theInvestiYunck, 1986a, b]. In the study described in this paper, it has
gation of Gravity Anomaly Recovery (CIGAR), 1989, 1990]
been investigated whether a GPS receiver has the capability
for gravity tensor components along the so-called sensitive
to also provide information about the high-degree part of
axes, i.e., axes in a direction perpendicular to the satellite
flight direction. Measurements in the flight direction are Earth' s gravity field, thus serving as a supplementarydevice
expected to be less accurate, because these measurements to the gradiometer. The expected accuracy of carrier phase
will be corrupted by unmodeled atmospheric drag fluctua- measurementsby future space-borne GPS receivers seems
tions. Because of the limited bandwidth of the gradiometer; to open the possibility to achieve this objective [Ambrosius
that is, the gradiometer can only observe signals with a et al., 1990].
The actual application of the GPS signalsto derive gravity
frequency between 0.005 and 5 Hz, the gradiometer is unable
field
information requires the availability of precise ARISto deliver information about the low-frequency part of the
orbits. Because the GPS satellites track the
gravity field, i.e., below degree 20-27, and about particular TOTELES
medium- and high-degree gravity field spherical harmonics, ARISTOTELES satellite continuously, the true orbit of this
i.e., from degree 28 to 180. The reason for this is that satellite is actually measuredby the GPS signals.Of course,
especiallyfor a polar orbit, high-degreesectorialgravity field the quality of this orbit depends on the type of GPS data
constituents may cause primarily low-frequency orbit per- beingprocessed.In the present study it is implicitly assumed
that when GPS data are used to derive gravity field informaturbations (as will be shown).
A supplementary and complementary device to obtain tion, the same GPS data are used to compute the ARISTOgravity field information will be an on-board Global Position- TELES orbit in a dynamic mode, applying the best geometing System (GPS) receiver, yielding Satellite-to-Satellite ric and dynamic models. That orbit serves as a reference
Tracking (SST) pseudo-range and carrier phase measure- orbit to which the "true" orbit, reconstructed in a more-orments. A pseudo-rangemeasurementis obtained by measur- less geometric mode from the continuous stream of GPS
ing the transit time of coded radio frequency signalstrans- data, is compared. The differences between both orbits
mitted by the GPS satellites and recorded by a GPS receiver primarily reflect the so-called epoch state vector errors of
and by multiplying this transit time with the speed of light the ARISTOTELES orbit that was computed in the dynamic
[Arnbrosiuset al., 1990]. There are two types of pseudo- mode and the effectsof errors of the dynamic modelsapplied
random noise codes modulated on carrier signalsat the L in the computationof that orbit. It is well known that the
band frequencies(L 1 = 1.575 and L 2 = 1.227 GHz). The epoch state vector errors and most dynamic model (e.g.,
first code is the so-called "civilian access" (C/A) code, surface forces) errors produce mainly errors with frequenwhich is primarily intended to ease the acquisition of the cies of up to 2 cyclesper (orbital) revolution (cpr). Therefore
second,more precise, P code. The first code is the only code the orbit differences mentioned above are assumed to be
ottlciallyavailableto "civilian" users.This codehasa "chip filtered, and the resulting signalwith frequenciesabove 2 cpr
rate" of about 1 MHz, and in conjunction with this, the is assumed to represent primarily the effects of errors of
highest ranging accuracy that can be achieved from these Earth's gravity field model. These filtered orbit differences
measurementsis nowadays of the order of 5 m. The P code are called the "orbit perturbations" in the remaining part of
has a "chip rate" of about 10 MHz, and therefore the this paper. The precision with which these orbit perturbaaccuracy of these measurements is of the order of 1 m.
tions can be measured by the GPS signal is, of course, a
Moreover, if P code measurements are available in addition function of the precision of these measurements. In this
to C/A code measurements, the so-called first-order iono- paper, the root-mean-square (RMS) of the history of the
VISSER
ET AL.:
ARISTOTELES
GLOBAL
errors in the measured orbit perturbations is called the
"accuracy of the (measured) orbit perturbations."
The paper describes a method to estimate the formal error
of the global gravity field harmonic coefficients from data
acquired on board of the low-altitude ARISTOTELES satellite and presents some results of the application of this
method. These data were assumed to consist of gradiometer
data and precise GPS SST range measurements. The GPS
GRAVITY
FIELD
RECOVERY
2843
inclination of 96.3 ø [CIGAR, 1989]. In that case, no ARISTOTELES measurements will be available for the polar
caps. To analyze the effect of a nonpolar ARISTOTELES
orbit, a few gravity field recovery error analyses were also
performed for inclinations of 92ø, 93ø, and 94ø.
,
SST measurementswere assumedto deliver precise infor-
Modeling of the ARISTOTELES
Measurements
If the gravity field potential is described by a spherical
mationaboutthe (filtered)ARISTOTELESpositionpertur- harmonic expansion, the unknown part of the gravity field
bations in the radial, along-track, and cross-track directiong.
These perturbations may be related to the (global) gravity
field harmonic coefficients by a linear perturbation theory
[Kaula, 1966; Schrama, 1989; Visser, 1992].
For a global gravity field recovery, the requirement of a
100-km resolution implies the setup of normal equations for
more than 30,000 harmonic coefficients. With the expected
abundance of gradiometer and GPS SST range measurements, it will be obvious that it is an immense task to
compute and solve the normal equations for this number of
unknown harmonic coefficientsin a straightforward manner.
Therefore various simplifying assumptionsare usually introduced. In this study the starting point for all simulations of
formal error estimates was a circular reference repeat orbit.
For the GPS SST range measurements, it was assumed that
the measurements
to each GPS satellite
in view
of ARISTO-
TELES (with the full GPS configuration always at least 5
GPS satellites) will be transformed to measurements in the
ARISTOTELES satellite along-track, cross-track, and radial
directions (the x, y, and z directions of the satellite localhorizontal, local-vertical reference frame). Also the gradiometer
measurements
local frame.
were
The time interval
assumed
between
to be available
successive
in this
measure-
ments was taken constant during a complete repeat period.
With the preceding assumptions it can be shown that the
normal matrix will have a special structure [Colombo, 1984;
Schrama, 1990, 1991; Visser, 1992]. This phenomenon will
be addressedin this paper, together with an investigation of
the signal contents of ARISTOTELES orbit perturbations
and gradiometer measurements. After this, the concept of
global gravity field recovery error analyseswill be described.
In addition, the issue of the quality of the ARISTOTELES
GPS receiver required to satisfy the mission objectives will
be addressed. As mentioned before, this GPS receiver must
deliver the information for a precise orbit computation and in
the secondplace for an accurate modeling of the gravity field
at low degrees. In the third place a GPS receiver will be
necessary to deliver the low-frequency gravity field information that will be missing from the gradiometer measurements
due to the bandwidth limitation of the gradiometer instrument. As indicated before, the bandwidth limitation of the
gradiometer hampers the gradiometer to observe signals
below a frequency of 0.005 Hz. Thus, with gradiometer
measurements alone, the low-frequency gravity field cannot
be determined. Therefore gravity field recovery error analyses were performed for several combinations of ARISTOTELES orbit perturbations information, as derived from the
GPS SST range measurements, and gradiometer measurements. The effect of the gradiometer bandwidth limitation
was included in the investigations.
For all analyses it was assumed that the ARISToTELES
satellite will fly in a polar orbit at about 200 km altitude.
However, in reality, the orbit will be slightly nonpolar at an
potential, denoted by T, may be expressed by [Kaula, 1966]
(AClm COSmA
d- ASlm sin mA)Ptm(Sinc•),
(1)
where /x is the gravity parameter of Earth, a e is the mean
equatorial radius of the Earth, Plm is the fully normalized
Legendre polynomial of degreeI and order m, and AClm and
ASlm are the (unknown) fully normalized gravity field harmonic coefficients. The satellite position in the rotating
geocentric coordinate frame is denoted by the radius r, the
longitude A, and the geocentric latitude •b. From (1), any
other gravity field ip_ducedsignal along a near-circular satellite orbit can be derived. In this paper these signals are the
gravitytensorcomponents
AFij or orbitperturbations
Ari,
where the indices i and j denote the radial, along-track, or
cross-track direction. In order to find these relations, (1) can
be transformed into an equation that contains orbital elements [Kaula, 1966]. The orbit selected for ARISTOTELES
is a circular repeat orbit; that is, the eccentricity e is equal to
zero. This orbit will be close to the real anticipated ARISTOTELES orbit [CIGAR, 1989, 1990]. For such an orbit, the
unknown part of the gravity potential can be written as
T=•ae t•
=2
FlmpSlmp(OO
+M,•- 0),
m=0p=O
(2)
where
Slmp(OO
q-M, •-
O)
A•lm
]l-meven
cos [(/-
--Ai•mJl-m
odd
2p)(w + M) + m(l•-
0)]
even
IA'•lml
l-m
sin
[(/-2p)(w
q-M)q-m(l•-0)].
q-LA•lmJl-m
odd
(3)
Equations (2) and (3) use the so-called Kepler elements.
These are the satellite's orbital semimajor axis a, eccentricity e, argument of perigee w, inclination i, right ascensionof
ascending node 1•, and the mean anomaly M. The Green-
wich hourangleis denotedby 0, whileElmp is a function
depending on the orbital inclination i only [Kaula, 1966;
Schrama, 1989]: The gravity field potential is truncated at a
certain maximum degree r/max. The value for r/maxwill be
specified later.
The Hill equations represent a set of linearized equations
2844
VISSER
ET AL.: ARISTOTELES
GLOBAL
for the motion relative to a circular referenceorbit [Dunning,
1973]. If T representsthe disturbingpotential, the equations
for the radial, along-track, and cross-trackorbit perturba-
GRAVITY
FIELD
RECOVERY
1+1
/x(1+1)(1+2)
(•_•)
I"zzImp= -'
ae
a
•
FlmpSlm
p
(9)
tionsbecomefor a generalterm Tlm
p [Colombo,1989;
Schrama, 1989; Visser, 1992;Zandbergen, 1990]:
4p-3/-
F zy,lmp
....
ae
2a
1
+
Arz,lmp
- a Flmpflmp
2limpq- 1
sini - F}mp
4p-t+l
1
q-2(limp
- 1)SImp
At'x,lmp
=a
Flm
p
q-
(4)
+
limp
4p-31-
{([m-(l-2p)
cos
i]Flmp.)S•l+l
p
([m--(l--2p)
COS
i]FlmP)S•l_l)
}
1
---
•
(10)
sini + F•mp
Fyy,lmp
- ae a2
Flmp((l-2p)2- (l + 1)2)Simp.
l-4p-1
f lmpq-1
f lmp- 1
Smv
(5)
Ary,lmp
=• a f-•mv
•,sin
i
Just as (4)-(6) for the orbit perturbations, (9)-(11) become
true Fourier seriesfor an exact repeat orbit.
Structure
Flmp
-Fjmp)
s•I+I)mp
- x,
sin
i[(l-2p)
cos
i-m]
of the Normal
Matrix
It hasbeen statedthat for a circularrepeat orbit the linear
relations connectingthe unknown harmonic coefficientswith
the gradiometerand GPS observations(or orbit perturbations) can be representedby Fourier series. If many observations are made during a complete repeat period and a
constant samplingrate is applied, it can be shown that the
normal matrix becomesblock diagonalwhen organizedper
order [Colombo, 1984; Rummel, 1990; Schrama, 1990, 1991;
Visser, 1992]. For different orders m of the gravity field
coefficientsthe frequencies of the orbit perturbations or
gradiometer signalcaused by such coefficientsbecome decorrelated (provided a perfect coverage). The greatest dimension of these blocks is equal to approximately half the
maximum degree of the gravity field harmonic expansion
+F}mp)S•l-1)mp],
(6)
where
m
f lmp--1- 2p
(7)
nF
S•mp(Oo
+ M, fl- O)
l-m
even
=[A•lm
l-m
sin [(/-
2p)(to + M) + m(fl-
0)]
odd
Ai•lm]l-m
even
cos [(/-
that is to be determined.
matrix transforms
dimensions much
The inversion
of the total normal
to an inversion of block matrices with
smaller than the total number of un-
2p)(to + M) + m(11- 0)]
knowns. This prevents the necessity of long and costly
computer runs. An additional advantageis that only a small
(8)
part of the total normal matrix (only the blocks on the
and F}mp is the derivativeof Flmp with respectto the diagonal)has to be stored. It is believed that the previous
inclination i. The factor nr is the number of nodal orbital procedure can also be applied in a real gravity field model
revolutions per day. These equations establish the relation recovery experiment using real gradiometer and GPS obserbetween the disturbingpotential T and the orbit perturba- vations. With the ARISTOTELES orbit being near-circular
tions in the radial, along-track, and cross-trackdirections.In and with the expected nearly continuousglobal coverage of
this paper these perturbations are assumedto be recovered gradiometerand GPS measurements,it is expected that an
with great precision from the GPS measurements.Because approach can be developed in which the normal equations
the orbit is an exact repeat one, theseequationsbecometrue for the gravity field unknowns can be forced into a block
Fourier series [Colombo, 1984]. Several studies have shown diagonalmatrix and an iterative procedurecan be applied.
the validity of these equations [Schrama, 1989; Visser,
[AClmJl-m
odd
1992].
Similar linear relations can be established for the second
derivatives
of the gravitypotentialFij for a circularreference orbit [Visser, 1992]. Because in the ARISToTELES
mission no precise gradiometer measurementswith a com-
ponent in the along-track (the so-calledless sensitiveaxis)
directionwill be obtained,as this is the directionof the
relatively large and fluctuatingatmosphericdrag, only rela-
tionshaveto be established
for Fzz,FzyandFyy.Rummel
[1990], Schrama [1990, 1991], and Visser [1992] derived
following relations:
Orbit Perturbations and Tensor Signal
A quantitative analysiswas performed to show the information content of orbit perturbations in the satellite orbit
frame (radial, along-track, and cross-track directions) and of
the gradiometric signal along the circular reference repeat
orbit. Use was made of the equations (4)-(6) and (9)-(11),
which are the relations between orbit perturbations and
gradiometercomponentsand a certain gravity field harmonic
coefficient. The OSU86F gravity field model [Rapp and
Cruz, 1986]truncated at degree 180 was used to computethe
VISSER
ET AL.:
ARISTOTELES
GLOBAL
Fourier series of the orbit perturbations and of the gradiometric signal along the satellite orbit. The cutoff at degree
180 is justified by the fact that 100-km resolution of the
gravity field model is to be recovered from the ARISTOTELES mission. The effect of the gradiometer bandwidth (i.e.,
that certain frequencies are not observable) on the signal
content was also investigated. The ARISTOTELES satellite
was assumedto be in a 200-km-altitude polar orbit (i - 90ø).
The repeat period was taken equal to 91 days, in which the
satellite completes 1479 revolutions.
Orbit
GRAVITY
FIELD
ARISTOTELES.
Figure 1 displays the frequency spectra of the orbit
perturbations for this part of the gravity field. It was found
that the root-sum-square (RSS) of all orbital perturbation
componentsin the radial, along-track, and cross-track directions is 14 m, 778 m, and 12 m, respectively. The high value
for the along-track perturbations is caused by very high
peaks close to 0 cpr, which represent very long period
perturbations. As shown in Figure 1, the frequency spectrum
of the radial orbit perturbations is more spread over the
higher frequencies compared to the frequency spectra of the
along-track and cross-track perturbations. Above about 30
cpr, the amplitudes of the cross-track perturbations are
smaller than those of the radial perturbations, while the
amplitudes of the along-track perturbations drop very
quickly for frequencies above 10 cpr. Therefore it is expected that if a gravity field is recovered from radial orbit
perturbations, the errors of the recovered coefficients, especially of the high-degree coefficients, will be smaller than
when the gravity field would have been recovered from
either along-track or cross-track perturbations.
2845
1000
100
10
1
0.1
1o
20
30
40
50
60
frequency (cpr)
Perturbations
The frequency spectrum of the orbit perturbations, as
described by the solution of the Hill equations, was computed using the OSU86F gravity field model from degree 37
to 180. The part of the OSU86F gravity field below degree 37
was neglected because it has been shown extensively in
other studies [Colombo, 1989; Smith et al., 1988; Wu and
Yunck, 1986a, b] that the GPS measurements acquired on
board ARISTOTELES can be used to solve this part of the
gravity field with great precision. Attention to low-degree
gravity field recovery from orbit perturbations will be given
below. Another important question that will be addressedis
whether it is possibleto recover the gravity field for degrees
above degree 36 from GPS SST measurementsacquired by
RECOVERY
I
lOOO
I
cross-track
100
10
1
0.1
10
20
30
40
50
60
frequency (cpr)
000
along-track
100
10
o.
..........
10
20
30
40
,50
60
frequency (cpr)
Figure 1. Spectra of ARISTOTELES orbit perturbations
using the OSU86F gravity field model from degree 37 to 180
for the (top) radial, (middle) cross-track, and (bottom) alongtrack directions.
orbit perturbations of the ARISTOTELES satellite. Figure 2
also shows that the amplitudes of the gradiometer compo-
nentFzz are largerthan for the other components.In reality,
the gradiometer can not observe signals with a frequency
below about 27 cpr due to the limited gradiometer bandwidth. Because gravity field terms with a degree below 28
almost only cause gradiometer signals with a frequency
Gradiometer
Measurements
below 28 cpr (equations (7) and (9)-(11)), the part of the
The Fourierseriesfor Fzz, I'zy, andI'yy werecomputed gravity field below degree 28 is unobservable from the
using the OSU86F gravity field model up to degree and order gradiometer measurements.To study the effect of the limited
180. The frequency spectra are displayed in Figure 2. From gradiometer bandwidth, the RSS of the amplitudes of the
Figure 2 it is obvious that compared to orbit perturbations, frequency spectrum was computed for both excluding and
the larger part of the power of the gradiometric signal can be including the part of the OSU86F gravity field model below
found at higher orbital frequencies (note that the vertical degree 28. However, not only the entire part of the gravity
scale of Figure 1 is logarithmic and that of Figure 2 is linear).
field below degree 28 is unobservable, also a part of the
This concentration at the higher frequencies becomes even gravity field above degree 28 is unobservable, because
more pronounced if, as in the computation of the frequency gravity field terms with degree higher than 28 also produce
spectra of the ARISToTELES orbit perturbations, only the gradiometer signals with a frequency below 27 cpr.
part of the OSU86F gravity field above degree 36 is used in
For example, sectorial coefficients almost only cause a
the computation of the gradiometer frequency spectra. Thus gradiometric signal in the very low-frequency band (at least
the gradiometric signal is expected to deliver more informa- below 27 cpr) for a polar orbit. Figure 3 supports this
tion about the high-degree terms of the gravity field than the conclusion. In Figure 3 the RMS of the ratio of the gradio-
2846
VISSER
I
ET AL.'
ARISTOTELES
GLOBAL
i
Fzz
0.04
E 0.02
0
20
40
60
80
100
120
140
160
I
Global Gravity Field Recovery Error Analyses
I
Fzy
0.02
0
20
40
60
80
100
120
RECOVERY
180
frequency (cpr)
0.04
FIELD
28 is included are equal to the RSS values for the case where
this part of the gravity field is excluded. This is in agreement
with the assumptionthat the gravity field coefficientsbelow
degree28 hardly produce any signalwith a frequency above
27 cpr. Therefore it is concluded that the GPS SST range
measurementswill be needed to improve our knowledge of
the low-degree part of the gravity field and to deliver the
low-frequency information of the higher-degree,especially
sectorial, gravity field terms.
0.06
"-'
GRAVITY
140
160
180
frequency (cpr)
Fyy
In this section, results of global gravity field recovery
error analyses will be discussed. In the first part of this
section, attention will be paid to gravity field recovery error
analysesfrom orbit perturbations, which are assumed to be
determined from GPS SST range measurements. In the
analysesdescribedin the secondpart, gradiometer measurements were the input of the gravity field recovery error
analyses, and the effect of the gradiometer bandwidth limitation will be analyzed. It will be shown that this limitation
will prevent a stable gravity field solution. This is in agreement with the results given by $chrama [1990, 1991]. In
addition, results for other orbit inclinations will be presented. In the last part, recovery experiments from combinations of orbit perturbations information and gradiometer
measurements
0.04
will be described.
It will be shown that the
latter combination is a very powerful concept that seems
capable of solving bandwidth-related problems. In all simulations, it was assumed again that the ARISTOTELES
satellite
0.02
was in a circular
orbit
at 200 km altitude
with
a
nominal inclination of 90ø, unless another value for the
inclination is specified.It was assumedthat one complete
repeat period of observations was available (i.e., 3 months).
0
0
20
40
60
80
100
120
140
160
180
GPS Measurements
frequency (cpr)
First, the question is addressedwhether it is possible to
determine high-accuracy low-degree gravity field models
Figure 2. Spectra of the ARISTOTELES gradiometer
from GPS SST range measurementsonly. Four cases have
components
(top)Fzz,(middle)Fzy,and(bottom)Fyyusing been investigated, which were selected to reveal the effect of
the OSU86F gravity field model complete to degree and
the accuracy of the recovered orbit perturbations and the
order 180.
addition of a priori gravity field information. The accuracy of
the (filtered) recovered orbit perturbations will depend on
metric signalabove 27 cpr and the completesignalcausedby
the applied GPS SST measurementtypes. For example, as
the harmonic coefficients is displayed as a function of the
stated before, the precision of C/A code measurementsis of
degree minus order (l - rn). It is concluded that for
coefficients close to the sectoffal band (l - m = 0) the
largest part of the gradiometric signal content is below 27
cpr. To study the effects of this phenomenon,the computalOO
tion of the RSS of the amplitudesof the gradiometerspectra
80
was performed both for the full spectra and for the case in "•
which a cutoff frequency(cof) of 27 cpr was applied, i.e., in
o
60
the computation of the RSS values those amplitudesthat
•
40
belong to a frequency below 27 cpr were not included. The
I
I
I
--
o_
RSS values for four cases are listed in Table 1. The exclusion
of the gravity field part below degree 28 obviouslyreduces
the RSS values by about 30%, if the cutoff frequencyis not
applied.For example,the RSS value for Fzz reducesfrom
0.40 EU to 0.27 EU. If also the frequency cutoff is applied,
the RSS value drops an additional 22%. For the other
E
2O
I
0
0
50
I
I
lOO
15o
degree minus order I-m
Figure 3.
Effect of the gradiometer bandwidth on the
components, the RSS amplitudes also show a decrease of powerof the Fzz gradiometerterm, per harmoniccoefficient.
about 50% if the frequency cutoff is applied. On the other The RMS of the ratio between the gradiometric signalabove
hand, when the frequency cutoff is applied, the RSS values 27 cpr and the complete signalis plotted versus the quantity
for the casewhere the part of the gravity field below degree I - m.
VISSER
ET AL.:
ARISTOTELES
GLOBAL
GRAVITY
FIELD
RECOVERY
2847
Table 1. RSS Amplitude of the Gravity Tensor Components at the
ARISTOTELES Altitude for the Cases That the OSU86F Gravity Model
Complete to Degree and Order 180 or From Degree 28 to 180 Is Applied
Degree 1-180
COF, cpr
0.0
27.0
Degree 28-180
Fzz,EU
Fzy, EU
Fyy,EU
Fzz, EU
Fzy, EU
Fyy,EU
0.40
0.21
0.28
0.11
0.24
0.08
0.27
0.21
0.20
0.11
0.17
0.08
Also, for both cases the effect of applying a signal cutoff frequency (COF) of 27 cpr is
shown.
the order of 5 m, while the P code measurements have a
precision of the order of 1 m. The inherent precision of
carrier phase measurements is on the millimeter level. In all
cases, it is assumedthat uncertainties of Earth' s gravity field
model are the only contributors to orbit perturbations with a
frequency above 2 cpr. The values for the orbit perturbations
quoted below therefore only refer to orbit perturbations at 2
cpr or higher, unless explicitly specified otherwise.
Case 1. For this case, it was assumed that future GPS
receivers may deliver P code and C/A code measurements,
from which it will be possibleto determine the ARISTOTELES orbit with an accuracy of 3 m in the radial, along-track,
and cross-track directions, including perturbations below 2
cpr. It was assumed that this accuracy level holds for all
state vectors spaced at 1-min intervals during a complete
repeat period. From this data set of orbit perturbations, a
gravity field model complete to degree and order 36 has to be
recovered. The results of the analyses, in terms of RMS
formal harmonic coefficient error estimates per degree, are
displayed in Figure 4. The results clearly show that an
accuracy of 3 m for the orbit perturbations is not sufficient to
determine a high-accuracy low-degree gravity field model, if
no additional information is available. For degree 20 and
higher, the estimated errors of the gravity field harmonic
coefficientsbecome larger than the magnitude predicted by
Kaula's [1966] rule of thumb, which gives an estimate of the
order of magnitude of gravity field harmonic coefficients as a
function of the degree.
Case 2. To show the current status of gravity field
modeling, the normal equations were used that have led to
the GEM-T2 gravity field model [Marsh et al., 1989a]. In
fact, the inverse of the calibrated GEM-T2 covariance matrix, kindly provided by the NASA Goddard Space Flight
Center (J. G. Marsh, personal communication, 1989), was
used. The part of this matrix that holds for the gravity field
complete to degree and order 36 was used to compute the
formal error estimates of the gravity field harmonic coefficients. As shown in Figure 4, the corresponding RMS error
estimates
10-6 -
a case
1
+ case
2
, case3
x case
:
4
•_ 10-7
o
•_
'•- 10-8
•_10-9,•*•;[•
10-11
lO
-12
lO
15
20
25
30
35
degree I
Figure 4. Formal RMS errors of low-degree harmonic coefficients determined from Kaula's rule-of-thumb (solid line),
the GEM-T2 normal matrix (case 2), and three different
applicationsof GPS measurements.The specificationsof the
various
cases are described
in the text.
GPS receiver
should be able to track the P code.
Case 4. To verify whether state of the art in gravity field
modeling can be improved significantly if also accurate GPS
carrier phase measurements will be acquired by the ARISTOTELES on-board receiver, the analysis of case 1 was
repeated, but now for the assumption that the ARISTOTELES orbit perturbations could be computed with an accuracy of 10 cm in all components. This optimistic accuracy
level is comparable to the specifications for the current
TOPEX/Poseidon mission [Wu and Yunck, 1986a]. As
shown in Figure 4, from such a data set of 10-cm accurate
ARISTOTELES orbit perturbations (i.e., no a priori information), the accuracy of current low-degree gravity field
models can be improved by an order of magnitude. It is
therefore
5
of case 1.
x xx•_
xxxxxxxXXXxXXXXxXXx
10-no
the error estimates
set was added to the GEM-T2 normal matrix, and new error
estimates for the gravity field harmonic coefficients were
computed. As shown in Figure 4, these error estimates are
only a little bit smaller than those of case 2. This means that
if the ARISTOTELES orbit perturbations are measured by
GPS signalswith an accuracy of 3 m, the accuracy of current
gravity field models can not be improved significantly.
Therefore it may be concluded that C/A code measurements
do not provide the required information and that the ARISTOTELES
10-5
are well below
Case 3. The question now arises whether the data set of
case 1 can be used to improve the current status of gravity
field modeling. To give an answer to this question, this data
concluded
that the ARISTOTELES
GPS receiver
should have the capability to track the GPS carrier phase
signals very accurately.
The question that arises now is whether it is possible to
recover a gravity field model for degreesabove 36 if accurate
GPS SST measurements are available. For that purpose,
gravity field recovery error analyses were performed, in
which it was assumed that a gravity field model complete
from degree 3 to degree and order 180 has to be recovered.
In these analyses it was assumed that it will be possible to
2848
VISSER ET AL.' ARISTOTELES
GLOBAL GRAVITY
FIELD RECOVERY
rr2(orbit)
At
10-7
O'2(•lm, •lm)
10_8
= const
(12)
10-9
10-10
where •(orbit) is the orbit accuracy, At is the measurement
10-11
yseswereperformedfor severalvaluesof cr2(orbit)
At, and
interval, and •Clm, Slm) is the formal error estimatefor the
gravity field harmonic coefficients. Gravity field error analthe results are displayed in Figure 6. For example, if the
orbit accuracy is 100 cm and the measurement interval is 100
10-12
50
100
150
s (cr2(orbit)
At - l0 6 cm2 s), thetoplineintersects
theline
degree I
of Kaula's rule of thumb at a degree equal to about 30. The
Figure 5. Formal RMS error spectrum of harmonic coefficients determined from GPS measurements acquired by the
ARISTOTELES satellite. The four curves indicate from top
to bottom the results from the measured orbit perturbations
in the cross-track, along-track, and radial directions and
from the combined perturbations in all directions. The
decreasingline denotes the spectrum predicted by Kaula's
[1966] rule of thumb.
determine ARISTOTELES orbit perturbations at frequencies above 2 cpr with an accuracy of 1 cm and that the state
vectors are available at 1-s intervals. Comparable values
were used by Colombo [1989], Schrama [1990, 1991], and
Visser [1992]. Because of the well-known existence of general types of low-frequency orbit errors, a zero weight was
assignedto orbit perturbations with a frequency below 2 cpr.
Formal error estimates of gravity field harmonic coefficients were computed using orbit perturbationsin the radial,
along-track, and cross-track directions separately, or all
together. Of course, the smallest error estimates are expected when the information from the orbit perturbations in
all three directions are combined. The result of the computations is displayed in Figure 5. The decreasingline denotes
the•spectrumof the gravity field as predicted by Kaula's rule
of thumb. The other four lines denote, from top to bottom,
the formal error estimates using only cross-track, alongtrack, or radial perturbations or the perturbationsin all three
directions in a combined solution. The curves for the alongtrack or radial perturbations are very close together, although the curve for the radial perturbations is for degrees
higher than 20 always below the curve for the along-track
perturbations. As predicted earlier, indeed the error estimates are smaller for the case where radial perturbationsare
used separately, compared to the caseswhere along-track or
cross-track perturbations are used separately, although the
differences
between
the error
estimates
for the radial
or
along-track perturbation cases are very small. As indicated
before, the combined solution will be the best one, but for
higher degrees the results are close to the solutions using
either the radial or the along-track perturbations. For the
combined solution, it seems possible to obtain realistic
harmonic coefficient values for degrees up to 120 if the
previously described requirements are fulfilled.
The results displayed in Figure 5 hold for a measurement
interval of 1 s and an accuracy of the orbit perturbations
equal to 1 cm. However, it may be interesting to know how
the error estimates change if these parameters are varied. It
can be shown [Visser, 1992] that the following relation exists
between the error estimates of the gravity field harmonic
coefficientsC tm, Slm and the samplingrate and accuracyof
the orbit perturbations'
bottomline is valid for cr2(orbit)At - 1 cm2 s) and is
identical to the lowest line of Figure 5.
Gradiometer
Measurements
Gravity field error analyses from gradiometer measurements for models complete to degree and order 180 were
performed. In the simulations related to the gradiometer
measurements,it was assumed that frequencies below 2 cpr
are not observable, just as in the previously described
experiments. Gravity field recovery from gradiometer measurementswill also be corrupted by orbit errors. For example, if an orbit error At(m) in the radial direction exists, the
interaction
of this error
with
the central
term
of Earth's
gravityfield leadsto a gradiometerF zz-errorequalto
AFzz- •- Ar• 10-3Ar(EU).
(13)
F
For a 0.01 EU measurement accuracy, (13) demonstrates
that the orbit must be reconstituted with an accuracy of
better than 10 m to fully exploit the gradiometer measurement information content. This showsthat the application of
GPS measurements to recover gravity field information
requires a much higher orbit computation accuracy than the
application of gradiometer measurements.
With a measurement
noise of 0.01 EU and a measurement
interval of 4 s the simulations led to the results displayed in
Figure 7. The monotonously decreasingline again showsthe
spectrumof the gravity field as predicted by Kaula's rule of
thumb. The other four lines denote, from top to bottom, the
formalerrorestimates
usingrespectively
Fyy,Fzy, Fzz, and
these three componentsin a combined solution. The fluctuations at the low degrees are caused by the frequency
10_7
c• 10-8
• 10-9
-• 10
-'•ø
10-11
• 10-12
10-13
50
1 O0
1 50
degree I
Figure 6. Propagatederror spectrumof gravity field recovery error analysis from ARISToTELES orbit perturbations
in three directions. The decreasingline denotes the spectrum
predicted by Kaula's rule-of-thumb. The other four lines
hold,from2bottom
to top,
for o(orbit)2 At equalto respec4
6
tivelyl, 10 , 10 ,and10 cm2 s.
VISSER
ET AL.'
ARISTOTELES
GLOBAL
GRAVITY
truncation at 2 cpr. This truncation causesan ill-conditioning
of the normal matrix, especiallyfor coefficientsclose to the
FIELD
RECOVERY
2849
10-7
sectoffalband. As expected,the useof Fzz yieldsthe lowest
10-8
formalerrorestimates
compared
to the useof r zy or ryy.
The solution in which all three gradiometer componentsare
10-9
combined led, of course, to the smallest error estimates. It
can be concludedthat with this simulationsetupthe gravity
field can be solved to at least degree 180.
In order to investigate the effects of a nonperfect polar
orbit, additional simulations (with a frequency cutoff of 2
cpr) were performed for inclinationsof 92ø, 93ø, and 94ø. For
inclinations
of 92 ø and 93 ø the curves for the formal
error
10-1ø
10-11
10-12
50
estimates are almost identical to the corresponding curves
for an inclination of 90ø, with the exception of the formal
error estimates in the "saw-tooth"
degree I
10-7
10-8
10-9
may be attributedto the fact that solvingfor harmonic
coefficients up to degree 180 means a resolution of 1ø,
whereas a polar cap of 4ø is not covered by the ground track
of the ARISTOTELES orbit. Although in reality the ARISsatellite
150
area, which has almost
disappeared(Figure 8). This is causedby a better sampling
of the fluctuations caused by coefficients close to the sectoffal band for the low degrees, which results in a better
conditioning of the normal equations. This also leads to
smaller fluctuations in the RMS error spectrum for the low
degrees. For the inclination of 94ø, the normal equations
became singular and no realistic results were obtained. This
TOTELES
100
will be in an orbit with an inclination
of
approximately 96ø, the problem of an unstablenormal matrix
may and will be overcome by including additional information in the normal equations. For example, a regularization
technique may be applied [Schrarna, 1990, 1991], in which
the spectrumof the magnitudeof gravity field coefficientsas
a function of the degree l, as predicted by Kaula's rule of
thumb, is added to the normal equations, or separateobservations over the polar cap or even simulatedpolar observations with a value equal to zero may be added to the normal
equations. In these cases, the gravity field model obtained
will be accurate over the entire globe covered by ARISTOTELES orbits but with a deteriorated accuracy over the
polar caps.
As indicated before, gradiometer signalswith a frequency
below 27 cpr will be unobservablein practice, becauseof the
bandwidth limitation of the gradiometer. In simulations
where a frequency cutoff of 27 cpr was applied, the normal
10-1ø
10-11
10-12
50
1 O0
15O
degree I
Figure 8. Propagated error spectra from gradiometer measurements if the orbit inclination is equal to (top) 92ø or
(bottom) 93ø. See also Figure 7.
equations, in this case for a gravity field from degree 37 to
180, again became singular and no significanterror estimates
could be obtained.
This result indicates
that even with the
normal matrix of a high-accurate gravity field model complete to degree and order 36 added to the normal matrix of
the gradiometer measurements, the bandwidth limitation
related problemscan not be solved. Therefore, in the next
section attention will be paid to the question whether the
addition of normal equationscomputed from precise ARISTOTELES orbit perturbations can help to solve this problem.
I
10-7
10_8
I
I
Combination
of Orbit
Perturbations
and Gradiometer
Measurements
As stated in the previous section, problems arise in the
gravity field recovery from gradiometer measurements if
frequenciesbelow 27 cpr are not observable due to system
limitations.An attractive option to overcometheseproblems
might be the addition of normal equations computed from
precise orbit perturbations.To investigatethis, first a simu10-12
50
1 O0
150
lation was performed in which the normal equations computed from orbit perturbations in all three directions were
degree I
added to the normal equations from the gradiometer meaFigure 7. Propagated error spectrum of a gravity field
recovery analysis from the ARISToTELES gradiometer surements(all three components) with an applied frequency
components
ryy, r zy, and Fzz and from all thesethree truncation at 27 cpr. The accuracy of the (filtered) orbit
components together (from top to bottom). The decreasing perturbationsderivedfrom GPS measurementswas assumed
line denotes the spectrum predicted by Kaula's rule of to be 1 cm and the orbit state vectors were assumed to be
thumb.
available at 1-s intervals. For the gradiometer, measure-
10-9
10-10
10-11
2850
10-7
VISSER
ET AL.: ARISTOTELES
•
•
GLOBAL
and these
10-8
½• 10-9
E 10-lø
c• 10-11
•
-
E
•- 10-12
10-13
RECOVERY
values
can
be used
to estimate
the
standard
deviation of the geoid undulation and gravity anomaly errors
averaged over Earth's surface. For the various cases of
global gravity field error analyses discussed above, the
standard deviations of the gravity anomalies and the geoid
undulationsmay be computed. Some resultsare summarized
-
o
FIELD
ances can be computed as a function of the harmonic degree,
•
.
•
GRAVITY
in Table2. The secondcolumnin Table2 specifies
•he figure
I
•
50
1 O0
•
in which the correspondingerror spectrumof the gravity
field is plotted and for which the assumptionsfor the data
degree I
rate and accuracy have been specified. Table 2 shows that
from GPS measurements only the gravity field can be
Figure 9. Propagatederror spectraof gravityfield models recovered to degree and order 120, with an accuracy of 4.15
recovered from the combination of the three gradiometer mGal for gravity anomalies and 24.8 cm for geoid undulacomp6nents and the orbit perturbations in the three directions (commissionerrors of the gravity field part complete to
tions.In all cases,•Fij) = 0.01 EU, At = 4 s and all degree and order 120). Above degree 120, the degree varifrequencies below 27 cpr were assumedto be unobservable.
From top to bottom,the three curvesholdfor a(orbit)is 3 m, ances become larger than Kaula's rule of thumb. If the
(omission)effectof the gravityfieldfrom degree121to 180is
10 cm, or 1 cm and At is 60 s, 60 s, or 1 s.
accounted for, the standard deviations of the gravity anomments.with a spacingof 4 s and a precision of 0.01 EU were alies and the geoid undulations for ttie global recovery
assu•ed to be available. This simulation led to a stable
become about 12 mGal and 50 cm, respectively.,
solhtionand demonstratedthat a highly accurategravity
The two lines for the gradiometer-only cases (inclination
field model can be determined (bottom curve in Figure 9). 90ø) show that for an ideal gradiometer with no bandwidth
Alsocasesin whichthe accuracyof the orbitperturbations
is limitation, an accuracy of 0.21 mGal for the gravity anomaequal to 10 cm (second curve from bottom in Figure 9) and lies and an accuracy of 1.5 cm for geoid undulations is
equal to 3 m (third curve from bottom in Figure 9) with achievablefor a gravity field model complete to degree and
measurement intervals equal to 60 s were included in the order 180. As was discussedbefore, in reality the bandwidth
investigations. In the latter two casesit was assumedthat the limitation of the gradiometer leads to problems in the soluvalue for the accuracy of the state vectors hold for the entire tion of the normal equations. It was shown that the addition
spectrum, i.e., also for orbit perturbations below 2 cpr. For of normal equations from the orbit perturbations can solve
the higher degrees, especially for degrees close to 180, the this problem. The last row in Table 2 refers to this case and
three solutionsconverge to each other, and the gradiometer shows that the commission errors of gravity anomalies and
signalcontent determinesthe quality of the solution. How- geoid undulations, for a gravity field solution complete to
ever, for the lower and medium degrees, large differences in degree and order 180, are 0.21 mGal and 0.9 cm, respecgravity field accuracy may be distinguished.The quality of tively. However, it should be realized that all numbers
the low-degree gravity field recovery is for the larger part presented are based only on pure formal error estimates;that
150
determined
by the accuracyof the recovered
orbitperturb
a-
is, it is assumed
thatthegravityfieldmodelcomplete
to degree
tions. From the results plotted in Figure 9 it is obvious that
a high-quality GPS receiver will be necessaryto fully exploit
the information content of the gradiometer measurements
and to facilitate a high-accuracy low- to medium-degree
gravity field recovery. Therefore it may be concluded that
the gradiometer really needs an additional measurement
device in the form of a high-quality GPS receiver, which will
deliver precise information of the ARISTOTELES orbit
perturbations, especially in the low-frequency band. Similar
conclusionswere also drawn by Schrarna [1990, 1991] and
and order 180 describesthe "real-world." In reality, omission
errors and errors due to mismodelingof, e.g., nonconservative
forces, errors in modelingthe gradiometermeasurementsdue
to uncertainties in the ARISTOTELES attitude, orbit errors
(equation (13)), etc., must be added. Nevertheless, it will be
clear that with the combination of gradiometer and GPS
measurementsthe missiongoals of a 5-mGal gravity anomaly
accuracy and a 10-cm geoid undulation accuracy, both at a
resolution of 100 km, can be realized.
Visser [1992].
Conclusions
The formal error estimates for the gravity field harmonic
coefficientscan be usedto estimatethe accuracyof gravity
Although a number of simplifications was applied in the
theory of formal error estimates, this theory is a valuable
anomalies and geoid undulations on Earth's surface. Vari-
Table 2.
Undulations
Standard Deviation of (Point) Gravity Anomalies and Geoid
With
the Formal
Harmonic
Coefficient
Minimum
Data Type
Error
Estimates
Maximum
Figure
Degree
Degree
Ag, mGal
N, cm
GPS
GPS
Gradiometer
Gradiometer
5
5
7
7
3
3
3
28
120
180
180
180
4.15
70.16
0.21
0.21
24.8
274.4
1.5
0.9
GPS + gradiometer
9
3
180
0.21
0.9
VISSER
ET AL.:
ARISTOTELES
GLOBAL
tool to gain insight in the gravity field recovery from ARISTOTELES gradiometer measurementsand orbit perturbations derived from GPS SST range measurements. It was
shown that with the implementation of a high-quality GPS
receiver on board of ARISTOTELES it is possible to improve current low-degree gravity field models significantly, if
such a receiver is capable of delivering information from
GRAVITY
FIELD
RECOVERY
2851
TELES, in Proceedings of the Anacapri Workshop, September
1991,Eur. Space Agency Spec. Publ., ESA SP-329, 137 pp., 1991.
Gurtner, W., GPS-papers presented by the astronomical institute of
the university of Berne in the year 1985, Mitt. 18, satellite0beobachtungsstationZimmerwald, Bern, Switzerland, 1985.
Kaula, W. M., Theory of Satellite Geodesy, Blaisdell, Waltham,
Mass., 1966.
which it is possible to determine ARISTOTELES orbit
Lambeck, K., ARISTOTELES:
An ESA mission to study the
Earth's gravity field, ESA J., 14(1), 1-21, 1990.
Marsh, J. G., et al., The GEM-T2 gravitational model, NASA Tech.
perturbations with an accuracy of the order of 10 cm at
measurement intervals of 60 s. If this accuracy would be 1
cm and the measurement interval would be 1 s, even
valuable information of the gravity field up to degree and
Marsh, J. G., F. J. Lerch, C. J. Koblinsky, S. M. Klosko, J. W.
Robbins, R. G. Williamson, and G. B. Patel, Dynamic sea surface
•topography, gravity, and improved orbit accuracies from the
direct evaluation of Seasat altimeter data, NASA Tech. Memo.,
order
120 can be obtained.
The
effect
of several
mission
parameters has been studied, including the inclination of the
satellite orbit and the limited gradiometer bandwidth. An
inclination
of the satellite orbit close to 90 ø leads to the best
monitoring
of Earth'sglobalgravityfi•!d. If the inclination
differs too much from 90ø, additional information of the
gravity field above the polar capsis necessaryto stabilizethe
normal equationsfor a high-degreegravity field recovery. It
was found that a deviation of 4øfrom the true polar inclination already resulted in an unstable set of normal equations.
The gradiometer bandwidth limitation (i.e., the fact that
the low-frequency band (below 27 cpr) of the gravity tensor
is not observable) requires additional information about
gravity field induced signals in the low-frequency band. A
GPS receiver seemsto be capable of delivering this kind of
Memo., TM-100746, 1989a.
TM-1000735, 1989b.
NASA, Gravity Workshop, Geophysical and geodetic requirements
for global gravity field measurements 1987-2000, report of a
gravity workshop, Colorado Springs February 1987, Geodyn.
Branch, Div. of Earth Sci. and Appl., NASA, 1-45, 1987.
Nerem, R. S., B. D. Tapley, C. K. Shum, and D. N. Yuan, A model
for the general ocean•circulation determined from a joint solution
for theEarth's gravityfield, paperpresented
at ChapmanConference on Progress in the Determination of the Earth's Gravity
Field,AGU, Ft. Lauderdale,
Fla., Sept.13-16,1988•
Rapp, R. H., and J. Y. Cruz, Spherical harmonic expansionsof the
Earth'sgravitational
potential
to degree360using30minutemean•
anomalies,
Rep. 376, 22 pp., Dep. of Geod.Sci. andSur•., Ohio
State Univ., Columbus, 1986.
Rapp, R. H., and N. K. Pavlis, The development of geopotential
coefifcient models to spherical harmonic degree 360, J. Geophys.
Res., 95, 21,885-21,911,
1990.
Rummel, R., A method of global recovery of harmonic coefifcient
from SGG data with peculiar geometry, CIGAR Phase II: Work
information in the form of precise SST range measurements
Package 510 A, Contribution to final report of "Study on Precise
to the ARISTOTELES satellite from which precise orbit
perturbations can be derived. The combination of a highquality GPS receiver and a gtadiometer on board of ARISTOTELES seems to be the optimum solution to achieve a
gravity field recovery with the resolutionanti accuracy
strived for. This will enable to answer many open questions
in the fieldsof geodynamics,geodesy,oceanography,climatology, and related scientific fields.
References
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(Received February 10, 1993; revised August 31, 1993;
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