Itdid not contain data from 6 low .... mass lunar, solar and planetary gravitation, solar radiation pressure,. Earth ..... orbits and force model parameters are associated with ...... (area to massratios of .00069 and .00096m2 kg-I respectively) covered ...... 1024. *. 1029. NO. OBS. 838. 904. 724. 752. 616. 1169. 978. 1303. 1359.
gr
NASA
Technical
An Improved Gravitational
J. D. B. T. R. N. S.
Memorandum
•
Q
4019
Model of the Earth's Field: *GEM-TI*
G. Marsh, F. J. Lerch, B. H. Putney, C. Christodoulidis, T. L. Felsentreger, V. Sanchez, D. E. Smith, S. M. Klosko, V. Martin, E. C. Pavlis, J. W. Robbins, G. Williamson, O. L. Colombo, L. Chandler, K. E. Rachlin, G. B. Patel, Bhati, and D. S. Chinn N87-29967
{NASA-TM-_019| AN IMPROVED MODEL OF THE EAETH'S GRAVITATIONAL FIELD: GE_-TI 'NASA) 351 p Avail: NTIS HC A|6/MF A01 CSCL 08G Hi/_6 JULY
1987
NASA I
I
Unclas 00998_3
NASA
Technical
An Improved Gravitational
Memorandum
Model of the Earth's Field: *GEM-TI*
J. G. Marsh, F. J. Lerch, B. H. Putney, D. C. Christodoulidis, T. L. Felsentreger, B. V. Sanchez, and D. E. Smith Geodynamics Branch S. J. O. K.
M. W. L. E.
EG&G Center,
Klosko, T. V. Martin, E. C. Pavlis, Robbins, R. G. Williamson, Colombo, N. L. Chandler, and Rachlin Washington Inc.
Analytical
Services
G. B. Patel, S. Bhati, and D. S. Chinn Science Applications and Research Corporation
JULY 1987
National Aeronautics and Space Administration Goddard Space Flight Center 1987
4019
TABLE
INTRODUCTION THE 2.1
OF
CONTENTS
...............................
GEODYN
AND
SOLVE
SOFTWARE
SYSTEMS
•
DESCRIPTION
°
°
2.2 3.0
OPERATIONS
REFERENCE 3.1 3.2 3.3 3.4 3.5 3.6
FRAME
.
°
II II
°
•
°
°
•
•
o
11 12
Design Philosophy Benefits .........
. . .
21
°
°
•
.
.
•
.
.....
•
•
•
•
°
°
°
•
....
°
IN
......
COMMON 4.1 .I 4.1.2
PARAMETERS
THE °
....
4.1.5 4.1.6 4.1.7 4.1.8 4.1.9 4.1.10 4.1.11
•
GENERATION •
°
°
°
OF °
°
°
°
•
•
°
•
°
°
°
°
°
°
°
43
....
43
•
.
•
.....
PR'_F-_DING
iii
41
THE
Earth Tides Ocean Tides Tidal Deformations Earth Parameters ................. Polar Motion and AI-UTI ............. Station Coordinates ............... Third Body Effects ................ Z-Axis Definition ................ Coordinate System ................ Relativity ..................... A Priori Gravity Modeling ........... .......
29 29 30 33 38
°
.......................
........
4.1.3 4.1.4
..........
25 29
....
.................
•
13 20
Gravity
.......................... DESCRIPTION OF THE CONTRIBUTING DATA .......... DISCREPANCIES BETWEEN DATA SETS ............. MATHEMATICAL FORMULATION DYNAMIC POLAR MOTION ..................... SUMMARY •
9
°
9
...................
........
A PRIORI CONSTANTS ADOPTED TOPEX GRAVITY MODEL 4.1
°
INTRODUCTION
•
4.0
•
°
II, SOLVE and the TOPEX ........................
....
•
.....
of SOLVE .............. GEODYN ................
GEODYN GEODYN GEODYN Models
°
.....................
Vectorization Evolution of
2.1.3
°
1
°
°
•
•
PAG_
_i___'[[_ItOf_ALLY
....
•
°
•
.
BLANK
°
°
°
.
43 43 44 44 44 45 45 45 45 46 46
b/._T _'R_t_
_LANK
TABLEOF CONTENTS (cont.) 4.1 .11 .I 4.1.11.2
5.0
TRACKING DATA ................................. 5.1 5.2
DATASELECTION............................ INDIVIDUALSATELLITEANALYSES .................. 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.2.7 5.2.8 5.2.9
6.0
Selection of an A Priori Gravity Model: General Vs. Several Tailored Fields ..................... 47 Simulations for Geopotential Solution Using Tailor-Made Vs. General A Priori Models .................... 49
Analysis of SEASATDoppler and Laser Data .... Analysis of OSCAR Doppler Data ........... Analysis of GEOS-ILaser Ranging Data ....... GEOS-3Analysis of Laser Ranging Data ...... Analysis of STARLETTE Laser Ranging Data ..... Analysis of LAGEOS Laser Ranging Data ....... Analysis of GEOS-2Laser Ranging Data ....... Analysis of Optical and Low Inclination Satellite Observations ................. Analysis of BE-C Laser Ranging Data ........
59 60 68 68 69 78 84 89 98 107 108 128
DEFINITIONOF A PRIORI GEOCENTRIC TRACKING STATION COORDINATES ..................................
133
COORDINATE SYSTEM DEFINITION................... INITIAL STATUSOFSTATIONCOORDINATES ............. THETRANSFORMATION MODELS .....................
133 134 134
6.1 6.2 6.3
Seven Parameter Transformation ........... The Linear Translation ................ 6.4
NUMERICAL RESULTS .......................... 6.4.1 6.4.2 6.4.3 6.4.4
6.5
DISCUSSION.............................. 6.5.1 6.5.2 6.5.3 6.5.4
6.6
NAD27 to SL-6 Transformation ........... GEM-9to SL-6 Transformation ............ GSFC-73to GEM-9Transformation .......... Other Transformations .................
Transformation Parameters and Accuracies ..... Precision of the Transformations .......... Error Sources ...................... Distortion in the NAD27 Datum ...........
SUMMARY OF STATIONDEFINITION..................
136 137 137 139 139 139 140 140 140 142 144 145 147
TABLE 7.0
FORCE
MODELING
7.1
CONTENTS
EFFECTS
7.1.1 7.1.2
Mathematical The A Priori
7.1.3 7.1.4
The A Priori Body Tide Model A Priori Ocean Tides Models
ATMOSPHERIC
DRAG
AND
Atmospheric
SOLUTION
8.3
DESIGN
9.0
THE 9.1 9.2 9.3
RADIATION
Formulation Model
of
PRESSURE the
Testing
GEM-TI
x 36
SOLUTIONS
SOLUTION
Models
EVALUATION
9.4.3 9.4.4
....
.........
170 !70 17] 173 177 177 179 183
CALIBRATION... A "SATELLITE-ONLY"
........................
183 187 198
......................
209
THE GRAVITY MODEL ........................ OCEAN TIDE SOLUTION ...................... STATION COORDINATE SOLUTIONS AND COMPARISONS
9.4.1 9.4.2
.....
Orbit Comparison Results ...... Evaluation of Apparent Timing... Errors .................. Conclusions ..............
RESULTS
149 152 154 154
..............................
9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.4
SOLAR
Potentials Models . . .
........... ...........
COLLOCATION ............................ STRATEGY FOR DATA WEIGHTING AND FIELD PROBLEMS AND ASSOCIATED BENEFITS WITH 36
149
Formulation of the Static Geopotential
Drag
7.2.2.3
8.1 8.2
149
........................
Mathematical
8.0
(cont.)
...............................
POTENTIAL
7.2
OF
......
209 209 229
Introduction ..................... GEM-TI Stations ................... Laser Station Solutions ..............
229 229 230
Doppler Summary
232 235
OF THE
Station Solutions ............. ........................ SOLVED
POLAR
MOTION
..........
Introduction ..................... The 1980-84 Solution ................ The Annual and Chandler Cycles ......... Summary ........................
235 235 236 240 248
TABLEOF CONTENTS (cont.) 10.0
A CALIBRATION OF GEM-TIMODEL ACCURACY ............... 10.1
11.0
12.0
THEGEM-TICALIBRATION OF A SATELLITEMODEL'SERRORS USINGGRAVITYANOMALY DATA...................
254
10.2
CALIBRATION BASEDUPONFIELDSUBSET SOLUTION TESTING
263
10.3
COMPARISONS BETWEEN GEM-TIANDGEM-L2...........
277
10.4
THENEEDFORLOWINCLINATION DATA--REVISITED ......
280
10.5
SUMMARY ..............................
287
GRAVITY FIELDTESTING ONGEM-TI ...................
289
11.1
ORBITTESTING..........................
289
11.1.1 11.1.2 11.1.3 11.1.4 11.1.5
290 302 304 307
GEOIDMODELING
11.3 11.4
ESTIMATED TOPEX/POSEIDON ORBITAL ORTHOMETRIC HEIGHTS COMPARISONS
....
SUMMARY
REFERENCES
APPENDIX
APPENDIX
Orbital Tests on Laser Satellites ....... Orbit Tests On Doppler Satellites ....... Tests Using Low Inclination Data ........ Radial Accuracy on SEASAT............ Tests Using the Longitudinal Acceleration on Ten 24 Hour Satellites ............
11.2
ACKNOWLEDGEMENTS
APPENDIX
249
°
...............
.
•
310 •
.
•
ACCURACY ........ ..............
313 318
..................................
325
.................................
327
..................................... I:
II:
IIl:
329
TOPEX GEODETIC TRACKING SITES
FILE: TRANET DOPPLER, ..........................
TOPEX
GEODETIC
FILE:
SITES
...............................
A PRIORI
OCEAN
TIDAL
OPTICAL
MODEL
vi
312
AND
EARLY
LASER,
S-BAND 337
DOPPLER
..................
TRACKING 343 347
SECTION
1.0
INTRODUCTION
Ground-based observational models
of
Analyses major
the
global
these
advance
effort
long
data
in
the
Geodetic
Observatory,
field
the
and
orbital
Department
observations
enhance
our
kinematics interior,
and and
in
has
field
Since
of
Space
the
creation
Astrophysical
point
tectonics,
the
positioning,
in
in
understanding
effort
Forschungsinstitut
of
the gravity
near-earth
geopotential the
study
the
earth's
in the study of global oceanic
and
-- to name a few) to
for modeling of
the
(GSFC) and
and a cooperative
de Geodesie Spatiale
knowledge
of
a
1960's, a continuous
Smithsonian
Geodaetisches
capabilities
earth.
provided
Flight Center
of Defense,
an
harmonic
the
have
the
in the middle
provided
spherical
to improve our understanding
Better
advances
Geodesy.
(notably
Deutsches
motion.
dramatic
satellites
gravity
at NASA/Goddard
Groupe de Recherehes
use satellite field
U.S.
of
Program
centers
Germany's
France's
wavelength
Satellite
research
between
artificial
by the authors and many others
has been underway
other
of
data set which has been used to develop
of
National
tracking
has
created
the
earth's
of
processes
satellite
theology
with
and
spaceborne
instrumentation.
The acronym, kept
geopotential
GEM, standing
pace
near-earth
with
satellites
of the missions 1990's mission (e.g.,
the
require
rapid
oceanographic
by
GSFC
Earth Models.
advances
made
in
further Of
The
10 to
15 cm radial
the
radar
altimeter
known
the
accuracy
gravity
model
improvement
most
immediate
concern
which
and
the
is at
their
least a factor
for the
achieve
for
of TOPEX, three
their support
the
for launch
of
which
requirements
geodetic
geoid)
requirement
by
foreseen to
is
is under development
orbit accuracy
system,
marine
by
precision
However, new NASA missions
computations
satellite
are
The GEM have generally
are tracked and the orbital
themselves.
orbit
developed
for Goddard
objectives. for
models
TOPEX
in 1991. driven
beyond
by the
capability need
of
for
an
earth's
gravity Interim
gravity
accuracy
to
which
is under
these
objectives
global
gravity
both
of
taneous
solution
stages field
harmonics
and
obtained
from
the
model
will
one
an
as
type
to
of
and a
the
simul-
will
large
the
research.
of
plan
solution
earth's
extract
that
a
is
experimental
the
combination
utilize
types
pre-launch
each
evaluated
in
Consequently,
degree
Therein,
of
analysis
geopotentlal
satellite-to-satellite
report
GEM-TI,
tracking
to higher
Both
observations
permitting
TOPEX,
final
the
in
gravity
data
is
optimal
to
be
subset
satisfy
the
TOPEX
of
of
these
all
amount
of
available
and
surface
preliminary
gravity
tracking
observations.
This models,
a
to
Model.
data
extensive
for
the
improvement
arduous
unknowns.
required
separately
the
of
Mission
NASA.
satellite
equations
requires
towards
by
Interim
diverse
thousand
model,
This
of an
of
Research
substantial
requires
systems
wavelengths
project
from
final
altimeter,
gravimetrie
describes which
is
observations.
degree
and
effort
which
Center
for Space
data
be
a
spanning
sampled.
The
data.
with
It
extending
and
flight
model
model
short
additional
knowledge
Geopotential
development
gravity
builds
completely
will
laser,
the
is an
present
and
new
satisfied
accuracy
which
solutions.
validated
only"
the
scrutinized
criterion,
a
There
our
orbiting
as
several
improved
devised
is more
gravity
low
numerical
achieve
carefully
a
consuming.
of
1985.
enhances
observations
large
with
a
in
intermediate
and
time
of an
been
be
of
of
building
has
can
modeling
numbers
To
at
consideration
and
preparation
existing which
support
recovery
costly
large
model
field
needed
The
models
model
has
selection,
been
This
developed GEM-TI
of
result by
to produce by GSFC
although
and
more
upon
harmonic of
the
GSFC
and
an
these
based
spherical
undertaken
Research
first
exclusively
36 is a direct
order
was
the
Interim is
model,
gravity the
field
This
herein.
in spherical
satellite complete
University
Model.
reported
complete
direct
to
improvement of
Texas'
"satelliteIn regard
harmonics,
to is
like
earlier
GEM-L2
GSFCmodels, for example, GEM-9(Lerch et
(Lerch
tracking
are
upon
the
The
GSFC
in the
demands
played
assessment. least
major
step of
ago
In the
data
data
in
definition
of
the
error
has
the
normal
and
order
available past, sets with
as
been
For
although the
in
creation
of
produced
which
GEM-TI. is
by
of
our
the
GEM-TI
generation in
a
data
sets,
have the
_ot
been
science
modeling.
science
and
data A
based
largely
have
with upon
been
improved the
more
a
re-
The
models
last
than
in
force
ten
all
the
to
GEM-TI.
avoided
by
parentage set
and
data
degree
most
of
aliasing
orbital
to
This
terms
the
in
degree 50
are
In
the
recent
data
associated
required
standard
of
modeling
inconsistencies
lag-time
was
performed.
solve
only
the
re-iteration
was
GEM
to
The
the
analysis model
total
extending
evolved,
improved
and
total
representation
used
the
1977).
evaluating
terms
at
resource
and
particular,
harmonic
a to
within a
occured
non-conservative
spherical
extension
activities.
In
be
of
processor
complete
earlier
consistently
recovery
Program
vector
a
all
GEM-TI.
from
et al.,
frame.
gravity
will
feasible
activities
the
the
GEODYN
model
and
"super-computer"
both
matrices
(Wagner,
reference
many
evolving
implementation
a
for
from
normal
treatment,
they
state
benefitted an
of
reduced
equations 36.
modeling
surface
their
205
Cyber
gravity
satellite
within
made
a
used
fields
practical
for
lacking
constants,
and
foundation
matrix
adopted
task
to the
for GEM-7
consistency
this
determination
least-squares
and
CYBER
orbit
previous
computation
the
us
the
later
required
upon
system
and
contained
and
of
making
solution
in preparation
a
in
our
laying
analysis
permitted
role
of all
the
These
1979)
satellite-to-satellite
missions
necessary
availability
our
recalculation years
model
include
stages.
future
exclusively
observations
orbital
squares
iteration
altimeter
imposed
in
will
of
Adapting
SOLVE
which
information
major
constraints
also
wavelength
The
a
which
planning
gravity
degree.
time
radar
long
accurate
higher
1982) Models
spaceborne
measurements
more
al.,
observations.
tracking,
built
et
al.,
for design has of
now
their in
the been
constants
adopted for the MERIT Campaign (Melbourne, et al 1983) with some significant improvements. Additionally, other NASAGeodynamicsresearch activities like the Crustal Dynamics Program, have provided improved a priori have
tracking
been
used
treatments
are
models
in
the
coordinates
development
described
planned
tracking
station
for
the
station
in
next
and
of
GEM-TI.
detail
within
few years,
adjustments
earth
rotation
These this
models,
report.
a simultaneous
with
the
series
gravity
which
values
and
In
subsequent
solution
including
field
will
also
be
explored.
Although is
more
than
one
deliberately more
models
were
model
keeping their present
and
to
to
new
track
of
differences, a brief
highlights
report model
of
for the
test
them
and/or all
these
models
description
specific
and
tables,
of
discussed
the
sake
of we
have
additional
specific new to are these
figures
test
in
an
the
fields,
and
sections
a cross where
we
have
were
I.
response an
aide
are
of in of
Therein
reference
they
in
these
understanding
Table
a
design,
which
As
easy
in
to
Generally,
show
We
permit
doing,
fields
and
summarized
4
to
contributions.
assist
and
In so
there
pages.
pursued
purposes.
points
data
its
completeness
many for
otherwise,
within
solution.
specifically
weights
indicate
GEM-TI
to
illustrate
might
approach
the
pertaining
developed used
this
brevity
discussion
material
cases
of
gravitational
calibrate,
presented some
title
sacrificed
thorough
compute,
the
the
we
which
used.
TABLE
FIELD
I. KEY
TO
GSFC
GRAVITATIONAL
DESCRIPTIVE
SUMMARY
CROSS
REFERENCE
AND
NAME
IGEM-TI]
FIELDS:
DESCRIPTION
is a "satellite-only" gravitational field model developed from trackina data acauired on 17 unique satellite orbits (Table 5.4). Asummary of the observations utilized is presented on Table 5.3 and the weighting used is shown in Figure 8.4. The spherical harmonic coefficients for GEM-T I are found in Table 9.1 and their uncertainties
are shown
in Figure 10.1. This model is the focus of this manuscript. GEM-TI had an internal GSFC field number of PGS3113. Note also, certain data sets were corrected to improve
PGS-T2
the overall model.
is an earlier model the American
presented
Geophysical
at
Union
Meeting in the spring of 1986. It did not contain data from 6 low inclination satellite (Section 5.2.8 and 10.4) and contained GEOS-2 problem
5
matrix
a serious
back-substitution
(Figure 8.3).
PGS-T2'
is the PGS-T2 the GEOS-2
field (above) with
problem
corrected.
GEM-9
is a pre-Lageos "satellite-only" model (Lerch et al, 1977).
GEM-L2'
is the GEM-L2
model
(Lerch et al,
1982) solved with theC,S(2,1) coefficients constrained to equal zero. This was
GSFC's
general recommended
"satellite
only" model of GEM-T I.
PGS- 1331'
prior to the completion
is the PGS-1331 model (Marsh et al, 1985), like Gem-L2 ; solved with C,S(2,1) constralned to equal zero. PGS-1331 wasa model "tailored" for the Starlette satellite orbital computations.
PGS-S4'
is thePGS-S4 model (Lerchet 1982b) solvedwiththeC,S(2,1) coefficients constrained to
al,
equal zero. PGS-S4 was a model "tailored" for SEASAT orbital computations.
GEM- 10B'
is the GEM-lOB model ( Lerch et al, 1981) solved with the C,S (2,1) coefficients constrained to equal zero. GEM-1OBisacomprehensive model
which
contained
and surface gravimetry.
altimetry
PGS-30
13
is the PGS-T2 data
model
weight
was
respect
the collocation used
matrix to gove
of the adequacy method
PGS-3167
was
normal
from
of GEM-TI
PGS-3163
and order and not 36.
the truncation (Figure
limit
8.7).
was a combination solution combining GEM-TI with SEASAT altimeter matrices.
The
altimetry
field was
given
a weak
of 0.1 (Figure Figure
PGS-3164
to
_' o,ze--being
to deqree
was
an example
but solved
(like GEM-L2)
which
8.2 )
the GEM-TI
be of a smaller
20
(Table
10.12.
equations
complete
by a to
of the calibration
in Figure
made
the
increased
factor of 5 with
and was
where
was
in this weight
8.5, Figure
10.3.1, and
IO. I0).
the PGS-3163
giving greater to the altimetry
field, solved
weight
of 0.5
(Figure
10.1 I).
SECTION THE
The Flight
Cyber
Center
continues SOLVE, This
in
to
software
to
describes the
enormous
development
SOFTWARE
The modeling
team
were
estimation
Martin
al.,
integrator,
a
spherical for
the
design
benefits
program
SOLVE
nutation,
point
Using
tools,
processing
decisions, accrued
Space
as
a
(and
GEODYN
and
capabilities.
status,
due are
lunar,
and
most
of
these
result
and
provided
for
diurnal
aberration,
center
of
gravity
offset.
parameters
and
Dynamic
include
axis data
is
measurement
9
and
timing
state
well
precession
tracking Tracking
is
and
stations
refraction, spacecraft
performed
biases,
and
measurement
and
solution
as
solar
Earth
parallactic
to
Cowell
includes
as
dynamical
and
editing
1980,
Cartesian
displacement
iterated
al.,
function
Earth
loading. and
et
geodetic
gravitation,
and
rotation,
programs.
and
gravitation
includes
antenna
estimator
forcing
drag,
gravity
high-order
planetary
tropospheric
of
Martin
spacecraft
Earth
ocean
TOPEX
system
1977;
The
and
Earth
tides
and
squares
solar
modeling
to solid
GSFC
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the
for
atmospheric
annual
least
[Putney,
representation
motion
the
GEODYN
orbit
derivatives.
Earth
by
and the
integrates
partial
mass
polar
utilized
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Observation
displacements
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Goddard
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analysis
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capabilities
model
tides.
Bayesian
by
immediately
vector
state-of-the-art
numerically
pressure,
corrections
Cyber's
the
harmonic
radiation ocean
the
tools
1987].
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force
models
was
principal
software
provides
parameter
the
obtained
activities.
primary
et
was
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SYSTEMS
DESCRIPTION
GEODYN
and
use
SOLVE
system
An
improve
efficiently
section
AND
computing
1982.
today)
importantly,
2.1
205
GEODYN
2.0
as
the
convergence. and
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all
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orbit
state
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normal
model
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inclusion
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large
analyses.
program
gravity
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formed
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provides
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mentioned
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rotation,
Program
as
error
computer
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earth
be
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for
motion,
well
GEODYN
estimations
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as
the
tracking
tides
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by
and
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GEODYN
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The
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rewriting has
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GEODYN,
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Typically,
number
program
intrinsically advantage
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redesign twice
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GEODYN
version
been
hours. the
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matrices
in
computer
vector
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improve
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computer
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205.
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limited
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20
the
solve
added,
last
past,
required
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the
tracking
the
and
clearly
In 1982,
over
capabilities
parameters
geophysical
more
In
the to
evolved
satellite
models.
constrained computer
has
typical
has For
been
GEODYN
years. run,
For I/O
2.1.1
Vectorization
The
SOLVE
the
Cyber
and
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time
The
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of
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Typically,
limit
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include
amount
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substitution solved
Initial
for.
parameters
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nature.
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matrices
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summing
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when
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Large disk
packs
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a second
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When of
alone
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The
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or
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tlme).
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interest
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matrix
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arc-parameters.
11
time;
the
smallest
parameters.
one
parameters
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expense
employed
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combfnlng
all
are
function
two
a single
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of individual
to
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affords the
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of values
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coefficients,
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set
packages.
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each
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The
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Minimizing
SOLVE
Into
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etc.,
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There
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satellite
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of
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matrices
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level
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of
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of matrices,
for
satellite
state
when
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at
of
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performance.
simultaneously
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data
I/O
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code
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eliminating
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combining
C-Matrlx.
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moved
many
are
CPU
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utilization
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1986]
section
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orbital
Major,
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led
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has
SOLVE
[Estes
205.
small
of
and
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matrix
parameters
unnecessary
arc
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SOLVE
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of
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tion
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time.
of
Cyber
a
in
of
of
GEODYN
the vastly
its
was
computing
was
116
77
of
minutes four
seconds
of
CPU
CPU
requires.
time
which
time
words time
improvement
user
in
1921 and
of
and
CPU
and
been
matrix.
31
On
minutes
of
memory)
seconds a
the
solu-
has
computer
142
or
(b)
parameter
x
on
Cholesky
solution,
a 1921
million of
speed
undertaken.
number
of
system
redesign.
system
this
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system
increased
approach
of
In
computer,
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90
computing
Other
matrices
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based
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GEODYN
advantage
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only
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vector
matrix,
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parameter or
values
of
factor
I/O
of
13
GEODYN
computers.
a
205
the
deviations
full
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mainframe
for
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on
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IBM 3081
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program
GEODYN
When
became
necessary
NASA
to make
Additionally
approximations
12
optimized
large
that
great
needed
be
IBM
take
full
process of
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I software to
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GEODYN
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Because
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205
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GEODYN
Cyber
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2.1.2.1
design
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individually
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commonly
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II
discussed
data
the
the
Processing
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of
and
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the
Facility
at
thorough on
This
in
such
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are
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on
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GSFC.
of about
impact
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user.
transition
more
This
designed
computer the
period A
was
intensive
vector
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I.
developed
and
IBM
for
version.
GEODYN
been
a smooth
I
environment.
computer
on
IBM
program
I/O
the
Because
5 years,
the
discussion
of
TOPEX
gravity
below.
Philosophy
of
key
system.
in the
formats
are
a development
number
GEODYN
has
-
GEODYN
the
as Cyber
speciflcatlon
Vector
the
from
GEODYN
and
which
philosophy
presented
a
II
permitted
in
of
vector-processlng
vector
the
version
to
new
either
at
II Design
were
the
required
design
are
All
to
to
was
II
the
was
environment
directly
GEODYN
performed
system I
the
functions
efforts
scalar
referred
of
those
a
a totally
computer
GEODYN
There
approach
GEODYN
computing
created
called
360/95
GEODYN
model
is
are
two
IBM
GEODYN
was
"front-end"
"front-end"
the
two-pronged
efficlent
replaced,
advantage
functions
These
be
effort,
that
the
a
historical
been
progr__m
on
highly
computer
full
new
from
to
a parallel
take
mind,
environment.
was
program
in
and
GEODYN's
360/95,
the
concepts
co_iderations
They
following
were
made
are
briefly
went
presented
into
the
below
and
paragraphs.
a uniform
13
that
64-bit
floating
point.
I/O
0
intensive
operations
were
off-loaded
from
the
vector
computer.
o
Observation
processing
0
Interpolation
and
was
adapted
partial
to
vectorizatlon.
derivative
chaining
were
fully
vectorized.
Force
0
model
evaluations
vectorized
where
Numerical
integration
and
parti al
deri vat i yes
were
appropriate.
of
the
orbit
was
vectorlzed
where
possible.
Numerical fully
o
integration
Large
of
fore
than
programs in
the
useful
in
2-I also the
the
data
"front-end", are to
be
partial
based
derivatives
system.
No
the
data
the
was
type
to of
an
allow problem
to
awareness
There-
optimization to
of
environment
references
vectorizatlon
solutions.
structure
operating
but
vectorized.
determination
flow
explicit
paragraphs,
fully
different
provided
upon
the
was
exhibit
orbit
were
indicates
be
the of
this
solved.
GEODYN the
figure
of
its
205
vector
of
contents
II
various are
made
may
be
reader.
Transmissions its
routine
presents
following to
model
equations
solutions
capabilities
vectorization
It
normal
parameter
problems
system.
force
vectorized.
Formulation
Figure
of
the used
of
data
Amdahl by
both
between V7
the
computer,
computer
Cyber require
systems.
14
data
These
computer
conversions
data
conversions
if
and the are
GEODYN-i
TRACKING
TRACKING
DATA
FORMATTER
GEODYN-II
REFORMATS OBSERVATIONS & ORGANIZES INTO BLOCKS
DATA
CONTAINING ONE TYPE
TRACKING
DATA OF ONLY FROM A SINGLE
DATA
PASS
GEODYN-IIS SET-UP CONTROL
R'EADS PLANET I
1_
DATA,
ARY COMPUTES
JPL EPHEMERIS
MEMORY
& OUTPUTS
REQUIREMENTS.
INTERFACE
FILE
&
LOADS SELECTED OTHER OBSERVATIONS NECESSARY DATA
JI_
_NS
................
............. GEODYN-IIE
OOSERV AT ION RESIDUAL AND S/C
TRAJECTORY FILES
READS
INTERFACE
FILES,
INTEGRATES
NORM AL MATRIX
& DATA ORBIT,
(E-MATRIX)
COMPUTES OBSERVATION RESIDUALS & STATISTICS, EDITS OBSERVATION DATA AND
ESTIMATES
GEOPHYSICAL
................
LINK
ORBIT
FILE
&
PARAMETERS
BACK
TO AMDAHL
80
...............
COLUMN
SUMMARY
RESIDUAL ANALYSIS
MASS UPDATED GEODYN-II
GEODYH-IIE
SETUP PROGRAM
CONTROL
TERMINAL SCREEN
PR INTED DATA
Figure
2.1
DEV ICE
OUTPUT
GEODYN-II
15
STOR AGE
KEYBOARD
Flow
Diagram.
greatly of
facilitated
one
and
all
point as
FORTRAN
the
into
a common
that basic
information component
system. used
with
the
GEODYN
GEODYN
This on
program
both It
operations
are
processing
environment,
utilizing
is
the
in
Cyber
all
This
205
of the
above
Beginning ment
pass.
and
data
are
observation
use
program
64-bit
has
all
the
all data
floating
been
designed
observation
performance
commands.
GEODYN and
II-S
data
reason
serial,
those
computations.
The
operations
this
program.
selecting
subsets
of
For
fundamentally
functions
the
is
performed
Thls
involves
subsets
of
GEODYN
II-S
and
transmits
of
the
system
to
the
computationally
data
program this
intensive
system.
is
the
in
computer
segment
of
GEODYN
II-E
and
computing
designed
type"
as
a
such or
the
engine a fashion on
system
has
of
been
the
that
Cyber
where
the
optimized
oonsequence_
is
the
II
it may
be
205 CPU
for
most
GEODYN
vector
intensive the
vector
efficient
when
computer.
has
made
the
been
possible
TDF,
tracking
observations The
the
data
all
converts
to
is
files
the
vectorlzed by
within
carrying
this
the
GEODYN
theme
II
throughout
programs:
with
type
contains
been
that
if
format.
which
been
this
(TDF)
pipeline
processing
has
exclusively
program
performed.
Observation system.
II
"IBM
processor.
205
numerical
has
the
Cyber
by
the
II-E
reasons,
system
bookkeeping
along of
The
the
speed
these
vector
input
higher
For
point
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Formatter
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processing,
perform
at
well-suited
of
V7
performs
Data
ls
various
to
the
floating
205
data
the
required also
64-bit
Amdahl
readlng
from
GEODYN
advantage
input
the
the
Cyber
take
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of
The
on
files
The
part
performed
variable
output
words.
and
observations
the
observations
station of
only
into one
within
16
data
data each
are
organized
blocks.
type block
from are
by measure-
Each a single
data
block
tracking
chronologically
ordered and the blocks themselves are chronologically respect to block start times.
ordered with
The GEODYN II-S program retains an observation block structure in the data that it selects and passeson to GEODYN II-E. However, at this stage the data blocks may be subdivided to facilitate later processing. GEODYN II-E processes data blocks by treating each observation identically within the same block. This allows the application of vector operations to the data processing algorithms. It further permits the vector interpolation of orbit and force model dynamlcal parti al deri vati yes obtained from the numerical integration of the variational equations and the vector chaining of partial derivatives. The primary time consumingalgorithms in the numerical integration of satellite orbits and force model parameters are associated with I) spherical harmonic evaluation of the Earth's gravitation field, 2) evaluation of variational derivatives, 3) numerical integration of the equations of motion, 4) evaluation of force model partial derivatives, 5) numerical integration of force model variational equations, and 6) the evaluation of other force model perturbations. The relative importance of each of these items dependson the specific circumstances pertaining to each problem. In the typical orbit determination problem items I-3 will be expected to dominate computation times. When a tide model including 300 pairs of coefficients is evaluated, item 6 will become a very significant factor. Or, if a full gravity field normal matrix is to be calculated, items 4-5 will have substantial impact.
17
Because all
of
the
above factors
enter
into
the numerical
integration problem, a very high level of vectorization is required in these areas. To deal in an efficient manner with these various problems GEODYN II-E has been vectorized in the following fashion: I)
Spherical
harmonic
including
2)
the
Spherical
evaluation
Legendre
harmonic
has
polynomial
been
fully
vectorized
recursions.
variational
derivatives
have
been
fully
vectorized.
3)
Numerical
integration
fundamentally
5)
Force
model
Earth
and
Numerical been
6)
Evaluation
For achieved
ocean
of
observations
and This
nature,
performed
have
Earth
problems,
in
this
for
motion
however
is some
area.
terrestrial
been
fully
vectorized.
of
force
model
partial
and
ocean
the
vectorization this
in
of
gravity
derivatives
and
has
vectorized.
large
in
equations
derivatives
tides
vectorized.
Computations
been
partial
fully
through
parameters.
has
integration
fully
the
sequential
vectorization
4)
of
area
proportional segment
of
greatest
of are
the
square
of
has
II-E.
18
been
of
has
improvements the
proportional
the
code
perturbations
speed
formation
linearly to
the
tidal
the
fully
normal to
may
vectorized
be
equations.
the
number
been
number of
of
adjusted in GEODYN
Problem-oriented within
GEODYN
For
orbit
parameters
is
observations matrix
generally
based
solutions
problem
each
data
arc
on
be
equations
number
a
been
based
Improvements
upon
of
the
of adjusted
problem
equations arrays
no
devices
performed
For
partition
in
addressed
by
the
normal
matrix
into
computer's thrash
the
matrix
that
the
for
each
with
the
the the
matrix partial
summation
adjusted
will
be
parameterS.
primarily
computation
GEODYN
from
time
memory
and
system
derivatives
number
paging.
19
of
left
paging
the
GEODYN on
segments
normal
to
of
been
its
its
own will
computer
optimized
II-E disk
and
system
amounts has
the
program
If
exorbitant II
when
the
memory.
problem.
partial
occurs
that
computer
summation
minimum
II
large
consuming
the
measurement in the
GEODYN
virtual
about
reason
without
be
block.
situation
result
between
will
parameters.
fit
this
the
the
partial
equations
normal
area
of
parameters,
such
of
that
combination
this
number
relationship
in
normal
In
model
number
summation
the
and
this
longer
stores
summation
the
sufficiently
interminably
arcs.
the
such
adjusted
later
force
circumstances
normal
dimensioned
become
the
these
of
for
chaining
achieved
linearizatlon
file
is
interpolation,
vectorized
that
data
derivatives
derivative
Another
a
other
and
of
than
dimensioned
number
complex in
number
of observations
large
put of
is
Under
chaining
the
with
must
partial
number
also
smaller
block.
derivatives
is sufficiently
normal
has
the
substantially
interpolation,
vectorized
time.
problems,
partial
derivative
of
determination
within
of
data
optimization
II.
simple
For
intelligent
to
temporarily
and
necessary
forms to
the allow
2.1.2.2
GEODYN II Benefits
The benefits
of this
extensive effort
to reconstruct
GEODYN for
the vector processing environment are several: The switch to the normalized Legendre recursion formulation in GEODYN II permits the numerically stable computation of gravitational coefficient accelerations and partial derivatives to degrees in excess of 360. The computation of the Right Ascension of Greenwich is performed more precisely, eliminating annual discontinuities on the order of 100 microns. o
Precession and nutatlon are included in the integration of the adjusted force model parameters resulting in more accurate force model partial derivatives.
o
Two-wayrange is strictly modeled as such, removing errors on the order of one micron for satellites at altitudes of one Earth radius. Errors of muchgreater magnitude are eliminated for more distant satellites. The JPL DE-200 ephemeris using the Wahr nutations year 2000 precession model has been implemented.
and the
o
Spherical harmonic contributions are fully computedautomatically output.
o
Time dependent non-conservative forces are now modeled.
2O
to the variational equations whenever normal matrices are
and last,
2.1.3
but not least,
o
Typical orbit determination runs are 6.5 times faster on the Cyber using GEODYN II than on the IBM 3081 using the original GEODYN I.
o
Gravity model normal matrix generations are at least 90 times faster using GEODYN II on the Cyber than original GEODYN I on the IBM 360/95. This factor of 90 is based upon duplication within GEODYN II, of the original GEODYN I processing of nonaltimeter, satellite only, dynamical normals for inclusion in the GEM-lOBgravity model.
GEODYN
The problem
II,
TOPEX
to
be
SOLVE
and
gravity solved
the
TOPEX
modeling
using
Gravity
effort
the
GEODYN
of
GEODYN
used
in
Models
presented II
system
the and
first the
large
Cyber
scale
optimized
SOLVE.
From satellite These
the data
classes
arcs.
The
optical
arcs
not
nearly
an
were:
and
the
typical
II
data
arcs the
2-2
number
the
the
on
observations analysis
of
TOPEX laser
derives
number
data
of
illustrates
in both
the a
and
of
The
parameters
but
data the
estimated
Doppler also
data
include
observations.
number
data
between
number
of
effort.
Doppler
difference
the
optical
2!
arcs,
the
of
classes
modeling
generated.
number
Cyber,
three
gravity
from
matrices
greater
the
operations,
performance
greatest
graphically time
of
arcs,
normal
of magnitude
running
for
in
include
order
data
II
the
computational
included only
were optical
laser
Figure GEODYN
arcs
primary
and
parameters
viewpoint
arc. and
relationship of The
adjusted numbers
laser
data.
between parameters shown
are
Similar
600
r:l
GEOS-I
1181
O
STARLETTE 596
A
GEOS-I
1889
OBS 5-DAY
LASER
OBS 5-DAY
LASER
OBS 5-DAY
LASER
5OO
400
p._.J I-0 i--
3OO _=
2OO
I00
w
1500
•
2000
NUMBER
Figure
2.2
Computer Time GEODYN-II.
•
Required
2500
•
•
3000
•
3500
4000
OF ADJUSTED PARAMETERS
for
22
Generation
of Normal
Equations
by
relationships
exist
fold
in
increase
weighted
profiles be
compared
the
360/95
and
number
with
the
the
Using the
Cyber
a
in
Figure
quadratic
normal
with with
an
the
IBM
3081.
the
speed
increase
is
the
the
on
in
approximately
ten-fold
IBM
is
scalar
GEODYN
ten-
increase
llnearlty
increased. time
in
on
of
the
the
This
such
90 for
IBM
of
as
following
with
the
GEODYN
360/95
all
should
associated
computers
factor I
3081),
strong
running
speed-increase
original to
2-2
parameters
matrices
conservative
in
associated
of adjusted
of
versus
comparable
time
note
the
generation
processing,
ons.
particular as
Doppler
running
observati
Of
for
II
IBM
on
(whlch
estimates
is
merit
co nsl der at ion.
o
Cyber
205
matrices 44
o
of
360/95
matrices 3,960
versus to
the
converge
tion
are
as
a
required
parameters
and
to
1380
generate
580
observations
normal
should
be
2000
time
required
parameters
to
and
1380
increase
for
generate
580
observations
normal
should
be
hours.
original each
computer
of
factor
of
6.5
GEODYN of
the
speed
I on data
the
arcs
IBM 3081, used
in
GEODYN
estimates
the
gravity
II
on
of
the
model
the
Cyber
resources determina-
follows:
Cyber
o
2000
time
hours.
IBM
Using
computer
205
data
arcs,
IBM
3081
data
arcs,
computer
time
using
iterations
12
computer using
time 12
required each,
required
iterations
23
to
to
each,
converge should
be
converge should
580 178
580
be 1156
satellite hours.
satellite hours.
Using that same factor of 6.5, the following are estimates of the resources required to converge 720 data arcs used to evaluate the test gravity model solutions. Cyber 205 computer time required to converge 720 satelllte data arcs, using 6 iterations each should be 110 hours. IBM 3081 computer time required to converge 720 satellite data arcs, using 6 iterations each should be 718 hours. Translated into other terms, the projected resource requirements for the convergence and formation of 580 normal matrices and the testing of gravity model solutions would require the exclusive utilization of an IBM 3081 computer by the project for the period of nine full months. This same computational burden, when placed on the Cyber 205 computer using the GEODYN II system, constitutes less than five percent of the annual resource allocation of the computer. In fact the total computer resource budget for this TOPEXgravity model effort was only 500 hours of Cyber 205 time spent over a period of approximately one year. This figure also includes the computer resources used by SOLVEto combine the 580 normal matrices, remove all arc parameters through back-substitutlon, and produce some 120 test gravity fields. Sucha concentrated effort to produce these TOPEXgravity models would not have been logistically possible using the original GEODYN I and SOLVEeven with a dedicated IBM 3081 computer.
24
2.2
OPERATIONS
With thousands of arcs to be processed by a dozen individuals at GSFC,the operation of the gravity field modelling effort was standardized as much as possible. This was achieved in several ways. Each satellite was given a two character abbreviation and a three digit number so that required data sets and matrix numbers could be related to the satellite automatically. Generic setups were created to provide commoncontrol language and commonmodel constants for ease of operation and quality control of input data streams. Naming conventions were defined for satellite observation data sets. The summarypage output of the GEODYN program was modified to include more summaryinformation. The normal equations were numberedto provide satellite and arc information as well as version number (see Figure 2-3). An on-line file was created to provide a reservoir of information for sharing and documenting the status of arcs completed and for combining arcs in the solution. The actual task of arc processing and matrix generation was divided into subtasks by satellite and data type. After the processing for an arc had been completed, matrix numbersand mass storage cartridge and backup tape location was stored in an on-line data file. The job submission was done on the Amdahl V-7, which is the front-end for the Cyber 205. It has an MVSoperating system with the TSO interactive capability. TSOcommandfiles, or CLISTs, were created for the job submittal. Typically, the submittal of any of the job steps in the GEODYN or SOLVEprogram required the typing of only one line of controlling input containing the epoch date of the data arc, the satellite identifier, and the type of processing to be performed. The CLISTs, given this information, filled in the required data sets and
E MATR I X
VSSSTDDDDDDER
13 DIGITS (USUALLY
EXAMPLE:
460176022701
VERSION
I GEOS-3
LASER DATE 760227
LEVEL I C MATRIX (CONTAINS
SSSTVAACCC
I0
DIGITS
ARC
PARAMETER)
EXAMPLE: BE-B
2403110205
OPTICAL
I0
VERSION
ARCS CARTRIDGE
I 205
LEVEL 2 C MATRIX
TTSSSAAVV - ONE SATELLITE EXAMPLE:
112601201
LASERBE-C
(NO ARC PARAMETERS)
9 DIGITS
12 ARCS VERSION
TTBBAAAVV
- MULTIPLE
EXAMPLE:
I10412001
LASER 4 SATELLITES VERSION
I
SATELLITES
9 DIGITS
120 ARCS
I
WHERE: AA OR AAA = NO. OF ARCS
SSS = SATELLITE
BB = NO. OF SATELLITES CCC = CARTRIDGE
NUMBER
VER OR VV OR V = VERSION
DODDDD = DATE
Figure
2.3
NUMBER
I"I" = DATA TYPE ENTERED TWICE
Matrix
Numbering
26
Scheme.
NUMBER
12)
submitted for
the
runs.
further
This
process
project.
Data
management
for
2400
Consequently, tape.
The
tapes
were
storage
The attached
store
the
onto
a Level
I C-Matrix.
device
as
were
point
The
were
produce
a
This
would
satelllte
and
combined,
since
optical
allow
data
and
numbering/naming
the
It
was
or
of
satellites,
number of
of
arcs
combined
important
date
In
computer
time. occur
that
were
at
the
each
numbered
it
in
to
successive
level
27
mass
the
true
of
and
management
the
satellite and
of fair
be the
keeping
different
a
by
also
number,
a
aim data
solutions
data
CAt
single
was
the
that
6
The
could
control
that
When
a
indicate
how
form
radiation,
record
requires
so
storage
large
maintain
necessary
the
with
version
shows
matrices was
a
GEODYN to
sets
number
2-3
the
matrix.
in
to
normal
2 C-Matrlx.
the
The
level,
used
solar
This
such
were
combined
a Level
alike.
matrix
on
data
prudence storage
C-Matrix.
matrices
332
mass
from
were
satellite
matrix
Figure
output
from
satelllte
combining
Therefore,
each
vital
combined
addition,
eliminated
satellites.
are
of arc.
matrices
problem.
compression
or
into
magnetic
combined
drag,
combined
of
words.
that
the
the
I
the
Consequently
on
stored
(state,
handled
laser
conventions
problem. number
be
was
arc
nontrivlal
8-byte
copies
six
Level
Some
would
of
These
the
from
a
Cartridges
were
weighting
type.
they
some
matrix
tapes.
two
equations
were
the
throughout
million
166
and
were
of
was
stored
matrix
collected
6250-bits-per-inch
equations,
through
parameters single
a
were
parameters
they
2.9
computer.
combined
maintained
arc
equations
cartridge.
arc
completed
the
V-7
normal
This
invaluable
minimal
used
normal
storage
well.
be
Amdahl
six
a mass
etc.)
type.
the
Typically,
Matrices
to
were
proven
requires the
output
continuation
onto
matrices
matrices
program
fit
of
or
requires
maintain
individual
equations.
to
1000 to
to
types
normal
matrix
matrices
of
required
device
biases
6
has
the
parameter
only
demanded.
was
automation
modeling
A
various
documentation
gravity
problem.
this
addition,
processing,
processing. TOPEX
In
the
levels the
data
amount
normal
a sufficiently
of
matrix small
number of matrices would be created to permit
a good turnaround of
experimental solutions. These operational concepts have paid off in providing a high degree of quality control, offering flexibility to the analyst in preparing arcs for inclusion in the gravity computation, and allowing control of the overall model and in the use of constants. The GSFCTOPEX gravity modeling project has benefitted
28
immensely from this effort.
SECTION 3.0 REFERENCE FRAME
3.1
INTRODUCTION
A uniform series for connecting the Conventional Inertial Reference System (CIRS) realized by the orbital dynamics, with the Conventional Terrestrial System (CTRS)realized by the global network of tracking stations was a requirement for our new gravity solution. This was one of the preliminary activities undertaken for the development of the TOPEXfield. A desirable technical constraint on the origin of these series requires that it be as close as possible to the average pole of the mid-70's to mid-80's interval. This required a redefinition of the origin to coincide with the LAGEOS estimated 1979-84 six-year average pole. The major characteristics of the new series are its uniformity, its new origin, and its consistency with other conventional models used in the transformation CIRS CTRS,namely the nutation model (Wahr's) and the precession model (Lieske's).
3.2
DESCRIPTION OFTHECONTRIBUTING DATA The polar motion and UTI-UTCdata available
to us were as follows:
(I)
the somewhat poorly documented but well maintained file polar motion values contained in GEODYN I,
(2)
two series cation),
(3)
the series resulting
based on BIH data
(Feissel,
private
from the LAGEOS SL6 solution.
29
of
communi-
The source(s) for the first data set is not clear, especially for the earliest years. The BIH series were obtained from the BIH Circular D data set with additional corrections to reference them to the IAU 1980 nutation theory (Wahr,1979) and contained some weak Vondrak smoothing to remove periodicities shorter than 35 days. The third and last set of data, that obtained by GSFCfrom LAGEOS,was used as the basis for unifying the series. This set was adopted for it is more consistent with the rest of the mathematical model than any other. Details about the periods covered by each data set are given in Table 3.1. The BIH series are shown in Figure 3.1.
3.3
DISCREPANCIES BETWEEN DATASETS
The discrepancies reconciled here were different for each of the data sets, even though for the most part, they all amount to a different origin of the local frame in which the pole coordinates are reported. As a first step we comparedeach of the above with the SL6 series. The origin of the BIH 1967-85 series could be easily and rigorously related to that of SL6 since the two series overlapped for a considerable time interval. The six year period (1979-84) was selected as the most appropriate for determining the transformation parameters between the two series for several reasons. First, this period is where the LAGEOSdetermined polar motion is the strongest due to the robustness of the tracking data set. Second it covers most of the period over which very accurate tracking data are available for analysis under this project. A six year period was selected to properly average both the annual as well as the Chandlerian cycles of the polar motion.
3O
Table
3. I
POLAR MOTION EARTH ROTATION
AND
• SELECTED DATA
SERIES
FOP POLAR MOTION
SOURCE
- OLD GEODYN
PERIOD
FILE
- LAGEOS
61
12
31
62
OI
05-
66
12
30
D (NEW
SYSTEM)
67
Ol
04-
78
12
27
7g
Ol
Ol-
84
12
30
58
0g
18-
61
12
31
62
01
05-
84
12
30
SL-6
• EARTH ROTATION
SERIES
FILE
- BIH CIRCULAR
18-
SYSTEM)
SOLUTION
- OLD GEODYN
og
(OLD
- BIH CIRCULARD - BIH CIRCULAR
58
D
• MAJOR DISCREPANCY THE REFERENCE BIH CIRCULAR
FRAME D SERIES
DIFFERENCE
BETWEEN
AND THE LAGEOS
31
SL-6
THE SERIES.
ERP(BIH POLE
POSITION
FOR
THE
SERIES
ERP[BIH
H]
H)85AO0
67/85A00
-"" !iF !i!iiii!V x, v ".., :,.: :._ v V !MiiV JD-2400000.S
'
[UAYS]
ERP(BIH POLE
FOR
THE
SERIES
ERPIBIH
_AIiA
. -!
POSITION
--
_=l.Ji_
H)
i!!"
:| :': :: .mrA
A i_. ::
#I
El
i"
"-
:!
:.
,.
!:
!!
: t /i
H} B5AO0
B7/BSAO0
B,,
:
i
#' | ! :.._ :
,-_j,rL__ ! ;, :.i _i _i V" |i _.i_
:
i!
vviv vVV *i*_.w -
w,,7
*e_*
,_lm
Figure
,-Ira ,-smm JD-2400000.S
3.1
BIH
32
m71 ,Trm [DAYS]
Polar
Motion.
_
,_m*
,sin
3.4
MATHEMATICAL
FORMULATION
The
theory
general
formations
is
(IAG/IAU
Joint
Conventional Committee
our
the
general
picture
fact
matical
in
of
model
transformation
the
the 3.2. two
Ax
is
of
provide the
and
reference
Xp,
report
yp
and
made
with
the
LAGEOS-derived
MERIT
changes
rotation series
are
eliminated
by
identical, the
e + _2
sin
virtue the
cos
e - B2
_I ' _2:
implied
inertial
81 , 82:
implied
terrestrial
Mean
frame
frame
Sidereal
33
misalignment
misalignment
Angle
of
mathe-
systematic
(3.1)
e - BI
A
analysis
GROUP MODEL
e + _2
Greenwich
thereby
follows:
where:
e :
series
whatsoever. this
to
a
Earth
We
in
discrepancies
of
Steering
uniform
series.
utilized
COTES
Maintenance
the
no
trans-
by
to
motion
notation
third
frame
a continuous
polar
Rotation
as
Since
series
the
WORKING
sin
System)
not
geometry With
our
Establishment
1982).
UTI-UTC
the
= -el
9,
that
Earth
cos
the
do
relating
Ay = el
based
recommendations
on
June
parameters
THE MERIT/COTES
we
Reference
to
BIH-provided
that
the
(UTI-UTC)
analysis
Figure
which
Group
Bulletin,
variations
adopted
the
in
Terrestrial
limited
is shown
Working
(CSTG
Rotation we
detailed
on
CIRS (I)
, CIRS (|I) ZII
ZI
ZSs Zs
T zs Ts
zS
ZII ZI I
Zx
ZI
Z II
l
Tl
ytS
Z 11 .
71z RlP,,P2'P3) CTRS (I)
Figure
3.2
Geometry
* CTRS(II)
for
34
Coordinate
Transformations.
s Is
Application
of
motion
discrepancies
angles
listed
of
BIH
the
reference
frame.
were
to
to
which
apparent origin we
therefore
and
the
in
Figure
average
two are
be
make
them of
The
motion
rotations
then
the
= XS-_
YT
ZT
frame
by
p
ZS
= YS
+ Yp
ZS
= ZS
+ _p
XS
subscript for
of
the
the
1978
The polar
axis
as
with
of a the
X-axls
systems
this
simple
(yp)
angles
in
be
to
Z-axis.
was
Y-axls
equal
radlans.
to
The
is this
reference, consistent
the
station
The
geometry
pole
are
shown
for
this
of
the
above
we
must
apply
(Xp). one
system this
realized
coordinates and
basically
of
SL6
table
Since
To
the
all
the
this
for
system
subtraction
station
in
motion.
at
origin
into
reason
rotations new
misalignment
coordinate
well.
this
polar
transform
listed
plane
refer.
coordinate
are
of
Xp,yp
appropriate
of
to
end
local
the
used
our terrestrial
through
cosines
the
that
these
case
the
dynamic
compatible
the
the
BIH-SL6
and
Since the
_he sines
transformatlon
are:
XT
the
the
the
about
small,
of
Z-axis
series In
of
of
were
to
of
pole
redefinition
approximated
equations
new
apply
3.3.
the
redefined
must
values.
angles
where
to
of
the
relationship
polar
these
with
1962
origin
6-year
determination
values
discussion
effect,
we
coordinates
can
a
coincides in
new
selected
parameters
from
average
the
the
the
These
series
coordinates
after
have,
new
The
to
in
3.2.
motion
define
the
model
resulted
in Table
polar
used
this
S
(3.2)
- Yp
stands
YS
for
the
TOPEX.
35
SL6
coordinates
and
the
T
for
the
TABLE
3 2
BIH (1979-84) TO LAGEOS (SL-6) POLAR MOTION SERIES TRANSFORMATION PARAMETERS i
131=
1.46
+0.3
mas
B2 = -3.80
+-0.3
mas
c[ 1= -0.22
+0.3
mas
o_2=
+0.3
mas
0.62
P,MS (Ax)
"6.5
mas
RMS (Ay)
"6.2
mas
SIX YEAR AVERAGE x = :38.2
_+ 0.9
y = 280.:3
+ 2.2
36
LAGEOS
SL6
ORIGIN/
Z-AXIS
Y
T
TOzPEX
Figure
_"
"...... ×
IOsRIG INI
3.3
Relationship
of Coordinate
37
System
Origins.
3.5
DYNAMIC
The temporal both
the
motions.
modeled
geometry
of
the
Earth's
motion,
with
the the
Most
of
[Lambeck,1971
and
geopotential
is
referenced
and
S(2,1)
It the
is
axis
reflected the
in
with
we
derive
on
inertia a
expansion
the
general
of
the
figure
variations
of
the
C,S(2,1)
by k
(to
be
the
the
C(2,1)
Moritz, some
second of
and
through
harmonics. the
I/3
to
that
Chandlerian The
when
depends
of
one,
[ibid.]
accounts
the
application
on
this is:
is the the
C(2,1)
the
Earth's
this
factor.
orientation
field
which
1972 ],
factor
of
harmonics
model
The
motion
the
order
Denoting
60
tides
for.
determine
frame
in
The
this
and
gravitational
resulting
38
that
arbitrary
S(2,1)
reach
[Gaposchkin,
parameterize
1967]
given
periodic
can
the
knowledge
to
figure,
3.4.
far
capability our
of
deformation.
Figure
about
degree,
the
formulation
determined),
is
prudent
to
of
concluded
our
equations
axis
in
so
than
only and
of
variations
factor
is
that
proportionality
CTRS
Because
respect
on to
the
to
accounted
rotational
developed
response
two
that
is
to
depicted
accuracy
values
Based
of
is
the
The
it
harmonic
moments
to
the
deformation.
motion
similar
wobble.
[Heiskanen
the
(C,S(2,1)).
period
in
axis
exhibits
tidal
this
in
Earth's
amplitude
the
1973]
higher
figure
spherical
can
of
known
of
to
an
manifested
inertia
momentum,
[McClure,
main
parameters,
well
response
properties.
is
elasticity
with
a
1972],
the
elastic
motion
theori es
to
of The
angular
response
proportional
Earth's
of
involved
the
clearly
moments
therefore
Earth's motions
is
deformations.
daily
and
planet
Earth's
axis
elsewhere
smaller is
is
the
our
tidal
principal
to
wobble,
of
and
There due
much
of
variability
is
meters
MOTION
non-rigidity
rotational
which
are
POLAR
of
reference
is
harmonics
of
of
the
body
relating
the
(through
C(2,0)),
for
temporal
of
the
proportional
proportionality
CHANDLER
POLE
Period - 14
4m
max _m max 12 m
F Pole
Figure
3.4
Dynamic
39
Polar
Motion
Model.
Figure Axis
\
F
^
C2,1{t ) : C2,1(to)
. C2,1(t-to)
+ kxp(t)
C2, 0 (3.3)
^ ^
S2,1(t ) : $2,1(to)
where
the
monics
harmonics
relative
clear
that
will
average series
center
of
in
as
larger
that for
gravitational average
negligible; analyze
Our to
the
recent
tracking
data)
degrees.
The
very we
still
would
use
harmonics
the
lose
been close
contain
in By
first
third
not
at t o equal
axis
polar
first
terms.
The
after
some
time,
becomes is
to
reference
the
will
much
included
second
average
to our
with
identically
years)
term
in
coincides
are
completely
developed. to the
the
of
most
above
BIH,
the equal
term
out
at
model
implications
if
is we
Additionally,
since
we
to zero.
4O
can
at
the
current pole
more
azimuth
the
the
same
when
the apply
for
important
of
would
and
always
effect.
average
and
therefore
origin and
an
this
with
the
accurate
off
the
model
Therefore
that
10 meters
software. nothing
the
were
terms
{several
redefining
these
third
be
term,
term
If we
should
the
offset
second
har-
cycles.
does
term
available
we
above,
about
avoided
the
time
be
significant.
implemented initial
{which
the
of
very
the
It
offsets
this
these
last
and
that
whose
then
it have
first
have
to,
extent
by
nonzero
slowly,
of
to .
mean
for
motion.
CTRS
Chandler
system
years
a
value
epoch,
any
need
part,
secular
software
implement
the
the
represented
migrates
period
full
a
periodic
argued
current
coordinate
CTRS
as
is
C2, 0
to
initial
cycle;
be
To
to
a short
over
SL6
to
future
at
the
Chandler
the
center
and
data
Plans
of
refer
which
motion
expansion
Over
at
offset.
this
wobble
to zero.
and
an
- kyp(t)
carets
part,
there
polar
accummulates than
CTRS
each
cause
the
compensate
the
the
periodic
out
motion
with
to
the
+ S2,1(t-to)
about be
270
nonzero
Z-axis
of
our
time,
we
can
full it
model with
is the
3.6
SUMMARY
The
methodology
coordinates 1984
of
based
system
on
has
modified
local
plane the
deviation
and
to the
ideal
over
the
provided
presented.
The
to
which
the
through
the
center
the
CTRS
still model
claim
a model
described
its
18,
of
no
the
Z-axis
pole
makes
with
Sep.
series
system
to zero
uniform
resulting
that
equal
period
a
by
sense
SL6
creating
primarily
the
from
motion
for
in
axis
identically
pole
series
been
CTRS,
with
the
used
BIH
the
through
Dec.
30,
LAGEOS
SL6
a modified
SL6
and
the
realizes
and
thus
the
Xp,yp
the
six-year
1979-84 to
._Jrther modeling
accuracy
41
for
coordinates
it possible
herein.
1958
series
which
origin
refer,
C(2,1)
for
the
and
This S(2,1)
dynamic
slightly
the
coincide
wobble.
set
is only
of
polar
inferior
SECTION 4.0 A PRIORICONSTANTS ADOPTED IN THEGENERATION OF THETOPEXGRAVITYMODEL
The constants that were adopted and used in the development of the a priori which
TOPEX
the
able
gravity
solution
thought
and
model
These
constants
debate.
Thls
brief
and
updates
(1985).
The
constants
following
meant
regarding
the
4. I
COMMON
4.1.1
Earth
A
other
of
eight
4.1.2
degree
were
The et
al.
solid
description
harmonics
were
tidal
flexibility
Ocean
are
after
describes
consider-
the
found
in
Marsh
listed
by
parameter
By
satellite
1979),
tides
Partials
complete
monograph
parameters.
not
tidal
(Wahr,
earth
2nd
some
are
chapter
procedures
common
chosen
common
within
adopted
and
Tapley type
parameters
dependent
(e.g.
In
it
is
parameters
Tides
frequency-dependent add
on
were
parameters
PARAMETERS
solid
lag.
and
which
model
combined
similar
physical
Earth).
total
dependent
a
section
parameters
the
exists.
parameters
the
delineate
of
providing
modeled
using
included
for
our
the
through
tidal tidal
earth
used
the
potential
earth
in
were
a
closed
k2zO.30 each
terms,
Wahr's
a p rlori
as
analysis
tide
from
standard. formula
and of
well (see
frequency
a
for
their
zero
the k2,
All
¢2
Section
phase
specific and 7.1
k 3, for
to a
modeling).
Tides
a_.riorl
(1986b)
in
ocean which
tide 600
model
was
individual
developed
terms
representing
FI_EASE1)[NG PAGE BLANK
43
by
Christodoulidis, 32
major
and
N(iT P17_E4_)
__N|L_'M_iO_,ALLY_
BLANk
minor
tides
were
semi-diurnal
harmonics
parts
of that
used.
The
using
the
graphic
a
priori
and
7. I)
in
frequencles.
4.1.3
Tidal
The
Love
h2 -
Earth
The constant
and
.609,
the
found
the
in
the
was
only
and
Partials orbital
were
out
were
each
band
(NSWC) found
oceanoelsewhere
computed
perturbations
in
terms
over
be
and
retrograde
prograde
Schwiderski can
diurnal
carried
admittances
algorithm
period
For
prograde
tides from
numbers
the
£2
for
for
the
6
of
12
each
h2
MERIT
and
£2 had,
Campaign
" .0852.
as
a priori
standards,
Partials
were
values,
(Melbourne
included
for
the
et
h 2 and
al.,
£2"
Parameters
a
priori
and
the was
Earth
was
at
set
values
set
are used
4.1.5
Motion
as
In cor_iunction gravimetric adopted.
adopted mass,
at
for (_,
the
was
299792.458
6378137m.
coordinates
Polar
value Earth's
adopted
been
both
period
document.
Shida
for
light
and
expansion
predicted
on
long
admittances.
Deformations
adopted
1983);
errors
giving
tldal
for
long
were
this
tidal 6
For
Details
terms
values
degree
terms
models.
prograde
These
to
point
the
expansion.
values
(Section
from
constituents,
spherical
4.1.4
calculated
The
Earth's with
the
a priori
values
for
The
gravitational
km3/s 2. semi-major
flattening adopted the
the
orbital
speed
axis
chosen laser
The
as
of
of the
1/298.257.
tracking
station
recoveries.
AI-UTI
with
reference Partials
of
398600.436
km/s.
consistent
and
product
a more frames, have
been
consistent a
zero-mean
calculated
44
definition
of
set
polar
for
of
average
the
geometric
motions
flve-day
has polar
motion and earth rotation values. used
for
in this
4.1.6
this
A
ence
priori laser
station station
u_,,1_v,,
mentioned
for
further
are
presented
and
The
4.1.9
found
elsewhere
axis
of
reference
J2000
used
that
constructed
SL-6. at
to
Station
The
McDonald,
comply
checks.
MERIT
Further
was
the
zero
partials details
upon
adopted
TX,
with
parameter
based
refer-
implemented mean
were on
the
pole
computed
this
subject
6.
the
IAU
the
the Wahr
The by
station
quality
for
System
adopted
were
perturbations
have
been
modeled
for
all
of
Definition
Coordinate
is
be
Pluto.
Z-reference
spin
files
rotated
potential
except
Z-Axis
taneous
can
Effects
Gravitational
4.1.8
coordinates
solution
before.
in Section
Body
planets
laser
were
analysis
Third
coordinate
the
coordinates
the
polar
coordinate
for
the
4.1.7
of
values
Coordinates
longitude
and
set
a priori
document.
Station
global
zero-mean
Details regarding the
have
gravity
is
provided
by
the
instan-
model.
epoch been
field
and
associated
utilized
of Wahr.
45
precession
throughout.
The
constants nutation
as
model
4.1.10
Relativity
Relativistic
4.1.11
effects
A Priori
Gravity
An a_priori orbits
and
to
correction
this regard.
of
Lerch
et
harmonic of
al
coefficients detail
model;
(Marsh
et
model;
a model
al,
al,
1982).
prepared
by
that
to
have
gravity
zero
has
other
using
the
models
different
data
following
study.
denotes a
has
also
GEM-10B'
were
This
prepared
contained
contributing
in
the the
et
that
approach
was
46
is
discussed
the
the
the
the
which models
S(2,1)
The
adoption
after
in PGS-
PGS-S4'
(Lerch
solution
and
parameters
S(2,1)
analysis
analysis
all
adopted
using
using
1981)
primes).
a priori
and
al,
C(2,1)
all
C(2,1)
by
by
SEASAT
(Lerch
spherical
STARLETTE
for
Note
differing
for
model
which
prepared
used
GEM-L2' (in
were tailored
were
model
the
the
differential
the
data
model.
that
the
solution
tailored
by
using
constraint
model
denoted
by
converge
models
this
data
to
linear
for
constraining
(as
a
constrained
been
been
order
gravity
that
new
now
satellites
resolved
sets.
but
Four
prepared
through
SEASAT
gravity
means
were
STARLETTE
The
values
solution.
prime
that
All
been
have
for
data
The
1985).
here
required
data
model
general
matrices
Justification
3.
preferred
in
obtained
zero.
a
neccessary
The
GEM-L2
to
applied.
is
gravity
LAGEOS
was
in Section
1331'
the
(1982).
original
model
a new
The
form)
the
gravity
to form
not
Modeling
construct
in
were
et were
is
GSFC's
mentioned
coefficients
were
of
several
used
with
conducting
the
4.1.11.1
Selection
of an Fields
Tailored
The a
question
priori
model
adopting hoc"
one
ones,
model
and
specific
Priori
gravitational
or
several better
question
was
"tailored"
satellite
the
of
fields.
orbit these
models
unrealistic.
The
general
models,
harmonic"
on
residuals
and
One single
set
and
approach
each
of
orbits,
along
to
orbit's All with
of
minimize
to the
single
equations. of
Nevertheless,
the it
is
a
a
one
individual
the
cally
best
specific is
series
set
"lumped
larger
in
data
orbits
as
points" for
"shifts") which could
actual
in
the
fields,
adjusted
to
already
ensure
that
the
and
residuals
possible.
note
47
This
made
answering
that
affect in
cemputed calcu-
approach
mis-modeling
fields
as
approximately by
sides was
avail-
are
latter
with
improvement,
adversely to
are
right-hand
required
this at
different
are
of
in
problem
associated using
implemented
several
equations
linear
particular
fields to
iterative
one
the
which
implies
of
with
a modification
(of
order
equations
method
"tailor-made"
in
all
However,
convergence
true
important
of
to use
chosen
problem
from
geophysi
result
Gauss-Newton
programs.
"starting
question
data
have
model
"ad
properties
the
hand,
The
non-linearities
field (or
A
the
poorly
to
the
linearized
This
various
transformations
aspects
the
other
new
observation
known the
the
whether
some
times,
was
sets
evolution.
the
for
projects)
which
the
prior
quicker
data
fit
at
linearization
values
approach
as close
these
the
estimation
previous
are
seeks
main
models
be
may
the
as
was
the
However,
can on
of
chosen
even of
Several
orbit.
correct
provide
second
the
from
lated,
field
or
sat elllte.
implies
is
models,
because
Vs.
be
concern
well.
they
accurate
starting
could This
able
a less
of
but
important
which
orbit
hand.
an
approach
adjustments, our
overall,
should
central
Tailored
very
General
linearlzation
concern
in
coefficients
The
for
Model:
field
specialized
coef f i ci ents
of
Gravity
investigated.
be
This
general
which
was
would
equations. of
of
A
preparation
a priori. compatible
way of
whether
these
the
of
linear
the
normal
non-linear
transformations. for
the
normal
equation generation, the use of "tailored fields" improved upon our ability to eliminate spurious data due to tighter editing than is possible when using a single, more general model. The main purpose here was to select a procedure that was likely to converge to the correct solution (within the accuracy allowed by the data). In order to clarify which of the two methods, the "unique starting field" or the "multiple, tailored fields," was likely to satisfy our needs best, a number of small-scale simulations of the problem were carried out. The idea was to reproduce the main characteristics of the adjustment for either approach in a reasonably inexpensive way. A more complete description of the results of the simulations is given in the next section. These simulations had the respective properties of tailored and global models. In the a priori tailored fields, were
some
clearly
a priori nearly
either
as well
the
method
choice
did as
the
situation
give
trivial
of
by
neighborhood, region.
was
should
non-llnear
As
each and shown
in the
fit
the
a
problem actual
normal all in cases
had
the the
such
solution,
equation different following
to
same
could
that be
section,
48
a
orbits
the
general
particular
perfectly
of
in
case.)
in
it,
regarded fields" this
which
view.
arc
seems
linear, case
would
the
in
a
hypersurface flat
fell to
are
happen
geometry
as
the
(There
This
well-behaved
"starting
simulated.
for
results,
this
a
hand,
being
point
in
accurate
model.
theoretical even
other
data
close the
provide
the
tailored
virtually
differences
the
On
sufficiently
from
to
simulated
corresponding
operational
neighborhood defined
the
adjusted
unrealistic.
not
problem
becomes
practical
coefficients
geophysically
model
If
if
potential
in
within have
been
this this the
4.1.11.2
A.
Simulations for General A Priori
Simulations
A
set
simulated from
a
to
be
to
perturbed the
model
satellite common
simulate plus
state
involving
same
of
perfect
recovered
observations
base
the
21 in
GEM-9.
Tailor-Made
vs
solved
a
each
data
data,
of of
applied
Terms
solution
but
the
to
is,
model
six
quality
that
on
(i.e.,
priori
three
the
with
of
Data
the
each each
were
satellite
range
no noise
the plus
arcs.
considered.
certain for
"shifted"
solving
One
observations
21
tailor-made
field)
elements
4
recovery,
permitted
then
order
contained
The
the
before
and
orbital
arc.
tailor
the
21 coefficients
model
arc
for
geopotential
degree
in the
orbital
were
resonance.
The
a priori used
for
using
employed
through
coefficients
consisted
CD ) for
it was
recovered
satellite.
general
adjust
general
were
consisted
coefficients to
order
stations
complete
When were
13th
on each
The
These
coefficient the
arc
model
perturbed
noise
Coefficient
were
laser
the
from
different
case
The
of
9-day
a
parameters
random
coefficients with
one
individually.
(CD,
Using
satellites
set
coefficients
Gaussian had
3
parameters
the
harmonic
coefficients.
state
values
parameters
B.
using
GEM-9
had
on
global
obtained
geopotential
The
spherical
data
recovered values
only
21
laser
used
with
of
equations
model
Solution
Design
simulated
normal
Geopotential Models
case and
each
to
the
normals. two Two had the
drag cases 5 cm other
applied.
Recovered
terms
of
consisted
the
spherical
harmonics
of:
Zonals
Tesserals
C(3,0)
CS(2,2)
cs(15,13)
C(4,0)
CS(I0,4)
cs(17, 13)
C(7,0)
CS(19,17)
cs(19,1 3)
C(16,0)
CS(25,23)
cs( 27,1 3)
C(17,0)
49
Resonant
Tes serals
that
were
Lumped
To terms
Coefficients
"tailor"
were
Solved
each
solved
for
individual on
Values
Except
for
values)
was
errors
in the
the
the
GEM-9
for
the
data
ANNA.
on
But
when fit
made
5BN-2. note
that,
"tailoring"
the
the
simulated
field.
In
the
greater
than
3a.
for
C(23,13)
C.
of
Satellite
same
although
constant The nearly
terms
true
for
the
Orbital
fields,
Section
than no 5BN.2 D for
used
Table
gravity
DI-D
7622
ANNA 5BN-2
on
tailored as
the
model
to
were
tailored
large models
constant
coefficient
fit for
a priori
errors
were
coefficient
error
comparisons.)
(rev/day)
Drag
Resonant Period
(CD;2)
(days)
m/day 2
e
I
.0848
39.5 °
13.05
8.4
70
7501
.0082
50.1 °
1 3.37
4.8
4
7462
.0058
89.9 °
13.46
2.4
10
km
very
adjustments
these
a
the
D the
adjustments
had
The
Characteristics
Me an Motion a
for
were
Primary
Satellite
_--25).
the
in
the
in GEM-9
for
C,S(23,13)
coefficient
for
published
solved
GEM-9
a local
field,
the
2 for
values
tailor
model
(See
were
from
better
(starting
recovered
(I0-5/_
to
a priori
50a.
not
example,
times
"tailored"
was
for
these
many
coefficient
model
represent
rule
satellite-specific
data
priori
a priori
Notice,
is
a
CS(27,13)
these
C(23,13)
The
the
Kaula's
model. on
lumped
Priori)
C,S(25,23)
obtain
of
Model
were:
a values
and
to
values
general
adjustment
from
C(17,0),
true
(A
the
Since
arcs
model,
These
terms,
where
computed
the
arcs.
Tailored
CS(27,13)
4x4
field.
satellite
and
large
base
C(16,0),
individual
priori
C(17,0),
+ 3a
was
coefficients
models,
the
for
satellite's
of Coefficients
GEM-9
g value
Satellite
individual
C(16,0),
Starting
by
5O
D.
A Priori
Satellite
Arc
Residuals
and Lumped
Coefficients
A Priori No. of Observations (±5 cm noise)
Satellite
Residuals
Tailor Model rms
General
cm
Model rms
DI-D
6937
110
ANNA
6124
133
730
5BN-2
3637
192
725
DI-D
ANNA
5 BN-2
Correct Answer: GEM-9
C(16,0)
- 5.4
- 7.8
-
18.5
- 8.5
C(17,0)
19.2
14.5
-
7.8
16.2
C(23,13)
-I 2.8
28.7
-202.2
- 7.7
S(23, 13)
-18.9
75.9
- 78.4
-10.7
Recovery
The
of
the
two
model 21
model
and
An
(i.e.,
significance Both
cases
without
starting of the
the
in
of
the
of
TOPEX
and A
solved
a
priori
GEM-9's
4.3)
in (b)
coefficients (c)
the
log
scale
of
noise
were
plotted
(GEM-9+3o),
goal
Topex was
Figure
on
the
since
plots.
5]
for the
also
in
the
the
errors plotted
data
of
(Figure
the
general
deviations
(error
goal
of 6
I/4
noise
orders
the
method. 4.2)
and
additional
was
GEM-9
the
GEM-9
show
each
(a)
where
of the
to
following
case
solutions
in
The
4.1:
over
tailor-made
comparison
solutions
standard
accuracy used
was
plotted.
the for
I/4
the
the
Errors
plotted
between
with
both
model.
uncertainties)
differences
been
using
were
accuracy
simulation,
recovered
the
general
(Figure
values
comparison. seen
of
were
coefficients
ideal
has
data,
the
25%
noise
information
equations
geopotential
methods.
10 -9
of Geopotential
normal
a priori
Units
cm
Adjusted Tailored Model Lumped Coeff ici ents
E.
Coefficient
383
a
priori
estimate) applied
error
o's
of magnitude
to for are
GEM-9_
TRUE
A PRIORI
(GEM--9
FIELD +_30)
10--6
TESSERALS ZONALS
10-7
SOLUTION
O '$
(+_s CM
RESONANT
4,0
7,0
c
c
160
170
3,0
c
s
c s
(2.2)
(10.4)
4.1
Geopotential
Simulation
52
c
s
(16.13) (17.13)(19.13)(27,13)(19.17)(_23)
21 COEFFICIENTS
Figure
TERMS
_'T--_'r-c--s-
RECOVERED
Information.
c s
F.
Summary
In
and
Figure
4.2
approximately value.
there main
(with for
most
feature
of
the
other
less
previous
set
the
most
terms) gives
errors
the
methods small
(Figure
of
the
and
has
a
a much
4.3)
zonal
larger more
4.2
the
based
is
the
but
spread
in
consistent to
the
where
the
upon
the
that
errors
for
between
terms
compared
Figure
data)
except
difference
when of
to
are
errors
significant of
two
significant
for
which
applied
differences
these
errors
model
much
a
noise
for
these not
of
(with same
is
smaller
tailored are
the
Moreover,
goal, The
Conclusions
errors than
These goal
was
model
larger
error.
noise
methods.
errors
TOPEX
TOPEX
general
with the
C(7,0)
the
two
are
the
errors
than
the
to
the
to
the
applied
data.
The simulated The
rms
solutions range
of
the
were
compared
observations residuals
gave
on
through
DI-D
the
using
following
Model
The
conclusion
slightly improvement TOPEX,
or
the
"true"
cm
"Tailored"
.025
cm
this
present
data
(no
noise).
results:
.116
is
fits
RMS
results
seen
the
General
of
better
post-solutlon
simulation
is
that
(especially
in
the
sufficiently
small
state-of-the-art,
that, either
53
the
tailored
perfect
data
considering method
may
model
gives
case)
but
the
goals
be
used.
the of
10-8
10-9
TOPEX
GOAL (1/4 GEM 90"s)
10-10
I&l
RECOVERY
0 i
ERRORS
• GENERAL
z r3 < • 10 -11
X TAILOR
X
MODEL
X
MODEL
O ee, uJ
I... Z tu
u_ Lk
_
10 -12
a N e J
0 z 10-13
10--14
X 10--15
C I
C I
C I
C I
3,0
4,0
7,0
16,0
C I 17,0
C
S
' I (2,2)
C
S
, n (10,4)
n C
q_ m
I I (15,13)
21 COEFFICIENTS
Figure
4.2
Gravity
Recovery
54
S
from
m
C mmS
q,_ w,m w
C S
* n I J (17,13)(19,13) RECOVERED
Noise.
m
_
' (27,13)
C
S
_ I (19,17)
C
S
n n (25,23)
104
10-0
10-10
10-11.
RECOVERY ERRORS • GENERAL MODEL
•
•
X TAILOR MODEL
_
•
•
10-12.
| 10-13,
10-14.
m
C 10-16
C 4_
C
I
I
,
I
3.0
4.0
7,0
16.0
C I 17,0
C
S
I
I
(2,2)
C I
S l
(10,4)
21 COEFFICIENTS
Figure
4.3
Gravity
Recovery
55
S J (15.13)
m,m
CS i,I (17.13)
RECOVERED
without
Noise.
_,S I
i
(19,13)
I (27.13)
c
s cs
I
i
(19,17)
I
I
(25.23)
G.
Interpretation
Even results
though
interesting non-linear
An
approach
is
the
converge
of
the
tailored
applied.
The gravity
coefficients
and this
been
employed
felt
safe
"tailored" a
it was
in
base of
drag
the in
is
of
it
clear
the
residuals
the
is
that
tailored
are
removed
iteration
used
when
to
the
geopotentlal
filtered-out
of work the
in
is
the
a
to
linear
common
is
values
the
in
adjusting,
40%
were
that reduce
there the
final
our
data
no
and
shift
development
of
Since and
GEM-TI.
56
orbit
was
of
the
the
21
resonant
methods We
have
however in using
resulting
normals
approach
convergence
with
of
order both
this
vs.
drag
most
ill-effects
the
residuals.
where
models.
inherent
solution. editing
practice
geopotentlal
were
data
in
drag
results
recovery
13th
orbit
tailored
since
the
the
the
the
to results
known
past
where
cases,
simplified
using
additional
geopotential
compared
since
that
from
Both
perfectly
of
of
effects
better
quite
important
as
results,
orbits.
as
well
adjustment
present
obtained
recovery
the
the
as
non-llnear
parameters
those
model the
the
considered
to
in
in
simulation
improving
adopted
it
different
coefficients
from
removing
have
concluding
models
common
benefit
in
in
in
significant,
results
remain
general
converging
models,
was
the
evident are
present
Yet,
the
effects
tailored
difference
First,
improved
effects
the
clearly
cause
the
benefit
is
of
field
terms.
both
of
a priori
not
not
coefficients
the
using
may
process
application
for
adjust
This
the
is
non-linear
apparently
general
that
model.
model
the
shows
system
These
approach
parameters
Investigation
difference.
(tailored)
to
parameters.
in
some
orbit.
general
The
the
explanation
is made
the
the
in
lumped
Future
approaches
interpret
that
and
simulation
two
effects
results.
shift
the
to
Results
the
between
with
of
had
to the
activities,
On the other hand, it was not necessary to compute tailored models for each satellite. Wewere able to adopt an approach of using available tailored fields for certain satellite analyses, and a general model elsewhere. One should exercise caution before accepting our conclusions as completely general. Wehave not attempted to assess the impact of using a truly poor model as a priori. Furthermore, the effects of nonlinearity
becomes
more
severe
degrades.
Hence
this
simulation
more of
complete the
of
simplified
provided model,
set
a and
ultimately
basis
coefficients
subset for
revealed
in the
actually
additional little
solution
would were
have
significant
adopted.
57
the
been
employed
used. insight
as
in
However, into
the
problem
matrix
more the the choice with
conditioning
conclusive
if
a
solution
instead
present
results
of the
an
a priori
approach
we
SECTION 5.0 TRACKING DATA The earliest satellite tracking systems were quite crude by today's standards. Cameraimages and Minitrack interfercmetric tracking yield satellite single-point positioning of from 10 to 100 meters. Although the observations themselves were somewhat imprecise, a large group of satellites having diverse orbital characteristics were tracked by these systems. Therefore, these observations (especially those obtained on twenty or so different orbits by a globally deployed network of Baker-Nunn and MOTScameras) have formed the basis for earlier gravity modeling activities at GSFCand elsewhere. In the early and mid-1970's electronic tracking of considerably higher precision than that obtained by cameras becamethe routine method for locating operational satellites. The main operational tracking network for NASAbecame the Unified S-Band Electronic Network. These electronic tracking systems acquired data in all weather conditions but provided data of significantly lesser precision the early laser technologies of this era.
than that produced by
Laser systems are currently the most accurate and advanced means of precision satellite tracking. These ranging systems have substantially evolved and have undergone nearly a ten-fold improvement in system precision every three years of the last decade. The evolution of laser systems typify the progress which has been madein monitoring the motion of near-earth satellites and has resulted in much more stringent demandsfor geopotential models capable of utilizing data which now are accurate to a few centimeters. The only limitation found with the lasers is their dependence on weather and the somewhat restricted number of satellites which carried corner cubes enabling them to be tracked by ground laser systems. Historically, there are ten satellites which have been tracked by NASA'slaser systems.
_U__L&NT¢_IONALL¢ 59
BL_NK
The
parallel
flexibility
within
provided
NASA
missions
which
utilized
in
operationally
data
either
for
data
SELECTION
There
are
to
gravity such
warrant
that
necessary. improved sets, for most
general
Such data
an
particularly selecting,
the
and older
qualifying
important
was
data
refraction by
and
developed
for
over
a
refraction
implementation or
some
received
inclusion
in
field
directed some
preliminary
step
with
at
is
greatly
existing
which in
GSFC
accuracies
manageable
data
the
requirements
accomplished
those
method
sufficient
determination
Therefore
processing as
be
stations
observations.
the
existing
only
been
effect.
which
orbit
has
S-band
model
this
validation
ones.
geodetic orbital
Ionospheric
pending
consideration
can
network
determination
range-rate
satellites
improvement
on
Network
The
GEM-TI
TOPEX
data
orbit
S-Band
average
improvement
handling
the
S-band
The
laser
operational
solutions.
corrupted
activities.
a four-fold
by
within
provided
Network.
ionospheric
their
quality
stringent
frequency.
sixty
tracking The
mere
single
used
perhaps
modeling
a in
been
less
S-Band
gravity
using
high than
obtained
significantly
DATA
tracking
are
not
a reliable
deleting
5. I
GSFC
significant
have
with the
laser
environment.
rather
missions by
and
obtaining
precision
tracked
are
of
tracking
past
S-Band
operational
supported
routine
of
means
required
were
The
of
the
Satellite
requirements
These
NASA's
with
accuracies.
effects
capability
the
data
framework were
deemed
creation
of
GEM-TI.
One sets
upon
objects reasonably
of
the
which which free
first a
had of
tasks
was
a selection
"satellite-only" geodetic large
field
quality
could
data
perturbations
6O
of
due
sets to
the be and air
most
important
data
computed.
The
sixty
orbits
which
were
drag
were
evaluated
according to certain
criteria:
(a) the quality,
quantity
and global
distribution of their tracking data sets, (b) the uniqueness of orbital perturbations on the satellite (d) the similiarity of the orbit to that anticipated for TOPEX(e) the distribution of the data set over the satellite's apsidal period and (f) the sensitivity of the satellite's orbit to present weaknessesin existing gravity models. The satellites which were considered are described in Table 5.1 which also shows their orbital characteristics. The satellite physical dimensions, shape and weight are also given in Table 5.1. Based upon an evaluation schemedetailed in (Marsh and Born, 1985) the ranking of the satellite data sets can be found in Table 5.2. GEM-TIhas been computed from seventeen of the top thirty ranked data sets. Almost all objects rated in the top ten have been utilized. To achieve a better sampling of inclinations, six satellites of low inclination were selected (see Section 5.2.8). Future models containing additional orbits, altimetry, surface gravity and satellite-tracking-satellite data are being planned. In all, 17 satellites were included in the GEM-TIsolution. A data summaryfor the GEM-TI solution is presented in Table 5.3. Table 5.4 describes the orbital characteristics of the satellites used in the formation of GEM-TI. The distribution of the selected satellite's orbital characteristics are shown in Figure 5.1.a. The temporal distribution of the data used is summarized in Figure 5.1.b. As is obvious from the summaries in Table 5.3, precise laser tracking played a dominant role in defining the GEM-TI gravity and tidal models. The LAGEOSand STARLETTE laser satellites especially, played a central role in both the tidal and gravity field recoveries. These satellites are completely passive orbiting objects whosesole functions are to serve as space-based laser targets. Both satellites are extremely dense spheres (area to mass ratios of .00069 and .00096 m2 kg-I respectively) covered by laser corner cubes and are in orbits designed to minimize nonconservative forcing effects. LAGEOS orbits at nearly an earth radius
6!
TABLE SATELLITE
NAME
DATE
AREA
5.1
CHARACTERISTICS
MASS
FOR
GEOPOTENTIAL
SHAPE
_
IMPROVEMENT
HI
PR
TELSTAR
621115
0.581
77.0
sphere
GEOS-I TIROS-9
651116 660115
1.23 0.6
172.5 138.0
oct. cylinder
SECOR-5 OVI-2
651201
0.288
18.0
sphere
661028
0.697
22.7
cyl.hemls.
4.839
414.80
ECHO-IRB
600920
0.23
23.0
cyllnder
2.976
BE-C DI-D
660405 670219
1.139
52.6
octagon
5.158
DI-C
670224
0.697 0.697
22.7 22.7
cylinder cylinder
5.372 5.913
ANNA-IB
640229
0.657
158.8
spheroid
GEOS-2
680310
1.23
211.8
oce.pyramid
OSCAR-7
660422
1.25
50.0
5H_-2 COURIER-1B
650426 670127
1.139
61.0
1.327
230.0
GRS
650623
0.889
TRANSIT-4A
610902
0.897
SE-B OGO-2
670316 660521
1.139 4.645
INJUN-1 AGENA-RB
610916 640615
MIDAS-4 VANGUARD-2RB
641110 660128
VANGUARD-2 VANGUARD-3
600505
0.19
AP
HI
ECC
INCL
1.986
955.89
5649.96
0.2426
44.80
-
0.659 2.165
1107.54 706.10
2276.53 2572.67
0.0725 0.1166
59.37 96.40
-
0.792
1140.15
2446.97
0.0801
3467.11
0.1835
1505.89
1702.09
0.0123
47.23
945.07
1321.12
0.0250
41.19
595.89 586.62
1888.31 1359.39
0.0848 0.0526
39.46 40.00
Sphere
69.23 144.27
2.970
1076.81
1151.81
0.0070
50.13
-
1.621
1092.09
1600.23
0.0330
105.79
cylinder
-
2.934
876.40
1222.86
0.0233
89.70
octagon
-
2.862
1096.16
1133.10
0.0025
89.95
sphere
8.230
963.38
1225.28
0.0175
28.33
99.3
cylinder
3.501
415.54
1309.79
0.0618
49.76
79.0
cylinder
-
0.694
902.89
1015.66
0.0077
66.83
52.6 486.9
octagon box
-
2.543 3.050
889.08 425.22
1087.64 1512.96
0.0135 0.0739
79.69 87.37
22.0
sphere
1007.86
0.0082
66.80
934.80
0.0004
69.90
-
0.6927
888.40
28.0
1000.0
cylinder
cyl.
-
1.276
929.08
84.5
1600.0
cylinder
-
0.980
3490.52
3752.47
0.0131
95.83
1.275
68.0
cylinder
5.273
572.15
3285.55
0.1634
32.89
600115
1.275 3.0
23.0 68.0
sphere roc.-eph.rod
5.256 4.859
573.94 513.84
3302.49 3754.57
0.1641 0.1904
32.90 33.35
ALOU-2
690721
1.0
145.0
oblate
1.906
507.65
2946.21
0.1505
79.82
LANSAT-I
720801
7.030
816.0
conc
2.728
924.20
938.78
0.0010
99.12
PEOLE SAG
710202 710103
1.539
70.0
sphere
13.121
520.93
745.25
0.0160
15.00
2.041
143.0
cylinder
14.914
522.09
563.62
0.0030
VANGUARD-I
581204
EXPLORER-7
671205
0.080 1.014
1.47 41.5
sphere double
4.421 3.417
652.11 562.75
3947.09 1080.22
0.1900 0.0360
34.25 50.31
TIROS-IRE
671106
2.168
24.0
cylinder
4.143
691.50
0.0030
48.39
AO4
661107 630101
2.168 1.883
24.0 78.0
cylinder oct.prlem
0.0170 0.2840
98.69 47.49
79.4
epherold
RELAY-I TELSTAR-2 MIDAS-7
630602
2.54
630803
42.412
SECOR-1
640204
0.496
LCS-I NIMBUS-2
650605
7.1 7.03
EXPLORER-39
660606 770407
LANDSAT-2
750202
42.084 7.03
LANDSAT-3
780403 810915
13.935
LANDSAT-4
7.03 7.03 9.935
2000.0
-
cone
-
3.012 1.213
614.92 1325.31
1.217
969.98
cylinder
-
1.001
3670.26
-
1.271
18.0
rect.box
34.0
sphere
922.92
10808.11
0.4010
42.73
3730.72
0.0030
88.41
952.11
0.0020
69.89
2875.39
0.0090
1105.93
1181.12
0.0050
2170.52 940.90
0.0950 0.0010
80.66 99.09
-
2.348
9.3 953.0
sphere cone
-
2.170 2.729
687.19 926.32
32.11 100.35
960.0
cone
-
2.730
914.89
929.46
0.0010
99.14
1496.86
cone
-
3.099
705.29
705.43
0.0001
98.20
827.0 832.0
cone cone
-
2.429 2.666
1098,47 959.37
1108.94 969.63
0.0007 0.0007
99.96 99.29
433.68
447.31
0.0010
22.76
494.37
508.11
0.0010
43.61
568.83
571.61
0.0020
28.51
531.27
535.41
0.0003
HEAO-I
770901
43.731
2720.0
cylinder
HEAO-3
791002
43.731
2720.0
cylinder
SMM SHE
800303 810701
28.903
2315.0
cylinder
STARLETTE LACEOS
750527
GEOS-3 SEASAT
750531
EXPLORER-38
12.835 6.222 10.570
437.0
cylinder
-
3.435
0.045 0.2827
47.25 411.0
sphere sphere
-
3.296 0.214
780921
1.4365 25.31
345.909 2213.6
oct.pyram£d cylinder
-
0.349 1.722
841.10 812.00
680801
4.58
190.0
tub.cross
0.152
5855.43
790812
856.79 7436.43
2710.42
cone
750705
19.97
734.04
3.04
3.623
414.0
NIMBUS-6 NIMBUS-7
781106
eph.
62
812.19 5834.25
1114.80 5944.82 857.55 818.59 5865.21
97.55
0.0206 0.0045
49.83 109.84
0.0011 0.0005
114.98 108.01
0.0004
120.64
' i"
]!
-
II
II
c
"-_ i_ --
NNNNNNNNNNNNNNNNNNNNNNNNNNN_HH_N__
d I
I
I
I-
63
Table
5.3
DATA UTILIZED Ill PRELIMINARY TOPEZ GRAVITY MODEL: 1906
SATELLITE
DATA TYPE
LAGEOS STARLETTE GEOS-I GEOS-2 GEOS-3 BE-C SEASAT DI-C DI-D PEOLE
LASER
SUB-TOTAL-
LASER
DOPPLER
SEASAT 0SCAR-14 SUB-TOTAL
- DOPPLER
GEOS-I GEOS-2 ANNA TELSTAR BE-C BE-B COURIER IB VANGUARD-2RB VANGUARD-2 DI-C DI-D PEOLE SUB-TOTAL
NUMBER OF NORMAL MATRICES
CAMERA
- CAMERA
TOTAL
58 46 48 28 36 39 14 4 6 6
144527 57356 71287 26613 42407 64240 14923 7455 11487 4113
285
444,408
15 13
138042 63098
28
201,140
43 46 30 30 50 20 10 10 10 10 9 6
60750 61403 4463 3962 7501 1739 2476 686 1299 2712 6111 38
273
153.140
580"
798,688
*PEOLE arcs contained l>oth optical and laser data. 64
NUMBER OF OBSERVATIONS
TABLE SATELLITE
SATELL ITE NAME
SATELL ITE ID NO.
ANN A- 1B BE-B BE-C COUR IER- 1B D I-C D I-D GEOS- I GEOS-2 GEOS-3 LAGEOS OSCAR PEOLE SEASAT STARLETTE TELESTAR- I VANGUARD-2RB VANGUARD-2
620601 640841 650321 600131 670111 670141 650891 680021 750271 760391 670921 701091 780641 750101 620291 590012 590011
5.4
0RBITAL
CHARACTER
SEM I-MAjOR AX IS 7501. 7354. 7507. 7469. 734 I. 7622. 8075. 771 I. 7226. 12273. 7440. 7006. 7170. 733 I. 9669. 8496. 8298.
* D -=Doppler L -=Laser 0 -=Optical
65
ISTICE
ECC
INCL. (PEG.)
DATA* TYPE
.0082 .0135 .0257 .0161 .0532 .0848 .0719 .0330 .0008 .0038 .0029 .0164 .0021 .0204 .2429 .1832 .1641
50.12 79.69 4 I.19 28 31 39.97 39.46 59.39 105.79 114.98 I09.85 89.27 15.01 108.02 49.80 44.79 32.92 32.89
0 0 L,0 0 L,0 L,0 L,0 L,0 L L D L,0 D,L L 0 0 0
0
0 0
66
o
.=_.,
[.-,
,.6 ,,4
67
above
the
tidal
effects.
1000
km,
and
stations
WEGENE
gravity
priority
and
have
extensive
Crustal
the
5.2.1
to
a
of
tidal
much
for
by
and
global
observation
sets
gravity
altitude
gravity
of
which
activities,
of have
of
of
these
network
about
long
satellites laser
and are
tracking
been
Project
and
perturbations
separation
Both
a
Project
lower
the
terms.
basis
wavelength
tidal
LAGEOS
was
Doppler
Doppler
laser
Orbit GEODYN-2
data
arcs
between
supported
MERIT,
and
the
arcs
of 8
maneuvers
the
orbital
using have
6-day
solution
also
has
listed
the
during
drag
coefficient were
28,
individual
which
in forming
Laser
four
analysis
were
undertaken
GEM-TI.
Data
1978.
parameters
are
and
atmospheric
pressure
it
orbit
covering
were
and
June
the
The
SEASAT
distinct
satellite
data
types;
is
of
S-Band,
altimetry.
program
August
to
on
by
activities
utilized
Doppler
computations
computer
laser
biases
and
reported
analysis
because
nominal
and
as
satellites
launched
laser,
The
data
of SEASAT
significance
daily
the
high-priority
Analysis
These
sub-sections
describe
major
of
spectrum
Dynamics
following
SEASAT
due
at
longest
R Campa i gn.
managers,
and
orbiting
and
high
The
for
the
a rlch
a
NASA's
only
complementary
wavelength on
senses
STARLETTE,
highly
tracked
by
and
experiences
is
short
earth,
for
in Table
the been
span
17, period
each
(CR)
determined
were for
gravity on
July
27,
with
the
which (Table
the
1978
model arcs to
arc, and
station
in
of
SEASAT
of
shortened In
the
six
a
single
or the
the
GSFC
Doppler
11,
1978.
those
arcs
lengthened computation
orbital
elements,
solar
radiation
Pass-by-pass in
the
both
October
exception
5.2.1b).
(CD),
68
14
were
determined. each
processing
5.2.1a.
performed
Doppler
coefficients
in
PGS-S4'
from
duration
September this
used
solution.
measurement
The Doppler data in the SEASAT orbital solutions were pre-edited by passing the residuals from the initial orbits through a residual edit analysis program. This program produced delete cards for passes of data that exceeded the maximumRMSvalue of 1.5 cm/sec, fell below an elevation cutoff of 5° and/or has a maximumtiming bias of 5 ms. Passes with less than 5 data points were also edited. The program also produced the initial measurementbias values for input into GEODYN-2. The laser Doppler
orbits
pressure at
and
their
nominal
of
determined
Doppler
relative
weight used
A
of
I meter
laser
orbits,
lasers
were
(KOOTWIJK),
(GRASSE)
An and
7834
estimate
of
10 cm
Doppler
for
orbits
(Table
was
at
and
the
drag
was
sampled 3rd
was
arcs
used
for
all
of
every
"true" data.
was
about
0.75
5.2.1d)
was
0.6
The overall cm/sec based
and on
was of
from
permit
proper
remaining
I cm/sec
the
all
stations
applied.
the
The for
laser
For
and 7804
the
the GSFC
(SAFLAS),
solutions.
cm/sec
for
RMS
fit
of
1.23 meters
the
constrained
observations.
Stations
deleted
noise
radiation
observation
observation. were
Solar
flexibility
2 meters
2nd
converged
to
laser
data
the
also
done
with
vs.
sigma
at
laser
and
were
Doppler
a
the
5.2.1c
parameters This
the
(WETTZEL)
the
data.
of Doppler
had
every
constraining laser
orbital
on
which
KootwiJk sampled
through
values.
and
sigma
by
atmospheric
sigma
7833
orbits
daily
computed
them
weighting
except
data
were
passing
laser
the
stations.
the
the
defining
7842
and
Doppler
combination for
orbits
a priori
the
Doppler
obtained for
the
PGS-S4
for laser
gravity
model.
5.2.2
Analysis
The navigation the
MEDOC
of
OSCAR-14
OSCAR
satellite,
satellites. Campaign,
Doppler
Data an
Data
launched for
this
international
in
1967,
satellite Doppler
69
is one were
data
of
the
obtained
program.
U.S.
Navy
as
part
of
The
data
is
Table
NOMINAL
ORBIT
5.2. la
PARAMETERS
FOR SEASAT
AREA:
25.31
MASS:
2213.6
ECCENTRICITY:
0.001
INCLINATION:
I08 °
PERIGEE
7171
km
7183
km
APOGEE
HEIGHT:
HEIGHT:
PERIOD:
1 O0
70
m
2
kg
minutes
Table
SEASAT
5.2. Ib
PRECISION
ORBITS
START ARC
NO.
YYMMDD
STOP HHMM
YYMMDD
HHMM
I
780727
O00O
780802
0000
2
780802
O00O
780808
0000
3
780808
O00O
780815
0730
4 5
780815 780818
0743 0749
780818
0748
780823
0921
6
780823
0922
7
780826
0928
780826 780901
0927 0000
8
780901
0000
780905
0000
9
780905
0000
780910
0105
I0
780910
0123
780917
0000
II
780917
0000
780923
0000
12
780923
0000
780929
0000
13 14
780929
0000
781005
0000
781005
O00O
781011
0000
71
LAUNCHED:
JUNE
28.
FAILED:
1978
OCTOBER
I0, 0
HEIGHT:
000
km
INCLINATION:
ALTITUDE
108
TKANET
II&C ANTENNA Ne.Z
$CATTE|OM| AN1|NNA$
T|II
SYNTHETIC APERTUIR| RADA l ANT|NNA
TT&C ANTENNA
No. MICIOWAV|
VIIIK IADIOMET|E
IUEDIOM|
'L|CTOIt
ALTIM|TEII
SAdl DATA LINK ANT[NNA
Figure
5.2.1.
72
SEASAT
T|II
1978
Table
NUMBER EPOCH
OF
OBSERVATIONS
5.2. Ic
WEIGHTED RMS
(cm/sec)
NUMBER OF STATIONS
780721 780727
7100 14860
1.7822 .7318
35 35
780802 780808
13511 15203
.7135 .7662
35 34
780815
6041
.6708
34
780818 780823
6723 5369
.7109 .6704
34 33
780826 780901
10808 7369
.7030 .7058
33 34
780905
8453
.8914
34
780910 780917
10404 9592
.7498 .7399
34 33
780923 781005
8934 6982
.7483 .7656
33 32
73
ARGUMENT OF PER IGEE (AT EPOCH) 180.573 193.017 153.474 116.081 146.012 141.374 124.192 51,376 99.272 292.5.90 115.672 93.448 122.805 56.247
Table
5.2.1d
ARGUMENT NUMBER EPOCH
OF
OBSERVATIONS
WEIGHTED RMS
(m)
NUMBER STATIONS
OF
OF
PERIGEE
(AT
EPOCH)
780727
676
1.4265
8
780802
986
1.3541
8
193.018 153.474
780808
1522
1.1539
8
116.082
780815 780818
424 483
1.3371 .9859
4 3
146.013
780823
355
.6760
4
124.193
780826
1129
.8644
5
51.377
780901 780905
627 664
1.0067 2.0218
4 9
99.273 292.591
780910
1289
1.7256
I0
780917
1725
1.2234
I0
141.375
115.672 93.449
780923
1785
1.3231
9
122.806
780929
1915
1.7240
9
281.185
781005
1343
1.8012
9
56.248
?4
of
particular
giving time of
complete a
global
strong
GSFC
polar
gravity
The were
importance
as
because
sampling orbit
has
of
nominal
been
orbit
25 m 2
Mass:
1000 city:
This
into
the
polar
orbit
is
first
the
determination
in
processing
OSCAR-14
data
.004 89 °
Perigee
1040
km
1085
km
Height: Height:
106 minutes
computations
for
Thirteen
7-day
computer
program.
The
October
24, arcs
1980
included
coefficient
(CR) , and
computed were
nominally
data
for
coverage
was
from
of
(743) the
a
from
the
and
six
PURPLE
solution.
the I,
orbital
each MT.
sigma
for
the
GEODYN-2
1980
for
elements,
radiation pass.
on
through
solutions
(7185).
The
gravity
GSFC
orbital
solar
for
GEM-lOB'
August of
single
biases
the
using
Computation
(CD),
observation
SHANGHAI
deleted
analyzed
adjustment
parameters
utilized
were
5.2.2a).
the
drag
OSCAR-14
arcs
(Table
atmospheric
pressure
Timing Data
all
daily
the
biases
from data
GRAZ was
I cm/sec.
An
estimate
largely
overall
(Table
a
kg
Inclination:
model.
The
used
Area:
Orbit
cm/sec,
in
field.
incorporated
parameters
Period:
(425)
the gravity
is
follows:
Apogee
were
satellite
fields.
Eccentri
these
the
RMS
due
of
the to
obtained
"true"
the for
large the
noise
variety 0SCAR-14
5.2.2b).
75
of
Doppler
receivers
orbits
was
data
was
which about
-1.2
tracked.
1.59
cm/sec
Table
OSCAR-1
5.2.2a
4 PRECISION
STOP YYMMDD
START ARC
NO.
ORBITS
YYMMDD
I 2
800801 800808
800808
3
800815
800822
4
800822
800829
5
800829
800905
6 7
800905
800912
800912
800919
8
800919
9
800926
800926 801003
I0
801003
801010
II
801010
12
801017 801024
801017 801024
13
800815
801031
76
Table
EPOCH
NUMBER OF OBSERVATIONS
5.2.2b
WEIGHTED RMS
(m)
NUMBER STATIONS
OF
ARGUMENT OF PERIGEE (AT
EPOCH)
800801
5867
1.4677
16
800808
5559
1.3992
16
800815 800822
6227 5635
1.4702 1.5358
17 17
800829 800905
58!2 5944
1.5332 1.5991
18 17
800912
5993
1.6518
17
240.671 209.115
800919 800926
6015 4519
1.6174 1.5773
16 18
187.183 187.551
801003 801010
5500 2251
1.5881 1.8217
17 13
136.816
801017
1881
1.6457
I0
801024
1895
1.7754
77
9
357.420 337.814 336.019 277.827 273.059
140.581 119.267 97.921
5.2.3
Analysis
GEOS-I 1978
have
cycle
of
The
been
of
stations
this
in
orbit
NASA
third
even
and
tracking
the
was
velocity
in
observation the
vectors
of
radiation
pressure
model,
the
and
convergence solar purpose
of
position
and
normal
solution, passes.
One
for.
(2) air
to
A
total
and
parameters
of
101
delete (C D)
(CR) arcs
passes
provide
data
and
little
and
of 104
the
arcs
or
number
survived
summaries
78
one
It
of
the
was
the
of
the
(I) to
the
used
position
and
drag,
In
air each
of
for
tide
converged.
creation
day
solar
ocean
more
nonrellable
this
more
obtain
in
each
a
an
for
to
select
get
air
vectors,
adjusted
to
to
for
were
velocity
for
decided
parameters, arcs
be
measurement/sec,
observation
coefficient
survived
of
values
were
to
coefficient pressure
of
estimates
preparatory
and
with
SAO
twofold:
("E"-matrlx) identify
periods
total
tidal
field,
is
time
data.
every
position
vectors
drag
radiation
SAO
earth
gravity
equations
solar
solved
velocity
and
the
nominal
convergence
one
coverage.
the
a frequency
satellite,
pressure
the
at
the
the
than
temporal
catalog
5.2.3b
Using
process,
more
14,
data.
and
solid
December
to
A
weighting.
and
to
spans
good
number
data
GEM-lOB"
radiation
the
and
for
was
period.
provided sec
period
those
to
5-day
1977
stations.
eliminating given
20,
providing
procedure
5.2.3a
NASA
balance
NASA
Tables
data
This
thus
the
any
one measurement/7.5
every
and
was in
January
analysis.
arcs,
involved
The
one
step
Attention
satelllte's
of
SAO
Data
period
perigee,
both
5-day
scrutiny.
with
of
Ranging
the
for
argument
it into
coverage.
from
chosen
first
no
Laser
data
involves
The divide
laser
the
data
of GEOS-I
the
drag
The
accurate the
matrix
gravity
the
procedure.
a
5-day
whole
and
arc.
field
measurements of
the
and/or arc arc
and were
Table
ORBITAL
Semi-major
DATA
5.2.3a
FOR GEOS-1
8080
axis:
km
Eccentricity-
.O7
Inclination
59?4
Perigee
I 135
km
Height-
2270
km
of Launch:
1965
Apogee Year
Height-
m 2
Area:
1.23
Mass:
172.5
Period-
120
minutes
540
days
Period
of Arg.
of Perigee:
79
kg
Table
TRACKING
5.2.3b
DATA
GEOS-
• SATELLITE:
• TIME
•
DATA:
•
ARC
SUMMARY
1120177-
PERIOD:
SAO
ARCS
(INCL.
• NO.
OBSERVATIONS:
+ NASA
5 DAYS
LENGTH:
• NO.
1
NASA):
1o i (SO) 129,371
8O
12/14/7B
LASER
Table SUMMARY
OF
ARC EPOCH YYMMDD
* * * * * * * *
* *
* * *
* * * * *
*Includes
5.2.3c GEOS-I
NO. OBS.
770120 126 207 213 311 321 329 403 4O8 413 418 423 428 503 508 524 603 608 613 618 623 628 703 708 713 718 723 729 803 8O8 818 825 830 904 916 921 928 1003 1008 1013 1024 1029
838 904 724 752 616 1169 978 1303 1359 1589 1061 1649 2084 1778 1525 1085 1520 1830 1331 1245 1637 1240 1235 1255 1238 1095 704 1512 1728 1513 1151 1614 1364 1739 1661 2343 1804 908 1207 1647 1706 1598
NASA data
81
ORBITS
RMS (m) 0.886 0.721 0.821 0.848 0.850 0,744 0.463 0,816 0.658 1.088 0.890 0.794 0.801 0,717 0.771 0.933 0.782 0.949 1.345 0.714 1.073 0.788 1.025 1.141 0.836 1,077 0.655 0.959 1.326 1.063 0.828 1.081 1,153 1,189 1.458 1.106 1.452 0.652 1.707 1.507 1.424 1.340
Table
5.2.3c
cont.
ARC EPOCH YYMMDD
* *
* * * * * * * *
* *
* * * * * * * *
* * * * * *
771103 III0 1116 1126 1201 1211 1216 780123 201 209 217 222 308 314 322 330 404 413 419 424 429 504 509 514 520 528 6O2 607 613 625 630 705 710 715 720 725 730 8O4 809 820 825 830 906 919
*Includes
NASA data
82
NO. OBS.
RMS
(m)
1195 1295 1359 961 1089 1114 801 1196 1075 1039 1280 1644 864 985 827 885 942 894 940 1465 960 1313 1810 1049 1065 1092 1443 1700 1533 1478 1329 1670 1440 1212 938 632 1329 1318 742 683 771 961 789 1770
1.815 0.742 1.137 0.859 0.649 0.915 0.876 0.8O4 0.880 0.798 0.868 0,783 0.806 0.754 0.767 0.821 0.804 0.761 0,681 0.937 1,010 0.815 0.932 0.838 0.789 0.871 0.860 0.982 0.841 0.949 0.805 1,199 0.928 0.697 0.997 0.773 0.925 1.112 0.933 0.852 0.793 0.488 0.529 0.718
Table
5.2.3c
cont.
ARC EPOCH YYMMDD * * * * * * * * * * * * * *
Average
780924 929 1004 1009 1014 1019 1024 1029 1105 1110 1115 1120 1125 1204 1209
NO. OBS.
RMS (m)
1315 1468 1620 1975 1890 1189 2034 1278 1169 1227 1380 1571 865
0.793 0.908 1.044 0.579 0.969 0.807 0.701 0.826 0.967 0.709 0.753 0.973 0.658
912
83
I'_10
dl, .
I_
A
0.843
*Includes
rms - 0.912 m
1
NASA
data
.,_'
Finally, for
each
the
overall
arc.
m.
tion
5.2.4
laser of
the
of
to
RMS
This
GEOS-I
E-matrix
RMS
fit
The 0.91
one
the
is
provided
Analysis
The
Geodynamics
9,
1975.
parameters
are
April
the
m
to
in Table
1.8
the
m,
indication
of
5.2.3c
with
an
of
the
data.
to
the
determina-
vintage
contribution
average
of The
field.
Earth
Ran_ing
and
satellite
Data
Ocean
Satellite,
GEOS-3,
characteristics
and
was
the
I. 4365
Mass :
345. 909
Eccentricity:
0.O011 4
Incl inat i on:
115 °
Perigee
84 0
km
:
860
km
Period:
102
minutes
1039
days
Height: Height
Argument
of
available stations
prepared
launched
nominal
on
orbital
following:
Orbital
tracking
0.4
was
an
Area:
Apogee
The
provide
presented
important
of Laser
The
equations)
arcs
considering
an
gravity
normal
the
are
from
good,
of
for
They
ranged
quite
data
values
data.
values
Earth's
GEOS-3
fit
(matrix
Perigee
data during
Period:
were the
obtained
years
1975
follows:
1975:
196916
meas.
1976:
193405
meas.
Total:
389421
meas.
(SAO:
18%)
84
by and
m2
both 1976.
kg
NASA It
is
and
SAO
distributed
laser as
Past experience at GSFCindicates that a 5 to 7 day arc length is optimum for the analysis of data acquired on geodetic satellites at 800 to 1000 km orbit heights. This time span provides strong gravitational information without excessive contamination from nonconservative force effects such as atmospheric drag and solar radiation pressure. A 5-day arc for GEOS-3covers approximately the period of the effect produced by the resonant 14th order coefficients of the Earth's gravitational field. This effect can reach magnitudes of 150 meters in the alongtrack component. The gravitational field used in the computations was the GEM-10B"model complete to degree and order 36, derived from satellite tracking data, surface gravity and altimetry. density was that of the Jacchla 1971 model.
The atmospheric
Forty-elght arcs covering the time period from May, 1975 to December, 1976, have been analyzed using the GEODYN Program. The editing applied to the data consisted of several stages. There was a preliminary selection based on existing knowledge concerning the quality of the data obtained by different stations at different times. The internal Finally,
consistency of the data was checked on a pass by pass basis. the dynamic editing inherent in GEODYN was applied also. The atmospheric drag model formulation allowed the estimation of
a daily drag coefficient (CD), and the force model for the solar radiation pressure incorporated a single coefficient CR for every 5-day arc. The solid earth tidal effects were modeled after Wahr's formulation, the ocean tides force model used a spherical harmonics approach due to D. Christodoulidis, et al. (1986b): the long wavelength components of approximately 600 constituents were used in the calculations and the coefficients when computing a solution.
of about 60 are actually
estimated
The trajectory generated using these estimated parameters was used to compute an RMSvalue for each 5-day arc, which provided an
85
Table
GEOS-3
ARC EPOCH 750519 750524 750614 750619 750629 750709 750724 750729 750828 750902 750907 750929 751118 751123 751216 760108 760113 760205 760210 760217 76O222 760227 760404 760409 760417 760422 760427 760502 760507 760523 760601 760606 760614 760621 760913 761004 761009 761018 761023 761028 761102 761107 761112 761117 761122 761127 7612U2 7612U7
ORBIT
5.2.4a
DETERMINATION
NO. OF NEAS. 356 435 910 662 926 1120 796 876 1705 1240 1501 336 537 488 1333 903 1533 1219 2078 1450 1184 1801 1009 1217 1178 1112 2307 1866 1193 1010 1003 974 900 804 848 1641 IO85 878 1031 1072 810 1634 984 1394 1527 955 610 839
86
RESULTS
RMS (METERS) 0.510 0.273 0.559 0.679 0.633 0.757 0.469 0.363 0.596 0.459 0.527 0.571 0.613 0.593 0.485 1.542 1.454 1,202 1.237 0.809 0.869 1.300 1.487 1.282 1.186 1.380 1.443 1.391 1.079 1.218 1.231 1.374 1.465 1.319 1.480 1.309 1.432 0.904 1.145 1.641 1.547 1.126 0.965 1.369 1.386 1.294 1.383 1.306
Figure
5.2.4a.
GEOS-3
87
Spacecraft.
,--
l,
?
©
t: 0
[]
c_
'lie
C ©
!
4Z J •
c_
0
B qm
e_ o
!
0_
O
c_
vJ
O
_Z _Z Z
88
O
indication given the
of
in
Table
normal
RMS
values
in the
The
SAO
stations
the
use
of
each
arc.
The
converged
arcs
were
used
1976 arcs
are
section
in
connection
fulfilled
which
for
data
results to
are
compute
documents
the
equations
editing
presence
accurate
than
global
coverage
which
the
of NASA
would
be
Data
various
STARLETTE
criteria
contributing
to the
less
Ranging
the
the
a
due
alone.
Laser
with
are
provide
of NASA
STARLETTE
This
normal
the data The
stations,
Analysis
which
to
below.
SAC
with
effort
fit
the
stations.
5.2.5
5.2.4a
higher
from
lacking
overall
equations.
The data
the
stages
laser
were
to
the
of
the
ranging
data
subsequently
estimation
data
reduction
set.
used
The
to
data
form
the
of
the
TOPEX
model
by
the
French
Space
parameters.
STARLETTE Agency
in
is
1975.
effort
each
station
Based
on
results
has
of the
with
those
more
data
of
of
data
that
August
range
1984
and
size,
shape, The
ranges
per
we
six
seconds that
using
normal
was
data
in
and
orbital
data
used
such
{whenever
this
way
that
produced laborious
We
first
in
available).
The
avoided. the
a
procedure
points.
thus
covering
mass
STARLETTE
sampled
decided
points
being
have
for
raw
the
launched
5.2.5a.
obtained
normal
analysis
passes
set
one
its
Table
experience
available of
satellite on
in a
about
to
much
through
tracking terms
of
forming
These 1984
given
consist
similar
completed 1984,
is
previous
process
geodetic
Information
characteristics this
a
have
eight
only
months
of
available.
been
selected
period.
analysis individual
for
Table from
89
5.2.5b
each
ranges
analysis
shows
station.
per
cover
station
the
the
The
January
amount
of
breakdown
in
gives
a
rough
Table
5.2.5a
ORBITAL fiND PHYSICAL CHARACTERISTICS OF STRRLETTE (7501001)
APOGEE
HEIGHT
PERIGEE
1105
HEIGHT
010
ECCENTRICITY
0.02
INCLINATION
49°.0
PER IOD
104
ASCENDING ARGUMENT
NODE
RATE
OF PERIGEE
km km
min.
-3.94 RATE
3.30
"/day °/day
AREA
0.04524
MASS
47.250
SHAPE
SPHERE
RADIUS
12 cm
ONBOARD
INSTRUMENTATION
m 2
kg
RETROREFLECTORS
9o
indication
of
the
prior
experience
priori
models,
a
edited
using
the
programs
to
with
analysis
in
looked
was
suspect
creatin_ shows
a
stable
a residual
number
of
edited give
plot
where of
and
the
Tables
5.2.5d
and
improvement
is
our highly station
models
the
previous
as
covering
sample
2.2
periods
ascending
node.
recent
data
to
ongoing
the
and
the
complicated
(at
records and
Starting
which
this
process was
the
of on
tedious,
with
the
resulted typical
RMS
until
to
1984
new
is
in a
through
the
very the
and
editing
1984.
2.6
stations
for
which
made
the
time
more
same
fits
at
significant "normal-
9]
the
have and The
of the
same
These
with
the
5-day
periods
intense large
be
fits
summary
process by
we
and
a
To
forty-six
August
perigee
to
PGS-T2.
data
analyzed
and
had
model
a
The
data
at
have
5 _ _
process,
gives
in
converged.
the
of
station
outlier
apriori
set
that
beneficial
it
model,
same
points
latter
campaign.
satellite at
the
and
_g,_o
this
5.2.5f
characterized
tracking
"raw"
fit
the
MERIT
any but
of
start
time)
through
We
argument
period
the
repeated
and
residual
curable
edited
The
show
data
was
quality.
the
tables.
an
and
problems
_.
see
TOPEX
on
January
the
chose
participation
performance
editing
of
We
since
from
process
t_k_ng
Table
again
two
a period
this
which
editing
The
data
a
were
constants
process.
for
generation
based
the
lacking
clearly
significant.
by
of
was
achieved
5.2.5e
summary
edit
process
was
first
statistics
arcs
whole
what
on
can
our
data
with
on
on station-by-station
to
_r
of
The
appended
was
_t
one
choosen.
locating
data,
Based
quality
here
questionable
into
based
of
bias-fr_
insight
included those
and
a
in
philosophy
abundance
was
determination
documentation
residuals
manually an
The
the
gives
network.
the
analysis
orbit
this
package
invaluable
where
Given
length
residual 5.2.50
in
considering
software
dynamical
outliers.
problems.
arc
Table
package
eliminating
nominal
post-fit
the
rates
and
GEODYN-II
basis.
used
repetition
STARLETTE
5-day
perform
pass-by-pass models
varying
arcs
of
the
the
more
tracking
due
amount we
editing
had
of no
effort
data prior more
important.
I-2
reduction
meters to
equation-forming"
level about
the 60
cm
stage.
A
Table
5.2.5b
STBRLETTE DBTB CRTBLOG JANUARY
1983 - AUGUST
SUMMARY
TOTAL TOTAL
BY STATIONS
NAME POTSDM ML0306 ML0502 ML050I ML0702 ML0802 ML0602 ML0201 MLOI01 ML0601 HOLLAS FINLAS HELNAN KOOLAS HETZEL GRASSE SHOLAS GRAZ RGO ARELAS MATERA DIOLAS
LOCATION POTSDAM, DDR SAN DIEGO, CA. AUSTRALIA GREENBELT, MD. GREENBELT, MD. QUINCY, CA. MONUMENT PEAK,CA PLATTEVZLLE, CO. HUAHINE, FR.POL. MAZATLN, MEXICO MAUZI, HANAII METSAHOVI, FINN. HELHAN, EGYPT KOOTNIJK,HOLLAND HETTZELL, FRG GRASSE, FRANCE SIMOSATO, JAPAN GRAZ, AUSTRIA HERSTMONCEUX, UK AREQUIPA, PERU HATERA, ITALY DIONYSOS, GREECE ZIMMERHALD, SNIZ
NO. NO.
1984
NUMBER 1181 7062 7090 7102 7105 7109 7110 7112 7121 7122 7210 7805 7831 7833 7836 7835 7838 7839 7860 7907 7939 7960 7810
OF PASSES = 2792 OF OBSERVATIONS =
92
PASSES 59 3 66 1 105 288 270 208 61 56 37 12 12 32 50 7 126 106 56 939 289 6 29
127662
POINTS 1271 30 3669 5 5699 18267 12059 8589 1033 2733 1661 209 376 619 1602 111 4690 3665 1609 66366 15089 81 691
Table
5.2.5c
DATA REDUCTION mODEL FOR STARLETTE DATA EDITING
PARAMETERS
GENERAL
3.98600436
GM SPEED
299792458.0
OF LIGHT
ae
6378137.0
II!
298.257
JPL
EPHEMERIDES
I0 t4
mZls 2
mls
m
DE-200/LE-200
ATMOSPHERIC
GLOBAL
x
DENSITY
MODEL
JACCHIA
Ig71
PARAMETERS
GEOPOTENTIAL
PGS
TIDES
APRIORI
TOPEX
MODEL
APRIORI
TOPEX
SERIES
POLAR
MOTION
STATION
ARC
&
EARTH
ROTATION
POSITIONS
133
LAGEOS
I"(36
SL6
PARAMETERS
STATE
VECTOR
6
DRAG
COEFFICIENT
SOLAR
RADIATION
MEASUREMENT
I I
COEFFICIENT
NONE
BIASES
93
x 36)
SOLUTION
OE POOR
STATION ITER 1 Z 3
NATERA1
MUSED 76 75 7B
NEASUREMEKT TYPE,
BASE IhqS 6.630 0.674 0.674
MBIAS 79396101
51
DELETE?9394101
51
NMHHSS.SSS RESID 02.5 321716.017 0.334 I 221724.016 0.665 I 221732.016 0.356 3 221748.013 0,262 I 221756.013 0.104 I 221804.015 0.636 I 221812.015 0.206 I Z21820.014 0.632 t 221828.014 0.230 1 221836.016 0.218 I 221844.013 O,ZO0 I 221852.013 0.066 I 221900.013 0.022 I 221908.012 0.152 I 221016.012 0.087 I 221024.012 038.530 IE 221932.012 0.784 I 221960.011 J.J08 I 221948.011 -0.033 I 221956.011 0.615 I 222004.011 0.028 I 222016.010 -8.124 | 222024.010 -0.255 I 222032.010 -0.167 I 222039.999 -0.228 I 222048,069 -0.145 I 2ZZ056.029 -0.121 I 222104.029 -0.203 I 222112.029 -0.206 I 222120.029 -0.315 I 222128.028 -0.361 I 222136.028 -0.570 I 222144.028 -0.345 I 222152.038 -0.405 I 222200.038 -0.420 I 222208.030 -0.668 I 222220.088 -0.305 I 222228.088 -0.394 I 222236.108 *0.620 I 222300.148 -0,510 I 222300.140 -0.436 | 222316.168 -0.625 I 222323.968 -0.401 I 222343.948 -0,310 I 222351.958 -0.602 I 222359.959 -0.260 I 222407.959 -0.264 I 222636.009 -0.261 I 222444.010 -0.231 I 222452.000 0.513 I 222500.000 -0.104 I 222507.990 -0.122 I 222515.981 -0.111 I 222523.901 -0.037 I 222531.991 -0,032 I 222339.991 -0,047 I 222548.002 0,007 I 222536.002 0.050 I 222604.002 0.020 I 22Z612.002 -0.014 I 222620.003 0.061 I 222628.003 0.172 I 222636.003 0.001 I 222648.004 0.160 I 222656.004 0.160 I 222704.016 0.300 I 222712.013 0.290 I 222720.005 0.233 I 222728.003 0.336 I 222736.006 0.275 I 222744.006 0.293 I 222752.006 0.206 I 222804.007 0.253 I 222812.097 0.351 Z 222828.008 0,604 I 222860.008 0.360 |
Figure
5.2.5a.
RANGE
MIDPOINT
NFJ4 RMS
BIAS
4.622 1.306 0.306
-0.998508 -0.482160 -0.482160 -0.6&21603
QL_.LffY
TIME
(YY_qDD
HHMMSS),
(860115
SIOf4A
TIME
0.114723 0.113527 0.115527
222258) IIAS
TIME
BIAS
SI_
0.024592 0.024806 0.024806
0.200471 0.077130 0.077130
860113221716.02860115222842. 840115221924.0109
860113221926.0129 0
02.5 I I I I I I I I I I I I I I I I I I I I I 2 I I I I I I I I I I I I I I I I I I 2 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
HillE
I I !
• •
I
i
:
! i
-
IM ! I nI I m m I I I It I • ! E I u •
"
! I I I I I I
" • • K •
:
1 •
! I
-
!
:
l
E
I
I
!
:
1 " " •
I I I I
•
"I n! "I ii N IN I " i= l = I • I m I • I • I •
I ! I I I
: • m = •
Example of Residual Analysis Package Matera Residuals Plotted Versus Time.
94
Diagnostic
Plot
from
Starlette:
ELEV 10.75 11.63 12.14 13.60 14.36 15.15 15.96 16.79 17.65 18.34 19.66 20,41 21.39 22.41 23.46 24.55 23.68 26.84 28.05 29.31 30,60 32.63 34.04 35,49 36.B8 38,51 60.07 61+65 43,24 64+84 46.42 47,97 49+47 S0.88 52,17 53.32 54.70 55,34 55.73 55.30 54,66 53.80 52.78 69.59 43.15 46.67 45.16 39.87 38.41 56.99 35.59 34,24 32,03 31.66 30.63 29.24 28.09 26.98 25.90 24.87 23.86 22.89 21.96 20.61 19,74 10,90 18.09 17.30 16.53 15.79 15.06 14.36 13,34 12.68 11,61 10,49
Table
5.2.5d.
STANAN
IOTA
NAZI WtAZl WtAZl NAZI NAZL OAAZ) NAZ1
78393401 ?8393401 78393401 ?8393401 ?839340| 78393601 78393401
Example of Residual Analysis for Starlette Laser Passes During Period of January 3 to 7, 1984: Statistics Based on Apriori Model (Pgs 1331')
NTYPE
YYIWIDO
RANGE RANG[ RANG[ RANG[ RANGE RANGE RANGE
840103 840105 840105 840106 840106 840106 840107
TOTAL POINTS THIS STATION, NMS OF UNADJUSTED DATA, RNS OF ADJUSTED DATA,
STANAN
IOTA
LAOUII02 LAOUII02
71100402 71100402
|STA
NATEHAI HJTERA1
79306101 79394!0!
IqTYPE
YYlqI¢DD
RANGE RANGE
84010T 84010?
IOTA
PL&TVL1
71120201
OUINL092 QUINI092
SIMOSATA
_rNIADO
NWqlASS
840103 E40104
50556 53318
NTYPE
YYNADD
HHNHSS
RANGE
840105
104850
ZSTA ?1090BOZ 71090802
ESTA TSSIKll
YARAO| YARAGI
|STA ?0900501 78900501
NUMBER NUMBER
IUqS
OF OF
YY1N_DD
RANGE RANGE
840105 840106
NIINHSS 85703 91659
-8.041 00.320 0.131 0.39? 0.$90 0.039 -i.B4S
SlOIAA 0.178 0.148 O.•6S 0.161 D.I34 0.206 0.621
T.
BIAS 0.539 0.223 0.097 0.147 "0.003 -0.273 0.337
SlORA 0.000 0.043 0.383 0.063 0.040 0.063 0.386
FIT
It•IS 0.157 0.008 0.046 0.044 0.091 0.045 e.R4g
BASE
RNS
2.211 0.883 0.417 0.502 0.486 0.923 0.641
IqAXEL 42.6 80.0 67.6 45.3 81.0 71.6 44.Q
NPTS
ITER
16 34
3 3
USED
8|AS
16 34
1.157 -1.EBB
BIWSA
USED
J|AS
S191_
7? 61
0.1S3 0.303
0.114 0.220
BIAS
SIOHA
T.
BIAS
8.$24 0.541
8.178 0.509
S[GHA O.OT8 0.106
FIT
ItHS
BASE
l. ET6 0.043
OHS
0.993 0.630
HAXEL 50.6 36.2
IIPTS
ITER
81 A6
6 S
T.
|%AS -8.020 J.iJ?
SIgP.A 8.030 0.83_
F%T
ItPtS
0.097 8.219
BASE
RHS 0.199 8.604
IU, XEL 39.4 39.2
NPTS
ITER
40
USED 4
38
"1.100
B.201
T.
JZAS -0.002
SI_qA 1.059
FIT
RHS 0.118
BASE
RNS
1.140
HAXEL 41.0
NPTS
ZTER
31 30
S S
USED
8|AS
S1OlqA
31 89
0.2S9 1.088
0.439 0.649
USED
81AS
SIOHA
0.64•
$.TS2
SEAS
SIONA
YYMIqDD
NHIBISS
UI4G[
BAOIBS
2SLOBS
BIAS "O.iiE -0.349
SIOHA
FIT
0.125 0.147
I_tS 0.O26 0.O6S
BASE
OHS
0.114 0.444
HAXEL 38.8 60.3
NPTS
ITER
•
3
•
T.
BIAS 1.6T3
SIOHA O.TIS
FIT
IUqS i.120
JADE
RNS
4.333
NAXEL t6.S
9 4.$33 0.180
NTYPE
YYNIqDD
NH_qSS
BARGE RANGE
840105 840106
154412 121436
NPTS ?Z B4
ITER
USED $ 7
?E 40
0.178 -0.09T
121 0.34| O.JSO
608
14
OF PASSES PROCESSED DELETED FJT|OELYs
T.
70 0.336 B.031
IqTYPE
UNADJUSTED DATA, ADJUSTED DATA,
i|AS
53 66 10 50 DA 85 $
3.140 0.118
ITTYPE
POINTS INPUT= USED• G_4 EL CUT = l OTHER EDITS 8
USED 3 $ 3 3 S 3 S
38
TOTAL POINTS THIS STATION, ORS OF UNADJUSTED DATA_ RM3 OF ADJUSTED ])&TAt
TOTAL TOTAL TOTAL TOTAL
ITEB
33 46 10 39 $6 ZS •
238 0.474 B.161
TOTAL POINTS THIS STATION, RHS OF UNADJUSTED DOTAl I_S OF ADJUST[D DATA,
STANAH
74440 95513
RANGE OSNGE
TOTAL POINTS THIS STATION, RHS OF UNADJUSTED DATAs IUqS OF ADJUSTED DATA,
STANAfl
NHNqSS
IqTYPE
TOTAL PoIwrs THIS STATION, I_S OF UNADJUSTED DATA: IUq3 OF ADJUSTED DATA_
STANAH
NPTS
SO 0.743 0.153
_OTAL POINTS THIS STATION* RHS OF UNADJUSTED DATA, RlqS OF ADJUSTED DATA,
STANAN
I8643 20521 35558 3600 22456 41648 B416
218 1.048 0.089
TOTAL POINTS THIS STATION, ItHS OF UNADJUSTED DATA, RHS OF ADJUSTED DATA,
STAHAH
M¢NIASS
•
IT I 0.877 0.188
95
|.118 0.144
T.
BIAS -LOST 0.095
SZOHA 0.026 0.039
FIT
RHS 0.063 0.020
BASE
RHS 0.317 0.380
HAXEL 63.0 40.4
Table
STAHAH
5.2.5e.
IOTA
GRAIl GRAZ1 ORAZI GRAIl ORAZ1 GRAIl aRAZI
78393401 78393401 78393401 78393401 78393401 78393401 78393401
Post Fit Residual January 3 to 7,
NTYPE
YYNqDD
RANGE RANGE RANGE RANGE RANGE RANGE RANGE
840103 840105 840103 8A0106 840106 840106 840107
TOTAL POINTS THIS STATION* RMS OF UNADJUSTED DATA, JUqS OF ADJUSTED DATA*
STANAH
|STA
LAGUII02 LAOUII02
71100402 71106402
IOTA
NATERAI IqATERAI
79394101 70394101
NTYPE
YYHNDD
RANGE RANGE
840107 860207
ISTA
PLATVL1
72120201
:STA
RUINIO92 OUIHI092
71090802 71090002
NTYPE
YYIg¢DD
RANGE RANGE
840103 840104
IOTA
SIHOSATA
YARAGI YARAGI
NUMBER NUHOER
ITER
50540 33518
104855
-0.087 -0.307 0.11S 0.024 -0.092 -0.199 -0.303
0.178 0.148 0.960 0.161 0.234 0.206 0.818
T.
lIAR
0.104 -0.041 -0.038 -0.122 -0.019 -0.103 -0.056
$|0K4 0.080 0.043 0.381 0.063 0.040 0.063 0.386
PIT
I_S 0.049 0.000 0.046 0.020 0.047 0.069 0.049
BASE
RMS MAXEL
0.242 0.428 0.064 0.320 0.126 0.454 0.233
42.6 80.8 67.6 43.3 81.8 71.6 64.0
USED
lIAR
16 34
°0.231 -0.042
SIN 0.324 0.341
T.
BIAS
-I.O01 -0.026
$101U 8.078 0.106
FIT
ImS 0.094 0.036
BASE
RMS MAX[L
0.202 0.130
50.6 36.2
76 61
2 3
USED
BIAS
SIOMA
76 60
0.158 0.130
O.I1S 8.230
BIAS
31WqA
T.
BIAS -0.018 0.023
SIOHA 0.050 0.033
FIT
RNS 0.095 0.118
BASE
RNS
0.201 0.196
HAXEL 39.4 39.2
NPTS
ITER
USED
38
2
HHI_SS
NPTS
ZTEO
85703 91659
31 59
38
-0.237
T.
RIAS
I.EOB
0.004
310MA 0.059
FIT
RNS 0.073
DARE RHS _UXEL 0.254
A2.O
38
YYIqI¢DD
RANGE RANGE
840105 840106
2 Z
USED
BIAS
SIOHA
31 39
8.103 0.270
0.438 0.648
BIAS
S|OMA
T.
BIAS
0.058 -0.033
SZONA 0.223 0.147
FIT
RK5 0.026 0.032
BASE
RNS iUXEL
0.512 0.135
38.8 60.5
70 0.227 0.029
RTYPE
YYNNDD
NHHIqSS
RANGE
840105
231608
NPTS 9
ITER
USED 2
9
-0.656
3.503
T.
BIAS -0.364
SIW_A
FIT
1.743
RMS 0.117
BASE
RflS
2.691
flAXEL 46.3
9 2.691 0.117
RTYP[
Y'Y14NDD
HHI_ASS
RANGE RANGE
840105 840106
134412 121436
NPTS 72 53
ITER 2 Z
USED 72 53
BIAS -0.051 -0.1S8
223 O.l?S 0.043
647
1
OF PASSES PROCESSED= DELETED ENTIRELY=
RMS OF UNADJUSTED DATA, RNS OF ADJUSTED DATA,
SlOIU
i.254 0.073
MTYPE
POINTS INPUT= USED: 646 EL CUT= 0 OTHER EDITSt
BIAS
53 46 10 39 56 23 9
ITER
NIAlg¢SS NPTS
HNI_qSS
IOTA
NPTS
2 2
840105
7090050! 70900301
USED
2 2 2 E Z Z 2
16 34
YYWqDD
TOTAL POINTS THIS STATIONt RMS OF UNADJUSTED DATA, RNS OF ADJUSTED DATA*
TOTAL TOTAL TOTAL TOTAL
74440 93513
RANGE
TOTAL POINTS THIS STATION, RMS OF UNADJUSTED DATAt RNS OF ADJUSTED DATA,
STAHAN
NHIg¢SS
NTYPE
78383601
ITER
136 0.190 0.205
TOTAL POINTS THIS STATION, RMS OF UNADJUSTED DATA, RRS OF ADJUSTED DATA,
STANAR
53 46 10 39 56 ZS 9
of
SO
TOTAL POINTS THIS STATION, RHS OF UNADJUSTED DATA, IU¢S OF ADJUSTED DATA,
STANAN
HPTS
12643 20521 35550 3600 22434 _1448 B¢16
Period
0.177 0.059
TOTAL POINTS THIS STATION, RNS OF UNADJUSTED DATA, IUqS OF ADJUSTED DATA,
STANAfl
HHflNSS
During
218 1.302 O.OSS
TOTAL POINTS THIS STATION, RMS OF UNADJUSTED DATA, RNS OF ADJUSTED DATA,
STANAM
Analysis for Starlette Laser Passes 1984: Statistics Based on Pgs-T2.
27 0 O. 367 0.067
96
SIN 0.118 0.138
T.
RIms
-0.015 B.046
5IOIU 1.026 0.038
FIT
RNS O.OAO 0.047
BASE
RNS
0.088 0.252
NAXEL 65.0 40.4
Tible
STARLETTE
RESIDUAL
STATISTICS
SUMMARY
APRIORI STANAN GRAZ1 GRAZI GRAZI GRAZI GRAZI GRAZ1 GRAZI LAGUII0Z LAGU1102 IqATERAI IqATERAI PLATVL1 RUIN1092 QUIN1092 SINOSATA YARAG1 YARAGI
ZSTA 78393401 78593G01 78393401 ?0393401 ?839340| ?8393401 78393401 71100402 71100402 79394101 79394101 71120261 71090802 71090802 78383601 ?0900501 70900501
5.2.5f
NTYPE
Y_gqDD
HHI_3S
RANGE RANG[ RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE
840103 840105 840105 840106 840106 840106 840107 840107 840107 840103 840104 840105 840105 840106 840105 840105 840106
12645 20521 35558 3600 ZZ454 41448 5416 ?4440 93513 50554 33518 104850 85705 91659 231608 134412 121436
MODEL
HPT5 33 46 10 39 56 25 9 16 34 81 64 40 31 39 9 72 $4
•TEA
USED 3 3 5 3 3 3 3 3 3 6 5 4 5 3 3 3 7
53 A6 10 39 56 25 9 16 34 77 61 38 31 59 9 72 49
(PGS 133 I') 51AS 02.001 -0.329 0.131 0.397 0.390 0.039 00.048 1.157 -1.209 0.153 0.585 -1,190 0.259 1.688 0.6_9 0.178 -0.097
SZGNA 0.178 0.140 0.945 0.161 0.134 0.206 0.821 0.324 0.341 0.114 0.129 0.208 0.459 0.649 5.?52 0.118 0.144
T.
BIAS
0.539 0.223 0.097 0.147 -0.083 00.273 0.337 0.178 0,309 -0.020 0,087 -0.002 -0.062 -0,349 0.673 -0.057 8.095
SIGMA 0.080 0.043 0.385 0.063 0.040 0.063 0.388 0.078 0.106 0.030 0.035 0.059 0.125 0.147 0.795 0.026 0.039
FIT
RMS 0.157 0.088 0.046 0.04_ 0.091 0.045 0.0_g 0.2?6 0.043 0.097 0.219 0.118 0.026 0.065 0.120 0.063 0.020
BASE
RNS NAXEL
2.211 0.883 0.417 0.582 0.486 0.923 0.861 0.993 0.630 0.199 0.68_ 1.140 0.114 0.444 4.333 0.317 0.380
42.6 80.8 67.6 45.3 81.8 71.6 64.0 50.6 36.2 39.4 39.2 41.0 38.8 60.5 46.3 65.0 40.4
TOPEX rlODEL PGS - T2
STANAM ORAZI ORAZI ,NAZI ORAZI ORAZl ORAZI ORAZI LAOUII02 LAGUI102 HATERA1 IN?ERA1 PLATVL1 QUIN1092 QUINI092 SIMOSATA YARAOl YARAG1
ISTA 78393401 78393401 78393401 78393401 ?1393401 78393401 78393401 71100402 71100402 79394101 79396101 ?1120201 71090002 71090802 78383601 70900501 70900501
NTYPE
Y_rNI490
HHIqqss
RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE RANGE
840103 840105 840105 840106 840106 840106 84010? 840107 040107 840103 840104 840105 |40105 840106 840105 840105 840106
12643 20521 35558 3600 22454 41448 5416 ?4440 93513 50540 33518 104855 85703 91659 •31608 134412 1214.16
NPTS 33 46 10 39 $6 23 0 16 34 76 61 38 31 39 9 72 53
|TFJtUS[9 2 Z 2 2 2 2 2 2 • • $ • Z • • • Z
01_S 33 46 10 39 56 25 9 16 34 76 60 38 31 39 9 72 S3
00.087 -0.397 0.115 0.024 -0.092 -0.199 00.303 -0.211 -0.042 0.158 0.136 00.237 0.103 0.1'71 -0.656 -e.931 -0.156
9?
SIGIqA 0.178 0.140 0.060 0.161 0.134 0.Z06 0.818 t.324 0.341 0.115 0.130 0.208 0.438 0.640 3.503 0.118 0.138
T.
BIAS 0.104 -0.041 -0.038 -0.122 -0.019 -0.103 -0.056 -0.001 -0.026 -0.018 0.023 0.084 0.054 -0.035 -0.364 -0.015 0.046
SxOIqA 0.080 0.043 0.381 0.063 0.040 0.063 0.386 0.078 0.106 0.030 0.033 0.059 0.125 0.147 0.745 0.026 0.038
FIT
RHS 0.049 0.080 0.046 0.028 0.047 0.069 0.049 0.094 0.036 0.095 0.118 0.073 0.026 0.032 0.117 0.040 0.047
BASE
RIq$ IqAXEL
0.242 0.428 0.064 0.320 0.126 0.454 0.233 0.262 0.130 0.201 0.196 0.ZS_ 0.312 0.135 2.691 0.088 0.252
42.6 80.8 67.6 45.3 81.8 71.6 64.0 50.6 36.2 39.4 39.2 41.0 3&,8 60.5 46.3 65.0 40.4
detailed in
picture
Table
5.2.5g.
adjustment
of
the
the
the
weighting shown
the
the
data
have
GEM-TI.
C-mat.
in
and
remarkably
models
performance
orbit
Its
sensitivity
to
these
the
edited
set
and
STARLETTE's
its
contribution
to
the
solution
earth's by
the
LAser
utilization
motions May,
1976
nominal 5.2.6a.
orbital The
in
short-wavelength
a
single
combination Table
a very
and
5.2.5g
are
and
the
serve
altitude
of
gravity,
This
gravity
and
coupled
is
model,
very
the in
a
encouraging
relatively
tidal
low
perturbations.
the
similarities
important
of
resulted
is
with
robustness
with
of
TOPEX
make
one.
Ranging
for
has
the a
for
tidal
98
NASA
space-based
LAGEOS and
reduces
effects,
for
satellite
to
target.
described errors leaving
the
enhanced
stands
laser
are
orbit drag
greatly
LAGEOS
first
LAGEOS
monitoring
been
satellite.
is
the
TOPEX
overhauling
STARLETTE
rotational)
as
have
analysis
data.
ranging
equations
interim
this
Laser
LAGEOS and
normal
general
of
laser
Observations
Satellite
of
to
by
Ranging
tectonic
STARLETTE
latest
orbital
characteristics
high
these
however,
Laser
Satellite
exclusively
used
altitude
forces
launch
the a
influenced
of
(both
GEOdynamlcs
launched
from
is strongly
the
(1984)
and
of
its
The
that
LAGEOS
from
tidal
subsequently into
easier
resulting
editing
and
of
recent
of
of
Analysis
values
fit
an
the
positions,
were
parameters)
of
The
light
5.2.6
arc
of
station
matrices
shown
for
subset
the
for
of
in
data
selected
allowed
RMS
arcs
data
fact
allowed
is
5.2.5b.
mathematical
the
equations
46
the
determination
improved
equations
the
The of
analyzed. the
normal
parameters,
This
The
five-day
Extensive
physical
orientation
in Figure
six
arc
harmonics,
elimination
been
contributed
normal
data.
pictorially
Forty
STARLETTE
parameters.
STARLETTE
of
5-day
Earth arc
(after
matrix,
individual
The
orbital
combined
the
geopotential
coefficients, and
of
in
the be The
Table arising
a strong
Table
NUMBER EPOCH
OF
5.2.5g
WEIGHTED
OBSERVATIONS
RMS
(m)
NUMBER STATIONS
0401o2
633
.5736
7
040107 040112 840117 840122
602 1043 1012 2270
.5172 .6436 .7107 .4651
9 I0 I0 9
840127 840201 840206
958 047 1499
.4331 .3903 .5625
I0 7 8
840211 040216 840221 840226
390 338 502 041
.6710 .4215 .8665 .7439
6 5 8 7
840302 840312 840317 840322
451 716 741 1289
.8990 .6586 .4125 .6363
5 5 6 9
840327 040401 840406
1971 2069 2212
.5744 .5924 .5219
8 7 6
840411 840416 840421 840426
3084 027 1437 093
.5851 .6289 .6400 .B068
8 0 7 9
840501 840506 840511 840516
619 074 905 574
.5879 .go00 .7750 .6051
5 4 4 0
840521 840526 840531
2250 1437 2012
.7150 .7178 .6031
8 0 0
840605 840610 840615 840620
1279 2160 2323 1480
.5656 .7684 .5638 .5611
II I0 12 9
840625 840630
3451 1429
.6866 .4409
I0 0
840705 840710 840715 840720
1197 550 406 024
.6200 .4866 .5503 .7427
7 5 5 4
840725 840730 840804 840809
350 754 749 921
.4617 .4867 .6397 .5161
3 5 6 7
040814 840819
I170 2849
.5073 .4891
0 0
46 [MATS
57356
.6120
99
OF
ARGUMENT OF PERIGEE (AT
EPOCH)
328.219 343.032 1.779 17.217 32.676 50.280 64.865 83.486 97.697 113.218 129.760 144.077 162.043 194.533 212.022 227.683 247.627 262.668 279.917 297.023 312.347 332.052 347.073 4.323 20.754 36.110 54.741 68.645 05.147 100.373 115.013 133.685 148.093 165.902 181.369 197.607 216.576 231.370 249.614 265.374 281.668 301.339 316.584 335.486 350.501 7.096
STARLETTE
E-MAT
SUMMARY
WEIGHTED
RMS
(APRIORI)
m_
I,D I
I,LI
)
I0
20
EMAT Figure
5.2.5b.
STARLETTE
100
30
40
NUMBER E-MAT
Summary.
50
signal
for
the
Furthermore, shape non-
longest
by being
(see
figure
extremely
5.2.6a),
conservative
radiation.
wavelength
forces
Therefore,
determination distinction extensive
of of
the
international
cooperation
coverage.
There
is now
laser
stations
which
is
These constitute
observations years
of
laser
two
These minute
outstanding end of
data
1979
so
data
from
tracking
data
have
over the next year. observations
in
our
important
dominant
polar motion
are
within
lengths
acceleration
analysis 4.5 to
as
time
laser
have
span
parameter
third
their
into laser
equations
A summary
is presented
priority
set of laser analysis,
in
5
the GEM-TI
"normal-points" the
1980 through
were first deployed The
Six
years
most
to the in late
additional
to the solution
the annual
tracking
and Chandler
to the definition
The LAGEOS data were radiation
pressure
to
along
adjust
of
to the beat period of
and
with
were carefully edited,
I01
the two periods.
in monthly
along
the
epoch
track state
and post-processing arcs
of the LAGEOS ares used to generate
in Table 5.2.6b.
is
of the pole
reduced
of these data indicate RMS of fits for monthly 10 cm.
global
generation
contains
will be added
contribution
allowed
used
the
to have at least six years of these
that of
solar
These observations
been
is
highest
unsatisfactory.
solutions.
our solution.
significant
missions
of
the years
it corresponds
terms,
a
A
to enhance
selected
systems
somewhat
gravity
because
with
the
In our present
been condensed
is now available,
And LAGEOS data make a strong
elements.
LAGEOS
It is desirable
somewhat
arc
on
The
mobile
sets
1985, which
for improving
satellite
LAGEOS
from re-
field.
network
spherical
and albedo
which has occured
set of these data encompassing
early
obtained
laser
field.
arising
the largest and best distributed
intervals.
1984. The NASA
errors
pressure
gravity
a worldwide
acquired
ranges
a perfectly
ideal satellite
which have ever been collected.
solution. at
is an
previous
laser
target.
radiation
wavelength
over
the gravitional
also minimizes
like solar
long
of
dense and having
LAGEOS
LAGEOS
LAGEOS
portion
of between the normal
Figure 5.2.6. LAGEOS Satellite.
Table
5.2.6a
LAGEOS .(LASER
GEODYNAMICS
Launch:
May
Spacecraft:
Spherical,
4,
1976
60
406.g65 426 Orbit:
SATELLITE).
cm
diameter
kg
laser
retro-reflectors,
Semi-major
3.8
cm
12265
axis
diameter
km
Inclination
10g.8
Eccentricity
0.004
Perigee
5858
km
5958
km
height
Apogee
Node
height
rate
Perigee Semi-major
rate axis
103
rate
degrees
+0.343
deg/day
-0.214
deg/day
-I.I
mm/day
Table
EPOCH
NUMBER OF OBSERYATIONS
5.2.6b
WEIGHTED RMS (m)
NUMBER OF STATIONS
ARGUMENT OF PERIGEE (AT EPOCH ) 345.174 338.042 330.814
791230 800129
1455 2319
.2065 .2210
13 14
800228 800329
2639 2231
.2475 .2228
14 14
800428 800528
1543 1926
.2396 .2336
I0 9
800702 800801
1801 3187
.2241 .2237
13 13
297.302 290.785
800831 800930
3496 3336
.1934 .2088
16 18
287.046
801030
2751
.2191
14
801129 801229
1413 794
.2022 .1736
II 8
260.453 255.325
810128 810227
1287 2739
.1784 .1787
9 13
253.457 240.940
810329
1943
.1913
II
232.084
810428 810528
1884 1944
.2057 .2512
9 II
226.531 221.412
810627
2187
.2555
12
217.269
810727 810826
2168 2821
.1948 .2065
13 14
201.207 199.978
810925 811025
3143 1972
.2308 .2095
16 12
194.745 188.166
811124 811224
1573 1314
.2126 .3018
12 12
181.017
820123
1878
.2427
12
820222 820329
1883 1926
.2125 .2007
15 12
820428 820602
3084 2488
.2055 .1811
12 II
153.177 148.207 142.263
820702
2980
.2022
II
134.020
820801 820831
2027 2720
.2197 .2154
13 14
126.356
820930 821030
3596 1938
.1788 .1604
15 12
821129
2041
.1788
II
821229
1699
.1990
II
104
321.579 311.512 313.865
281.014 271.071
168.490 172.349 162.371
127.720 118.145 110.051 104.642 101.347
LA GEOS
coara ....
EPOCH
NUMBER OF OBSERVATIONS
WEIGHTED RMS (m)
NUMBER
OF
STATIONS
ARGUMENT OF PERIGEE (AT EPOCH)
830128
1494
.2204
12
97.008
830227 830329
2010 2187
.2378 .2079
14 14
87.259 79.935
830428
2405
.2180
13
79.208
830627
1920
.1511
8
64.706
830727 830831
2751 2520
.1796 .1425
8 II
57.853 54.654
830930 831030
3761 3!77
.1760 .2306
17 17
48.845 36.054
831229
2729
.2583
17
30.879
840128 840227
2425 2437
.2172 .2519
16 22
23.527 16.220
840329 840428
3817 4129
.2267 .2554
20 22
9.126 1.119
840528
4541
.2468
20
840627
4372
.2724
19
349.233
840801 840831
4857 4611
.2617 .2408
22 21
344.696 338.433
105
3.869
LAGEOS
E-MAT
SUMMARY
WEIGHTED
RMS
(APRIORI)
0.4
0.3
iv
0.2
0.!
i
I
0
I0
ii !
I
20
30 EMAT
Figure
5.2.6b.
I
NUMBER
LAGEOS
106
40
E-MAT
Summary.
I
SO
6O
5.2.7
A_nalysis
The
GEOS-2
GEOS-2
satellite and
of
was
served
one
of
globally
distributed
Geodetic
Satellite
geocenter.
tion
of
(both
1977;
This
geometric
and
system.
secondary
the
were
calibrated
laser
ranging
5.2.7a.
middle
for
uation arcs use
of
served
the
We
of
of
the
thereby
appear
in
the
SAO
data
to
early
range
had
(NASA, Of
Network. and
fences and
were
declination GEOS-2
opportunity
biases
have
1976. which
a
a sparse
found
Some were
data
these of
seen
a
also
for
early
in
Table
107
low
given priority
apparently set
to
laser
only
a
limited
large
sample,
systems
were
be
a
1975 function
basis
ceased
utilize
Consequently,
the earlier to
are
generation
reasonably
although
on
Tracking
timeframe.
we
orbit
tracked
third
1975
catalogues,
1975
have
GEOS-2
1970's.
from
To
1975
ascension
of
global
installations
electronic
target
a
Minitrack
Fortunately,
a
the
intermittently
GEOS-2.
late
these
solutions
objectives.
Minitrack
right
as
NGSP
to
reduc-
observations
NASA's
worldwide by
within
early the
of
simultaneous
and
much
data
for
the
obtained
investigations to
satisfied
photographically.
was
1977.
modeling started
of
locations
respect
in
a
world's
accurate
use
by
National
the
with
for
of these
calibration
all
those
cubes
GEOS-2
of
positions
carried
The
an
NASA
GEOS-I)
unify
through
camera
which
was
uncertainty
accomplished
of
(as
to
by
GEOS-2
observatories.
of
Thls
initiated
foremost,
objective
analysis
characteristics
lasers
level
1968.
experiments.
The
the
at
acquired
corner
10 m
the
cosines
measurements carried
an
was
against
an
be
1973)
located
direction
had
28,
missions
be photographed
photographic
was
al,
interest
Cameras
to
April
and
optical
dynamic)
It
et
geodetic
to
of
5 to
was
satellite
Marsh
it
Program the
on
First
allowed
these
reference
earliest
network
to
Data
launched
purposes.
which
datums
was
the
several
lamps
the
Tracking
satellite
flashing
tracking
Laser
gravity
systems
which
number
not
SAO of
the
for
after
we
in
by
of
were
an
possible forced
upgraded
data
was
range.
eval-
to
until
found The
to SAO
data
taken
found
during
to be
day
normal
equation
solar
radiation
permitted
to
generated
and
arc
over
heavily
for
were
solutions.
In
coefficient
summarized
lasers
in many
this
time
arc
in of
the
nantly satellite
borne
for
Height
balloons
for
the
first
the
Smithsonian
observations solution orders laser
of
Optical
km km
105.8 Period
and
Low
observations Nunn
60
of
observatories the
era.
of
they
by
a
1960's. rocket
day,
were
a
all
of
these
arcs
were
Note
that
even
when
3 or
4 stations
were
important
have than
GEOS-2
global
A
an
reasonable
that
1980's.
108
fragments,
gravity 1966.
to
obtained
the
was
space-
data
base
that
Surprisingly,
noise is
and the
in
set
solution,
contributions
which
predomi-
data
provided
observational
of
state-of-the-art
bodies,
in
Observations
network
the
observations
Observatory
making
greater the
28
per
state
Satellite
satellite-based
Astrophysical
though
and
parameter
only
were
These
comprehensive
still
reduction
degrees
Inclination
satellites,
this
were
11 2. I minutes
acquired
throughout
magnitude
tracking
1569
O.O3
are
even
of
city
over
for
them
5.2.7a
Eccentri
Baker
data
orbital
arcs,
1077
of
drag
5.2.7b.
Height
tracking
acquired
the
Perigee
optical
SAO
and
Characteristics
Anomalistic
The
a
of
solution.
GEOS-2
arcs,
1975
subset
GEM-TI
the
Table
Incl ination
Analysis
our
a
period.
Orbital
5.2.8
but
equations
Table
Apogee
in
these
normal
are
in
used
per
The
edited,
inclusion
lengths
adjust.
SAO
tracking
were
satisfactory
Five
including
1975
of
these gravity
which
is
four
by
the
best
Table
5.2.7b
ARGUMENT EPOCH
NUMBER OF OBSERVATIONS
WEIGHTED RMS
(m)
NUMBER STATIONS
OF
OF PERIGEE (AT
EPOCH)
750708
595
1.3994
4
55.162
750803
638 472
1.6999 1.0250
3 3
14.673 354.021
732 416
.8124 1.0606
5 4
337.992
573 357
.6148 1.5540
5 5
785
1.8013
5
301.713 289.163
475 923
1.4644 I. 1042
4 4
268.194 244.037
1351 1204
2.1 442 2.0522
6 6
233.716
544
1.4113
5
95.276
894 1435
2.0713 1.6547
5 4
49.825
I 184
1.7588
7
1389 1418
1.9487 1.9838
7 6
1364
1.0963
7
349.358 341.704
1 475 701
1.2160 1.5675
7 5
331.432 222.343
750815 750825 750901 750906 750915 750923 751006 751021 751027 751102 760829 760927 761009 761019 761025 761103 761108 761115 770120 770320
327.452 319.665
223.o_9
33.373 17.469 7.638
784
1.4755
6
125.612
770403 770409
1412 1277
1.2887 1.5900
6 5
103.939 95.076
770425
1040 881
1.4304 1.1608
3 3
70.440
I 196
1.2060
6
1098
1.6737
4
770430 770607 770613
109
59.898 1.945 351.478
The
reason
for
objects
which
have
samples
the
certain
perturbative
this
been
earth's
harmonics
to
lite
can
orbits
a
sufficiently
large of
continue
tion
found
gaps
systems.
In
sources will
of
point
gravity
have
in
later
(m=O
coefficients).
one
of
which
inclusion
in
the
TELESTAR-I, optical
BE-B,
only
present
to
limited
laser
satellites visibility two
GEOS
satellites. solar
flashing observing
were
illumination
collection
was
taken
inclination
very
values
and
ANNA-IB
ANNA-IB's
other
satellites
restricted
to
robust
the
yet
to
flashing
than
those
were
passively
against dusk
110
period
a
dark or
have
terms
been
data the
observed, sky.
ANNA-IB,
the
at very GEOS
nighttime sets other
from four
requiring
Therefore, dawn.
an
other
used the
of
unlimited
from
for
solely
and
lamp
only
from
Both
before
observa-
:
exists
used.
The
best
selected
is
tracking
permitted
the
Results
were
BE-B
be
tracking
of
were
them
Doppler
which
more
objects
and
inclina-
satellites,
TELESTAR-I for
the
optical
harmonic
satellites
data
are
tracked
GEOS-2.
The
these
zonal
tracked,
These
lamps
the
optically
instruments.
the
for
spherical
one
role
a
accurate
objects.
important
many
yield
other
are
satelon
an
in
from
for
camera
BE-B
in filling
satellites
much
of
are
model.
available
low
"lumped-
individual
optical
six
for
on
they
permit
gravity
role
sets
tracking
data
to
determined
global
the
GEOS-I
GEM-TI.
data
These
the
While
obtain
for
a
solution.
BE-C,
optical
flew
well
exclusively
satellite.
systems,
harmonics"
accurate
gravity
although
(or
motions
Initially,
was
sums
be
spherical
mean
showing
resolving
These
sense may
the
and
data
fact,
to
inclinations
information
be discussed
tions
the
of
of
of orbit
perturbations
combination
important
the
it
analyzed
comprising
within
causes
data
into
an
satellite
that
signal
play
diversity
shown
"lumped
to
the
given
these
field.
has
in
which
determined
of
coefficients
way of
gravity
of
found Any
linear
accurately
set
this
satellites
the
range
a
Each
experience
wide
deconvolution harmonic
very
Past
over
in
some
represent
be
specific.
field
as
is
tracked.
frequencies.
described
harmonics")
optically
gravity
mathematically used
importance
data
A
summary
stations
found
through in
of in
5.2.8f.
the
PGS-T2
The arc.
The
Tables
the each
of
These field
weighted to
data
data
RMS
the
optical
et al,
have
the
1986)
were
w
are
which
of
shown
and
in
is
data
a precursor
approximately RMS
number
tables
optical
(whose
calculated
A6
coverage
total
residuals
declination:
right
perigee
arcs
a precision
observation 5.2.8f)
of fit,
comprised
(Marsh
optical
5.2.8a
data,
5.2.8a
set
of
two
values
of
found
GEM-TI.
seconds
are
of
given
in
as:
_ A__ 2
Ow: i_ _J -A_--
ascension:
cos_
where
A6,
A_
are
the
right
A6 w,
Figure
5.2.8a
obtained
from
compared
to
field. ties
This
tions absent
was from
we used low
those
of
is
a
in
are
obtaining
between
of an
for when
times these
inadequate
PGS-T2
with
objects.
111
for
result two
the
zonal
sets
than (see
coverage significiant
values
of
the
of
be
GEM-L2
uncertain-
for
within
the
minus the
field can
harmonics
(PGS-T2)
terms
PGS-T2
from
values
greater
and
the
These
the
and
residuals.
the
The
declination
fit,
weighted
solution.
GEM-L2.
many
in
orbital
similiar
confirmed
found
inclination
is
the
accuracy
determination
in
of
the
that
the
uncertainties
different
differences
concluded
from
the
which
strikingly
to
GEM-L2
Therefore,
5.2.8b
residuals
corresponding
covarlance
degradation
harmonic the
their
degradation
compared
zonal in
is
the
PGS-T2. are
figure
ascension
presents
a scaled
What is
are
Ae w
observation
zonals (GEM-L2)
uncertainty
Figure
5.2.8c).
orbital
inclina-
information
being
0
o,
X ¢0
0
¢:
o0
L_
W U.I ¢3
_d_d_NNNNNNNNNN_RR_RRR ,e"
Z ,,v
112
N
O0 N
err
N
N tn N
O
O N O N N N
O
O N O N f.j. =,=,,I
r-I
¢0 =,-I
e,=l
0
_J ¢'.,I
i,.,.l
I
url
'I'..,,,
X u)
O r=,l
_NN
0
N i-I
r-I =.4
I,-.I
i,.,,.I
UU
N iml
E r_
i,=l CO U't
__N__
¢".1
,,,.; U_
U't e,,.I err =,=1
_N_N_N__
I#l
N
=,..1
I.U I,U ee,
e_ 0q
s-
dRR_=_Rd=R_R¢_¢#RRdd_N_h_N
,,v
113
l_-,m I t
t
.._..t
f
,' ,'
I
\ / /
\ 0
0 t_
kL. d
CD
/
!
cN
CN ...J
+
/
U3
60l
X _'
114
Table ANNA-1B
NO.
1 2 3 6 5 6 7 8 9 I0 11 12 15 14 15 16 17 18 19 20 21 22 25 26 25 26 27 28
EPOCH
621101 621115 621122 621129 621213 621220 631107 631116 651121 631128 631205 631212 631219 631226 660102 660110 660117 651128 660116 660125 660130 660215 660222 660501 660508 660515 660329 660610
AVERAGE
TOTAL
5.2.8a
OPTICAL
NO. OF OBS.
HEIGHTED RMS ARCSEC/2
157 126 156 158 258 262 66 98 78 36 118 183 252 56 56 82 162 150 102 120 184 250 96 167 318 152 264
1.294 1.413 1.212 1.221 1.201 1. 155 1 • 149 1.109 1.479 1.028 1.293 I .360 1.577 1.175 0.960 0.875 I. 226 1.017 0.905 1.076 1. 122 0.994 1.065 1.169 0.899 1. 152 1.311 1.079
168
1.160
6151
115
7-DAYS
ARCS
NO. OF STATIONS
9 10 6 9 10 11 4 10 9 6 7 8 7 6 6 9 8 11 6 6 7 7 6 4 6 6 7 7
7
ARGUMENT OF PERIGEE (AT EPOCH)
207.7 268.1 274.5 296.2 565.6 358.1 228.6 245.9 269.3 295.7 516.6 336.0 5.2 17.5 59.8 65.9 75.2 296.9 85.4 101 .I 119.9 163.3 188.8 206.9 227.9 255.6 297.7 525.6
Table BE-B
NO.
1 2 3 4 5 6
1 11 12 13 14 15 16 17 18 19 20
5.2.8b
7-DAYS
NEIGHTED RMS ARCSEC/2
EPOCH
NO. OF OBS.
641026 641102 641109 650112 650203 650323 650406 650415 650424 650613 650627 650716 670226 670305 670312 67O319 670507 670514 670521 670528
38 60 38 52 32 54 30 46 30 50 40 30 211 56 128 228 60 154 232 170
1.427 1.309 1.021 1.173 1.139 1.005 1.329 1.555 1.300 1.357 1.166 1.451 1.181 1.258 0.909 1.109 1.148 1.461 1.064 0.983
87
1.217
AVERAGE
TOTAL
1739
116
ARCS
NO. OF STATIONS
8 11 3 6 4 9 6 8 6 5 8 8 9 4 6 6 4 5 12 8
ARGUMENT OF PERIGEE (AT EPOCH)
104.7 85.3 74.8 266.7 213.1 92.0 59.4 41.2 14.7 242.7 196.1 149.2 100.2 88.8 65.7 52.6 284.8 269.2 245.7 233.4
Table BE-C
EPOCH
NO.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 55 36 37 58 39 4O 41 42 43 44 45 46 47 48 49 50
650619 650626 650705 650710 650717 650724 650751 650807 650814 650821 650828 650904 650911 650918 650925 651002 651009 651016 651023 651030 651106 651113 651120 651127 651210 651217 651225 660101 660108 660115 660301 660308 660315 660322 660329 660405 660412 660419 660426 660505 660510 660517 660524 670312 670319 670526 670402 670410 670417 670424
AVERAGE
TOTAL
5.2.8c
OPTICAL
NO. OF OBS.
7-DAYS
HEIGHTED RMS ARCSEC/2
ARCS
NO. OF STATIONS
ARGUHENT OF PERIGEE (AT EPOCH)
327.6 1.5 38.7 73.9 109.0 145.3 180.3 217.7 253.7 135.8 237.9 4.8 38.9 77.4 109.2 147.5 182.1 218.9 255.7 293.3 329.0 6.0 41.4 77.5 142.5 179.8 219.2 258.7 293.7 331.4 201.5 258.2 275.6 311.4 349.5 24.2 60.7 95.7 130.9 167.8 201.9 241.4 275.9 346.0 23.5 57.8 94.0 135.7 169.4 206.4
64 56 52 56 94 155 80 48 62 74 50 38 66 64 58 38 42 66 54 56 68 58 58 34 48 32 54 73 92 67 216 301 374 544 269 235 27q 299 346 210 270 257 189 185 327 207 472 235 250 204
1.381 0.998 1. 326 1.113 1.104 1.225 1.080 1.079 0.871 0.985 1.190 1.12_ 1.002 0.848 1.08¢+ 1.188 1.220 1.16q 1.200 0.965 1. 346 0.940 1. 155 1.060 1.114 0.865 1. 357 1.079 0.970 0.985 1. 107 0.985 0. 957 0.897 1.096 0.992 0.85q 0.99_ 1.051 1.145 0.986 O. 858 0.886 1.089 1.090 1.062 1.116 1.173 1.187 1.074
4 6 7 8 9 9 11 10 8 9 5 7 9 8 11 5 9 9 9 8 4 8 6 9 7 8 9 9 7 6 9 10 9 6 7 7 9 8 8 9 9 9 7 9 9 7 8 10 10 8
150
1. 071
8
7501
117
Table GEOS-1
NO.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 50 51 52 55 34 55 56 57 58 59 40 41 42 43
5.2.8d
OPTICAL
7-DAYS
ARCS
NO. OF STATIONS
EPOCH
NO. OF OBS.
NEIGHTED RMS ARCSEC/2
651108 651115 651122 651129 651215 651220 651227 660105 660110 660117 660124 660151 660207 660214 660221 660228 660507 660514 660404 660411 660425 660502 660509 660516 660523 6607O9 660716 660723 660730 660806 660815 660820 660827 660905 660922 661006 661013 661020 661115 670226 670505 670512 670519
244 331 1692 885 1177 1426 1291 769 1524 1722 1296 838 364 773 1249 967 1506 2673 1781 1879 2034 2079 1471 743 263 3485 5780 3435 3059 1791 1506 1091 594 702 2218 2378 1721 1446 1141 214 575 575 286
0.920 1.051 0.727 0.785 0.829 1.001 1. 126 1.251 1. 056 0.980 0.862 0.961 0.901 0.954 0.836 0.889 1. 058 0.823 0.865 0.805 0.778 0.771 0.770 0.724 0.649 0.780 0.857 0.781 0.792 0.688 0.667 0.704 O. 585 0.615 0.919 0.892 0.805 0.809 0.707 0.987 0.951 0.928 0.971
9 10 17 22 22 25 30 24 29 26 27 22 18 21 25 26 36 30 30 50 51 28 24 17 11 31 50 28 25 28 20 16 11 15 9 22 24 24 14 10 8 11 7
1413
0.854
22
AVERAGE
TOTAL
60750
118
ARGUMENT OF PERIGEE (AT EPOCH)
150.5 154.7 159.9 164.4 175.5 177.3 182.2 187.5 191.4 196.0 200.9 205,2 209.4 214.8 218.6 225.7 228.8 252.9 246.6 250.8 260.6 265.0 270.3 274.7 280.0 310.5 515.6 519.9 524.5 329.7 333.9 358.2 545.5 348.0 559.7 9.8 15.7 18.6 35.1 101.9 106.2 110.1 115.1
Table GEOS-2
NO.
I 2 3 4 5 6 7 8 9 I0 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
5.2.8e
OPTICAL
7-DAYS
ARCS
EPOCH
NO. OF OBS.
NEIGHTED RMS ARCSEC/2
NO. OF STATIONS
680515 680322 680329 680405 680412 680419 680426 680503 680510 680517 680524 680531 680607 680614 680621 680628 680719 680814 680828 680904 680911 680918 680925 681002 681009 681016 681023 681116 681217 690128 690204 690211 690218 690225 690304 690311 690318 690325 690331 690407 690414 690421 690428 690505 690512 690519
1378 1938 1664 1613 1607 2132 1772 1696 1427 1619 1390 1196 2098 2775 2978 417 1712 1172 1220 1795 1242 2863 1650 2007 1954 1254 1616 869 463 729 908 912 579 429 760 908 847 675 861 1068 839 1259 778 1160 491 685
0. 857 0.865 0.803 0.753 0.986 1.040 0. 737 0. 826 0.798 0.720 724 O. 702 O. 754 O. 723 O. 709 O. 702 0. 727 0.668 0.922 0.920 0.808 0.766 0.829 0. 932 0.851 0.850 0.852 0.832 0. 970 1.030 1.099 0.995 1.085 0.969 0.931 O. 927 0.851 0.874 0.770 0.758 0.762 0.816 0.774 0.761 0.669 0.778
26 27 32 33 32 36 35 30 27 24 26 18 30
1535
0.846
22
AVERAGE
TOTAL
O.
61403
119
1
34 17 30 15 30 29 29 35 28 29 30 29 29 14 13 9 13 12 9 II 13 13 12 12 19 22 II 23 18 20 9 9
ARGUMENT OF PERIGEE (AT EPOCH)
67.1 53.5 44.6 34.6 21.7 11.0 357.4 347.7 358.7 324.3 313.2 301.3 289.1 279 o8 266.6 255.0 220.0 177.2 154.9 143.3 134.2 121.8 109.5 100.2 87.4 77.4 67.6 28.5 336.4 269.1 256.0 244.6 235.3 221.3 210.I 198.3 186.9 178.2 167.9 155.4 143.3 133.5 121.7 110.7 100.5 87.4
Table TELSTAR-1
NO.
1 2 3 4 7 8 9 10 11 12 13 15 16 17 18 19 2O 21 22 23 24 25 26 27 28 29 3O
EPOCH
620713 620725 620801 620808 620816 620823 620830 620913 620920 620927 621004 621018 621025 621101 621108 621115 621122 621206 621213 630207 630214 630221 630228 630307 630314 630328 630414 630421 630526 630616
AVERAGE
TOTAL
NO. OF OBS.
5.2.8./"
OPTICAL
HEIGHTED RMS ARCSEC/2
39 80 74 128 138 106 116 153 105 166 209 154 210 124 94 138 114 68 58 64 147 139 122 129 193 144 118 110 180 342
1.096 1.211 1.112 0.989 1.482 1.113 0.936 1.127 1.102 1.043 1.122 1.225 1.171 1.037 1.256 1.187 1.004 1.405 0.898 1.047 0.840 0.965 0.853 0.806 0.783 1.095 1.033 0.767 0.884 0.764
132
1.045
3962
120
7-DAYS
NO. OF STATIONS
5 10 7 9 7 7 5 6 7 10 9 11 11 10 7 9 7 9 7 6 10 10 11 7 8 8 10 10 5 12
8
ARCS
ARGUMENT OF PERIGEE (AT EPOCH)
170.1 193.9 207.8 221.8 237.7 251.7 265.5 293.2 307.2 321.2 335.2 3.0 16.9 30.8 44.5 58.5 72.4 100.2 114.1 225.3 239.3 253.2 267.0 280.9 294.7 322.7 356.5 11.0 79.9 121.0
L4_
O"b
CO _D
_D CO J
O
aD O
_0
_D
O
O o
W r_ w
wv
N _D
_D
¢0
O
O
O
I
_.i w )--
_D
m
B-4
o c_
i.'! X
v
C.D
_D
m
.a g
I-
z c_
O LL
l-.CO
z ._J z
:]¢ O
_D ¢O _D
_D
Q_
O0
_D
a_ i.-..t W Z
Z
!
|
|
|
O O.
121
Table PEOLE
NO.
1 2 5 4
EPOCH
710225 710504 710507 710527 710610 710623
AVERAGE
TOTAL
LASER
5.2.8h
+ OPTICAL
7-DAYS
NO. OF STATIONS
NO. OF OBS.
NEIGHTED RMS ARCSEC/2
736 663 815 1594 104 239
2.840 1.730 1.400 2.810 4.270 0.680
4 4 5 4 1 2
2.29
3
692
4151
122
ARCS
ARGUMENT OF PERIGEE (AT EPOCH)
104.7 191.6 324.5 220.4 55.5 222.3
Table DI-D
NO.
1 2 3 q 5 6 7 8 9
7-DAYS
670219 670226 670505 670512 670519 670450 670507 670514 670521
164 250 432 275 174 1005 1567 1592 854
1. 158 1.113 1. 066 O. 957 1.050 0.967 1.020 0.954 1.360
7 10 7 8 7 11 11 12 14
679
1. 065
10
DI-D
WEIGHTED RMS ARCSEC/2
ARCS
NO. OF OBS.
TOTAL
1 2 3 4 5 6
OPTICAL
EPOCH
AVERAGE
NO.
5.2.8i
NO. OF STATIONS
LASER
+
OPTICAL
7-DAYS
ARCS
NO. OF OBS.
WEIGHTED RMS ARCSEC/2
710423 710507 710514 710705 710710 710719
3465 1824 2027 1604 2368 347
1.040 1.950 0.950 1.q80 1.870 1.890
6 9 10 2 2 4
1939
1.530
5
TOTAL
156.2 194.5 252.1 270.1 308.1 175.7 211.4 249.5 287.1
6111
EPOCH
AVERAGE
ARGUMENT OF PERIGEE (AT EPOCH)
11635
123
NO. OF STATIONS
ARGUMENT OF PERIGEE (AT EPOCH)
108.1 183.4 221.5 132.5 169.7 218.7
Tabte 5.e.Sj VANGUARD-2
NO.
1 2 3 5 6 7 8 9 10
EPOCH
660202 660209 660216 660225 660502 660309 660516 660525 660350 660_07
AVERAGE
TOTAL
NO. OF OB5.
7-DAYS
HEIGHTED RMS ARCSEC/2
ARCS
NO. OF STATIONS
_2 70 156 170 136 165 2_9 231 6_ 38
1.121 0.868 1.192 1.039 1.2q3 1.003 0.885 1.221 1.19_ 1.165
6 6 8 8 9 9 6 8
130
1.093
8
1299
124
ARGUMENT OF PERIGEE (AT EPOCH)
252. ¢_ 290.0 326.9 3.8 Col .3 77.9 11¢t.9 152.0 188.8 231.3
Table 5.2.8k VANGUARD-2RB
NO.
EPOCH
1 2 5 to 5 6 7 8 9 10
600402 600409 600417 600427 600505 600512 600519 600526 600608 600717
AVERAGE
TOTAL
NO. OF OBS.
7-DAYS
NEIGHTED RMS ARCSEC/2
42 50 40 50 74 9Z 124 94 55 105
I. 275 0.8_6 1.6_5 I. 007 1.298 1.6,27 1.020 1.175 0.920 1.259
69
1. 187
686
125
ARCS
NO. OF STATIONS
ARGUMENT OF PERIGEE (AT EPOCH)
357.5 31.7 71.3 120.7 160.3 194.6 229.4 226.3 328.6 0.0
6
Table DI-C
5.2.81
OPTICAL
7-DAYS
NO.
EPOCH
NO. OF OBS.
1 2 3
670220 670227 670306 670313 670320 670416 670425 670430 670507 670514
164 158 300 201 127 2(_4 40O 720 196 202
1.061 1.195 1.071 1.049 0.949 0.921 1.055 1.001 0.902 1.003
271
1.021
5 6 7 8 9 10
AVERAGE
HEIGHTED RMS ARCSEC/2
ARCS
NO. OF STATIONS
DI-C
LASER
+
OPTICAL
NEIGHTED RMS ARCSEC/2
7-DAYS
ARCS
EPOCH
NO. OF OBS.
710401 710608 710615 710622
751 698 3783 2582
0.780 1.320 2.580 2.230
4 I0 8 8
AVERAGE
1905
1.720
7
TOTAL
7614
1 2 5
217.9 259.0 301.5 343.6 2q.8 185.6 226.7 267.8 508.8 351.9
4 7 i0 7 4 8 8 9 9 10
2712
TOTAL
NO.
ARGUMENT OF PERIGEE (AT EPOCH)
126
NO. OF STATIONS
ARGUMENT OF PERIGEE (AT EPOCH)
165.6 213.0 255.9 297.8
Table COURIER-1B
NO.
1 2 3 5 6 7 8 9 10
EPOCH
66122q 670107 67011q 670121 670128 670602 670609 670616 670625 670708
AVERAGE
TOTAL
NO. OF OBS.
5.2.8m 7-DAYS
HEIGHTED RMS ARCSEC/2
334 507 568 501 237 97 97 151 258 326
1.130 1.183 1.072 1. 087 1.059 0.971 1.150 1.074 1.010 1.2_q
248
1.098
2q76
127
ARCS
NO. OF STATIONS
ARGUMENT OF PERIGEE (AT EPOCH)
9 8 8 10 9 5 5 7 7 7
95.5 211.8 273.6 332.1 27.8 343.6 40.5 9q.1 150.2 276.6
8
To were
remedy
selected
COURIER-IB, The
for
three
systems
summarize
the
will
be
positive
5.2.9
in
the
on
Explorer-C
ring
motion
studies.
United large
a
States
data
short
in
set
to
be
was
be
utilized
in
robust
magnetic
stabilization
and
beyond
equatorial
The
orbital
on
This
day,
vector
were
a
were
BE-C
BE-C
the
5.2.9a. per
end,
were
globally
these
radiation
adjusted quite
was
good,
deployed
to
had
a
As
dramatic
of
the
using
being support
data
In
from
the
128
its
inclina-
American
crustal
sites
located
not
visible
interval
determination network
However, of
in
Therefore, time
laser
set.
also
low
position
its to
tracked
given corner
lasers
this cubes
located
Hemisphere.
BE-C 5 day
are
and
general, third
LAGEOS
presented
arcs.
coefficient
arc.
magnetically
short
location
Southern
studied
each
data
for
of
a
global
the
Facility,
fortunately
laser
station
was
pressure
within
5.2.9m
satellites.
revolutions. in
the
characteristics
satellite
solar
studies,
unfortunately
region
to
acquired
a reasonably
lowest
through
was
North
successive
yielding
its
generation
Flight
and
early
visible
BE-C
at
Wallops
panels
for
To
satellite's
PEOLE.
solution.
Because
solutions. often,
support
were
and
first
data
satellite
solar
four
could
to
these
from
target
times,
three
arcs
of
The
large
at
the
5.2.8g
gravity
1965.
favorite
on
GEM-TI
DI-D,
inclination
retro-reflectors.
BE-C
BE-C
enabling
laser
became
inclusion
launched
reasonably of
low
satellites
DI-C, by
satellites
Observations
was
Virginia
had
BE-C
Laser
2,
Tables
these
additional These
tracked
1970's.
the
six
model.
were
of
of
VANGUARD
satellites
resulting
BE-C
sets
the
body,
early
the
of
Island,
tion,
a
these
later,
stabilized,
the
of
discussed
Beacon
a
rocket
contribution
Analysis
carried
2
data in
data
impact
Wallops
situation,
inclusion
VANGUARD
later
laser
it
this
the
A the
mission.
drag
data
systems
Since
Table
parameter
orbital
laser
generation
in
this
state taken which object
was
and
remains
sufficient
to
a have
modeling
solutions.
data
are
shown
used
in
for
the
field
of
this
satellite
The
in Table
GEM-TI
normal 5.2.9b.
solution
interest, well
equations In
data
all,
from
1979
represented generated
39
arcs
of
in from
BE-C
wlth
other
additional
the
drag
parameterization
onwards
arcs
our BE-C
laser being
were
gravity tracking data
were
available
testing.
Extensive performed
satellite
and
are
tests found
of
summarized
in
129
Section
7.2.2.
on
BE-C
were
Table
5.2.9a.
Semi-Major Apogee
Orbital
Axis
Height
Perigee
Height
Eccentri
city
Characteristics
7507
km
1320
km
940
km
of
O. 0257
Inclination
41.19
degrees
Mean
Motion
1 3.35
revolutions/day
Beat
Period
5.5
130
days
BE-C
Table
5.2.gb
ARGUMENT EPOCH
NUMBER OF OBSERVATIONS
790320 790402
1153 1535
790411 790417
2472
790426 790501
3596 3265
790512
1904 3136
790523 790813
i173 614
791022 791112
1254 1765
791202 791217
986
WEIGHTED RMS
(m)
OF
NUMBER STATIONS
OF PERIGEE ( AT EPOCH)
1.2126
8
1.7486 1.4003
8 8
18.204 81.950
1.2484
9
128.830 161.207
I.I 535 1.3096
8 6
207.915 232.713
1.2258
6
i .4735 1.3281 1.1893
4 5 8
291.352 349.258 51.989
I. 1033
7
54.306 161.403
1.4961 1.3430
9 7
265.595 344.681
.6662 .7459
7 I0
133.182 168.528
1.1481
7
206.047
.9070 1.2113
7 8
239.858 206.400
I. 1468
8
349.147
2.1713 1.2983
4 6
106.631 131.798
1.5013 2.0744
8 10
62.832 89.221
1.4970
7
99.180
1.5275 1.6996
I0 I0
359.756 63.421
1.7794
9
101.679
632 I010
1.0837 1.4706
5 6
1076
1.2099 1.5659
7 9
319.695 355.343 67.447
I. 1450 1.3487
7 7
181.514 149.842
1.3525
5
810924 811006
1266 2039 3997
1.4846 1.4363
7 8
254.153 92.630
811012
2717
811019 820201
2258
1.7980 1.0116
8 7
221.105
1135
1.2684
6
46.323
800115 800122 800129 800205 800408 800505 800528 800602 800728 800802 800915 800923 801006 801013 801124 801201 801215 810303 810317 810728 810817
1002 973 1022 2202 1710 1460 1551 644 1197 1215 1175 1683 1564 1412 1419
1911 1760 1357
131
111.785
150.636 182.613
SECTION DEFINITION
OF
TRACKING
In for
the
order
to
compute
TOPEX
mission,
stations
must
be
reference
frame
for
this
section
bring
as
existing
station
made
TOPEX
work
6.1
COORDINATE
in
The positions laser the
is closely
coordinate SL-6
adopted Thus
see from
all
system
the
were
zero
mean
rotation zero
pole
elsewhere referred coordinates
used
pole in to
as are
gravity
al.
(1985)]. MERIT
was
station
put
a
typical
of
required
to
existing
from
various
chosen
for
system,
SL-6.
This
issue
the
The
in Cartesian
form
rotated
System
(TCS). in the
al.
was
{1983)]. the
SL-6
longitude
to
definition.
A
mean
figure
further in
coordinate
___ 133
the
as
coordinate
into in
considered
for use
et
the
known
definition
meridian model
resulting
Coordinate
laser
station upon
tracking,
arcsec
were is
based
transformed
reference better
priori
is
[Melbourne,
+0.144525
to
a
longitude
were
coordinates
document. TOPEX
LAGEOS
campaign
by
the
project
The
that
adopted
for
from
of
Observatory
origin.
the
systems
The
course
The
coordinate
model
by GSFC
rotated
all
this
system.
system
developed
coordinates
McDonald
and
unique
laser
description
in the
position
axes,
mean
et
station
the
system
TOPEX
a
ultimately
accommodate
in the
transformations
coordinate
to GSFC's
tracking
system.
described
of coordinate
The
developed
[for
that
of
the
Smith
contributing
model
DEFINITION
system
system
solution,
years.
and a
field
coordinate
be briefly
into
gravity
all
unified
variety
related
for
of
procedures
coordinate
needed
one
will
in a
SYSTEM
unified
to
coordinates
past
preliminary
coordinates
the
are
GEOCENTRIC
COORDINATES
improved
work
as
station
coordinates
solutions
an
referred
well
A PRIORI
STATION
the
this
6.0
to
more
frame The
and this
detail will
be
station
data-reduction
and
|_:IgI,Ii_NAkLY _LANF,
the E-matrix generation runs, but, for the purpose of cataloging, the coordinates have also been transformed to geodetic form. The geodetic coordinates refer to an ellipsoid with a semi-major axis of 6378137 m and a flattening of 1/298.257.
6.2
INITIAL STATUSOF STATIONCOORDINATES
The station positions to be transformed into the TCS exist in a variety of coordinate systems. These include local datum coordinates and dynamically derived coordinates from solutions such as GEM-9[Lerch et al. (1979)], and GSFC-73[Marsh et al. (1973)]. The meansfor determining the transformations is provided by a set of laser sites for which both the SL-6 coordinates and the datum or dynamically determined coordinates are known. Table 6.1 lists the laser sites and their unmodified SL-6 coordinates that were used in this these stations is 1982.
work.
The approximate epoch for
6.3
THETRANSFORMATION MODELS
first
Two transformation models were used to complete this task. The model utilizes the coordinates for widely distributed laser
stations known in both coordinate systems, the SL-6 system and the other coordinate system of interest (e.g., local datum or dynamically determined system) for which we wish to establish a rigorous transformation. The second model employs a simple linear transformation for stations which are in close proximity to one of the laser stations listed in Table I. By "close proximity", we mean that station separations
do
committed
by
a size
a few
of
not
exceed
ignoring meters.
100
scale This
and
kin.
Beyond
rotation
aspect
will
134
this
distance,
parameters be
described
can
grow
shortly.
the
errors
rapidly
to
Table 6.1 Laser sites known from the SL6 dynamic Station NAME I no. QUINY EASTER SANDIE STALAS GSFCLS BDILAS GRKLAS RAMLAS BEARLK OVRLAS GOLDLS FTDAVS YARLAS HAYLAS KWJLAS SAMLAS OSFIO0 GSF101 GSF102 GSF103 0SF104 GSF105 QUILAS MONLAS PLALAS OVRLAS GOLLAS MUILAS HUANIL MAULAS FINLAS KOOLAS WETLAS GRALAS SHOLAS RGOLAS FORLAS QUILAS VANLAS HOPLAS XUMLAS ARELAS HOPLAS NATLAS MATLAS ORRLAS ARESAO HOPSAO NATSAO ae -
7051 7061 7062 7063 7064 7067 7068 7069 7082 7084 7085 7086 7090 709i 7092 7096 7100 7101 7102 7103 7104 7105 7109 7110 7112 7114 7115 7120 7121 7210 7805 7833 7834 7835 7838 7840 7885 7886 7887 7888 7894 7907 7921 7929 7939 7943 9907 9921 9929
d
latitude m s
39 58 24.5710 -27 8 52.1650 32 36 2.6580 39 1 13.3620 39 1 15.1040 32 21 13.7620 21 27 37.7710 28 13 40.6520 41 56 0.8960 37 13 55.6560 35 25 27.9630 30 40 37.3040 -29 02 47.4100 42 37 2i.6890 9 23 37.6890 -14 20 7.5170 39 1 15.4510 39 1 16.2050 39 1 14.3800 39 1 14.6070 39 1 17.0820 39 1 14.1640 39 58 30.0020 32 53 30.0020 40 10 58.0010 37 13 57.2120 35 14 53.9000 20 42 27.3920 -16 44 0.6830 20 42 25.9960 60 13 2.2880 52 10 42.2450 49 08 41.7770 43 45 16.8840 33 34 39.7210 50 52 2.5610 30 40 37.3060 39 58 30.0180 34 33 58.3570 31 41 6.3150 32 56 20.9340 -16 27 56.7010 31 41 3.2220 -5 55 40.1350 40 38 55.7930 -35 37 29.7560 -16 27 56.7010 31 41 3.2220 -5 55 40.1350
6378144.11,
f
-
1/298.255
135
longitude d m s 239 3 37.5530 250 36 58.9940 243 9 32.7810 283 10 19.7950 283 10 18.6050 295 20 37.927 288 52 5.0330 279 23 39.2980 248 34 45.5370 241 42 15.1130 243 6 48.9170 255 59 2.4810 115 20 48.1070 288 30 44.3390 167 28 32.4860 189 16 30.3570 283 10 47.6350 283 10 Li2.8350 283 10 18.7920 283 10 18.7950 283 10 36.8380 283 10 20.1580 239 03 18.9490 2.13 34 38.2580 255 16 26.3360 2.11 42 22.2150 2'13 12 28.9490 203 44 38.1020 208 57 31.7780 203 44 38.6000 24 23 40. 2110 5 48 35.1190 12 52 40.9670 6 55 15.8640 135 56 13.1890 0 20 9.8620 255 59 2.4780 239 3 18.0180 239 29 57.9780 2.19 7 18.5000 245 47 48.6070 288 30 24.6030 249 7 18.8370 324 50 7.2190 16 42 16.6860 1.18 57 17.1240 288 30 24.6030 2.19 7 18.8370 324 50 7.2190
solution
ellipsoidal height 1052.8800 110.5550 981.4700 12.1670 10.1530 -30.1170 -25.7760 -30.6690 1955.9060 1171.0190 958.3230 1954.3160 234.2260 84.9250 25.7920 41.8820 3.1100 1.3140 10.8910 10.8330 2.8980 12.0840 1099.2260 1831.8602 1494.4826 1170.9230 1031.5171 3060.6295 40.1250 3061.2004 71.2110 86.4620 654.0907 1315.9275 94.3156 68.2651 1954.2694 1102.4716 597.2122 2327.6088 234.6146 2485.1860 2345.8548 32.4910 528.8756 941.8380 2485.1860 2345.8548 32.4910
6.3.1
Seven
The
Parameter
seven
Bursa/Wolf
Transformation
parameter
transformation,
transformation
transformation rotations
[Leick
relating
are
two
involved.
&
van
geodetic
The
also
sometimes
Gelder
(1975)],
coordinate
transformation
systems
has
the
_X
I
AY
Y
m
-_
a
when
-_
the
rigorous
only
small
form
Y
+ (I + _L)
AZ
Z
is
as
- - dat X
m
- - SL6 ×
known
(6.1)
Z
I
i
where
is the
- - dat
to
ith station's
the
local
Cartesian
datum
(or
other
coordinates
referred
coordinate
systems,
5( depending Z _,
on
the
case),
. _, and
e
are
small
Euler
rotations
about
the
Z,Y,X
axes
(or
other
respectively,
AL
is a scale
AX,AY,AZ
are
translations
coordinate
The
seven
comparing
parameters the
transformation in
Rapp
laser
are station
is desired.
factor,
and
between
systems)
determined
in
coordinates Further
and
a in
details
(1983).
136
the the
local
SL-6
least both and
system.
squares systems
a
datum
solution for
derivation
which arc
by the round
6.3.2
The
Linear
The SL-6
Translation
approximate
system
linear
is found
•SL6
dat+
.SL6 _I
.dat+ : AI
HSL6 i
=
translation
of
the
• SL6 t@j -
(.SL6 .dat. _ Aj - Aj
(6.2)
Hal.at . SL6 Hdat i ÷ [Hj J. )
known
in
both
coordinate
systems
Some
errors
be
expected
to
scale
and
rotation
parameters.
i and
j
when
made
to
using
the
neglecting
true
stations ascertain
the
to
longitude
grow
most
large
3
primarily system
6.4
when
to
they
NUMERICAL
This
were
(e.g.,
in
these
a
and
the
distance
far
errors
optical
near
coordinates
and
the
local
in this
model
This
a
was
A
kin.
of
that
the
error
linear
doppler
tracking
especially
computation
found
The
datum
primarily
is
function
of
and
laser
its
apart.
magnitude 100
having
in
as
It
of
older
SL6
arise
transformation.
situated
will
used
parameters GEM-9. regarded
information
_t_i_,o _ ^_
was
distance errors
in
can
as
be
method
sites
in
was
our
new
aspects
of
the
a
station
stations.
RESULTS
formation
better
laser
relatively
of
determine
Table
currently
are
size
at
positions.
to
jth
rapidly
section
transformations
GSFC-73
can
SL-6
meters
used
near-by
the
NAD
as
the
)
denotes
to
into
dat
j
due
station
from
where
(dat)).
i th
to
6.2
highlight establish
lists
the
relating: The as
TOPEX
becomes
the
NAD
the
27
137
of
used to
station
best,
available.
numerical
table
stations
a priori
being
the
but
TOPEX to
SL-6;
determine GEM-9
coordinates they
priori
may
the to
given be
trans-
SL-6; here
changed
and are when
I
Table
6.2.
I
Stations used in least-squares determination oF the seven parameter transformations. (i.e. solutions From proqram STC)
i
7062 7069 7082 7086 7091 7105
: : : : : :
SRNDIE RRHLRS BERRLK FTORUS HAYLRS GSFI05
7109 7 ! I0 71112 7114 7115 7921
I 1038 7063 7067 7068
: : : :
: QUILAS : MONLR5 : PLALAS : OURLRS : GOLLRS : HOPLAS
"L-°I 7907 7921 7929 9012
IORORL STALRS BOILAI GRKLRS
: : : :
FIRELAS HOPLRS NRTLRS 1HRUIO
I GSFC-73-4 SL-6 1 g001 9002 9004 9005 9006
: : : : :
IORGAN IOLFRN ISPRIN ITOKYO INATAL
9007 9009 9011 9012 9021
138
: : : : :
IQUIPA ICURRC IUILDO IHRUIO HOPKIN
6.4. I NAD
27
The 12
to
NAD
stations
These
to
since
pole
the
GEM-9
were
SL-6
GEM-9
of
next
paragraph.
or
mean
pole
NAD The
determined
shown
27
27
considered
and
Doppler
in
and
more
Doppler were
acc_ate
solutions
to
6.1.
coordinates
definition
stations
from
Figure
optical
NAD
are
the
than
made
and
complete
zero the
in mean
trans-
could
are
relates
GSFC-73
not
was be
because
It to
to
SL-6
parameters the
of
stations
these
Again,
were
the
discussed
was
used
are
small
from
then
with
are
stations the
were
globe
transformation
definition
considered
positions.
European
determined
8
used
excepin
the
since
the
not
very
well
rotations
for
zero
applied
to
bring
these
Transformation
and
derived
around
most
solution
data
located
were
TCS.
GEM-9
GSFC-73
These
origin.
into
parameters
globe.
to
for
longitude
GSFC-73
as
longitude
these
The
GEM-9
dubious
to
the
Europe.
and
transformation
positions
for
stations
in
of
GSFC-73
terrestrial
camera
to
around
The
are
The
SL-6.
transformation
coordinates
coordinates
transform
coordinates
rotations
to SL-6
stations
known
to
States
were
Transformation
tracking
tion
datum
United
into
applied
distributed
local
used
parameters
TCS.
to
transform
the
from
Small
into
The
direct
then
determined
formation
6.4.3
over
terrestrial
years.
stations
transformation
coordinates
definition
6.4.2
SL-6
were
station
previous
Transformation
distributed
coordinates
to
27
parameters
tracking used
SL-6
GEM-9 SL-6
used
determined the
more may
because
European reliable
appear
rather
due
transformation
139
to
European
to
than odd
dynamically
the that
SL-6.
could
Datum
insufficiencies
GSFC-73
rather
than
a
not
This be
GEM-9 the was
to
SL-6
in
the
derived
dynamically
transformation done
established
since due
a to
insufficient
data.
used.
transformation
The
determined were
from
then
GEM-9
to SL-6
of
10 stations
again
were
definition,
some were
tracking
stations
(used
determining position
the data
determine
to
into
as
6.5
DISCUSSION
6.5.1
Transformation
the
described
in
one
SL-6
the
zero
mean
into
the
for
stations
such
Six
case.
above
The
of
the
were
SL-6
our
stations
S-band
network tracking SL-6
from were
stations
by
S-band used
were
system. to
and
the
into
employed
Small
The
stations
S-band
the
TCS.
exclusively
S-band
other
into
a few
by
pole
residuals.
found
Thirteen
followed
transformed
parameters
mentioned
that
were
stations
paragraph.
were
systems.
European
previous
the
was
GEM-9
in
anticipated
parameters
to
then
Likewise,
these
sites
to
TCS.
Parameters
determination
Comparison).
was
to
parameters,
apparent
than
procedure
these
these
SEASAT)
both
these
them
performed
bring
parameters.
bring
in Table
in
via
accounting
larger
GEM-9
rotations
systems
The
it became
track
via
The
globally.
stations
known
these
transformed similar
distributed
mentioned
to
a two-step GSFC-73
GEM-9
analysis, causing
problem, relating
to
After
this
parameters
applied,
Transformations
S-band
were
into
Other
positions
around
transformation
longitude
6.4.4
get
transformed
rotations new
To
a
The
of
the
least-squares transformation
in
the
previous
and
Accuracies
seven
parameters
based
program
parameters sections
6.3.
140
in
as
the
known relating
computed
transformations as
STC
the by
STC
(STation coordinate are
given
Table
6.3.
TransFormaLion
ParameLers
parameter
NAD -) SL-6
X
(m)
-31.4005
-0.9451
2.5460
(m)
172.5176
-1.7602
2.6820
Z (m)
182.7296
0.8776
-0.2535
1.6015E-6
-3.5305E-7
9.0237E-8
-0.77041
0.32384
-0.00924
-0.01160
-0.08520
-0.02139
-0.31404
0.04528
-0.04434
L_,Y
,_L
m (") (") E (")
Table
6.4.
QualiLg (RMS
parameter
X Y
(m) (m)
Z
(m) (") ('-)
H
(m)
oF Lhe abouL
NAD -_ SL-6
3.158 2.422 2.826 0.1161 0.1166 1.784
GEM9
--.) SL-6
GSFC73
-.) GEM9
transFormaLions
Lhe
mean.
GEM9
_ SL-G
1.404 1.133 0.469 0.0464 0.0233 1.537
141
see
texL)
GSFC73
-) GEM9
4.663 3.014 3.128 0.1615 0.1080 3.158
The are
large
center
of
with
(i.e.,
and
AY is
the
be
since
the
cally
active
present
least
which
AX
6.5.2
coordinate
The
likely
the
systems.
magnitude
AY.
of
the
scatter
is
--
with
the
determined
the
upon
our
This
determinais
the
case
the
more
tectoni-
far
frcm
optimal,
of
adequate
and
is
In the between
all
three
However, due
in
The
AZ
the
and
to
than
center
to
systems
are
differences
differences
translational
plane,
GSFC-73
coordinate
significant part
smaller equatorial
SL-6
ar_
in
component
the
is
equatorial
the
SL-6
center
of
mass
of
can
be
gauged
the
the at
plane of mass
GEM-9
and
systems.
the
precision
of
_c
most
half-way
Precision
scatter
and
since
systems.
of
to
635
dependent
is
GEM-9
a
The
in
needs
not
investigators.
States.
our
is
consistent
we
6.1,
distribution
transformations,
mass
of
order
nearly
GSFC-73
are
other
in Figure
suited
p.
value
concentrated
the
27 is
(1980),
highly
United
is
NAD
translations
by
is
transformation
below).
parameters
of
origin
components, falls
two
STC
SL-6
The
be noted
has
discussed
the
found
western
Although
smaller
an
the
coast.
center
longitude
can
to
since
Bomford
that
network
other
have
supposedly
be
in
27
meters)
of
by
transformation
(to
The
than
As
NAD
component.
tracking
west
of
reported
in
the
magnitude
larger
stronger
resulting
GEM-9,
The
stations.
LAGEOS
precision
hundreds
determination
of
will
in
AY translation
15 meters
parameter
tion
RMS
system.
of
distribution
the
tens
investigations
exception
seven
parameters
mass
other
for
translation
Transformations
of
residuals given
the
transformations
after
the
transformation
has
from
been
the
made.
RMS The
by
[ (_ifi
T (AX,AY,AZ,m,,,e,AL)
142
Xia )
(6.3)
'-:""_'-r:-._ ............ " .
/ /
OUILAS:
- BEAli_.K
"_
-
-
_-
........
o_....-........ ,.( _
i
i
"-..
• ..\(',._,'-....
6.1.
_
•
.........%.....-..,-j," v - _.--.. "-:""":_HAYLAS
- _
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¢ .: ".
!
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.._
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5
.-"" _"_
-,. _...---",):,.--_ ......
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•.oP,,S _
_...........:" _ _"
_..............;-:'"'" --; ....... , " -_.......-_---..."':""::Z. "'" "'-_
i
ii
:
-...,
.-,_.._........_
Laser Seven
Tracking Parameter
.:="_ _ ' ':/
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:
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Figure
!,".._
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AS .0 "_" • _._oMONLAS
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Station Locations Transformation
143
. • -
.-
used in Determining the between NAD 27 and SL-6.
where
m
×if
are
the
known
unified
coordinates
coordinate
for
system
tracking
(e.g.,
station
1
in
the
SL-6),
m
Xi a
are
the
(e.g.,
T
This to
denotes
actually
a priori
residuals
are
to
SL-6
Finally, meter
GSFC-73
GSFC-73
abandoned
goal
in
some
much
of
Error
Errors as
their
a
a
based 5
when for
the
can
be
is
the
strongest
meter
range.
in
system
is
early
meters.
As
not camera
coordinates therefore
must
that with
The
NAD
27
SL-6
3
3.5
since data
earlier, large
best
range.
with
laser
and
to
meter
weakest
seem
the
6.4
three
surprising
European are
in Table the
and
may
their
transformations
to
mentioned
coordinates the
too
to
used
of
2.5 the
were
applied
seen
the is
that
three
is
result
and
T
It
upon
coordinate
stations
6.4.
(especially
data,
priori
the
transformation
GSFC-73
same
uncertalntl
agree
1.5
latter
cases
well
residuals
GSFC-73
the
the
I to
GEM-9
of
the
sites),
large
to
are
standards,
Table
a
transformation.
how
T
the and
quantities
with
coordinates
of
RMS
the
This
uncertainties
6.5.3
in weaker
accuracy
of
in
residuals.
solution
of
transformation
is
the
station
given
averaging
transformation
used
a measure
The
in
etc.),
parameter
parameters
here
GEM-9
GEM-9,
seven
coordinates.
described
4
the
the
coordinates
27,
provides
determine
the
known NAD
other
with
in
the
a
the
today's
remote
available.
share
the
though by
to
or
GEM-9
resulting
es.
Sources
in few
priori
the
coordinates
meters. coordinates
This
of is
the
stations
especially
determined
144
from
in
true an
the
for
early
TCS
can
stations dynamic
be
as
having solution.
Stations
in
and
never
will
this
observations. directly to
an
better
errors
the
SL-6
the
seven
all
SL-6
solution
the
most
will
sub-decimeter tracking
have
mating
seven
their
the
their
range:
This
with
known
strong
histories
early
laser
is
an
true
have
come
accuracy
tracking
coming
determined
especially that
to
tracking
stations
coordinates
Stations
coordinates
stations
available
have
histories.
the
such
coordinates
of
of
histories
a
running
for from
I to will
2 be
weights
priori
the
For
the
map
and into
Program,
stations
were
was
maintained,
determined.
the
and
The
stations
when
STC
esti-
remaining
stations
parameter
uncertainties,
transformation
coordinates
a priori
be mapped
well
for
the
susceptible
will
distribution
parameters. to
STC
were
thus
both
errors
the
SL-6)
equal
are
in
coordinate
and
uses
addition
the
stations
geographical
transformation in
the
In
priori
unfortunately,
errors
directly
into
the
to
be
resulting
coordinates.
The
linear
are
meters
(a
transformations
These
directly. I) good
the of
systems.
that
transformed,
scale
of
coordinates
parameters
the
unified
parameters
coordinate
Program
not
when
only
a
this
way;
of
Distortion
STC
geographically,
them
in
program
the
of
suffer These
stations
number
all
after
translations
considered.
the
small
The nates
of
those
in
selected
6.5.4
hand,
other
tracking
determined.
to
the
the
solution
The
2)
On
robust
again,
limited
from
in
meters;
have
accurately
accuracy
GEM-9
likely
positioned
the
with
very
be
from
stations the
category
had
the
are
and
station
NAD
the can
fact
that
grow
separated
as
by
doppler
stations
separations
of
large
100 were
less
rotation
than
km.
as
and three
However,
transformed
in
3 km.
27 Datum
provides
transformation can
errors
involved
optical
from
illustrate
the is
residuals applied.
the
145
for These
relative
each
station's
residuals, distortion
coordi-
when between
viewed two
CONTOUR
Figure
6.2.
Longitude
Distortion
CONTOUR Figure
6.3.
Latitude
INTFRVAL:
Based
INTERVAL:
Distortion
Based
146
I meter
Upon
SL-6
vs. NAD.
I meter Upon
SL-6
vs. NAD.
datums.
The
classical
NAD
27
geodetic
squares.
The
determined
surveying
United are
shown
negative
distortion than
comparing
as
27
to
results
agree
6.6
SUMMARY
OF
Station
positions
a
of
mission.
a
I
geodetic
Appendix
longer
active.
station
coordinates
cally The
NAD
uncertainties determined
recent
and
possible,
their
good
27
and
data
as become
an
a
stations I
has
older
in
can
the
longitude
6.3.
and
Regions or
center
mass
of
of
latitude
similar
present
be
in
published
maps
system.
analysis.
have
available,
in
in
Table
of
a
have
few
the
made resulting
estimated
file this
in
studies. is
an
file
147
be
Doppler
project. updated.
are
to
no
yield
coordinates
for
the
dynami-
SL-6
system.
coordinate
they
have
been
have
been
sources
eliminate,
limited
in
all
as
transformed a
TOPEX
TOPEX
which
have
since
Maintenance
ongoing will
to
Error to
in
the
found
and
modified
assessed
solutions.
are
of
7 meters
centimeters
been
on been
I are
the
anticipated
the
in
maintained
many
5 meters
into
aid
laser
sites,
3 to
to
system
active
2 to
transformed
TCS
are
transformed
support
currently
of
and
been
TCS)
to
the
used
stations,
previous
geodetic
model
the
of
have
(the
optical
laser/dynamlc
with
SL-6
longitude
sources
field
accuracy
range
effects
agreement
the
system
transformations
attempts
distortions
coordinates
of
appeared the
of
primarily
coordinates
identified
NAD
the
consists
transformed
in in
gravity of
with
which
of
least
stations
and
27's
(1975)
those
a variety
Appendix
The
determined
6.2
satellite
coordinate
2 consists
27
stations
from
2.
which
with
to
distortions
NAD
Gaussian
by
DEFD;ITION
lists
and
file
the
well
preliminary
sites.
for
quite
by
distributed
Gelder
Doppler
established
respect
Figures
where
& Van
NWL9D
with
The in
areas
geocentric
Complete
Appendices
maps
STATION
unified
creation
contour
the
27
network
adjusted
densely
6.1).
Leick
Their
into
more
indicate
and
NAD
(Figure
SL-6's.
NAD
the
the
States
latitude
larger
of
utilizing
determined
techniques
distortions
by
western
is
is a terrestrially
best
as
coordinates.
region of As
and the
new
Since
are
in
station solutions
the
station
coordinates be
assigned
improves, the
come
tracking
a
variety
generally. epoch
stations
motion
from
It
dates
will
parameters. histories
be
can rotated
The
is
to
in
sources,
planned
either
effects
lengthen
of
be a of
an
that
assigned
particular plate
time.
motion
associated
as
the
to
individual
epoch will
TCS
using
epoch
cannot
geodetic
file
stations a set
continue
to
of
or
plate
grow
as
SECTION 7.0 FORCE MODELING
The force model used for the GEM-TIdevelopment consists of the conservative geopotential forces and the non-conservative solar radiation pressure and drag forces. This section describes the specific application of the models and provides the general basis for the details of the modeling.
7. I
POTENTIAL EFFECTS The geopotential
consists of both a static
part, which is defined
by the unperturbed mass distribution of the Earth, and a dynamic part, commonly known as the tidal potential, which is due to the mass deformation of the Earth caused by the gravitational forces of the Sun and Moon. The force is computed as the gradient of the potential.
7.1 .I Mathematical The
us
to
+
_
as
of
form
geopotential
of
the
the
Potentials
I! oaxn laln
+ __ r
where
standard
Formulation
is
[ n=2
the
GM),
geocentric
r
[ m=O
__e _r _
associated
the
geopotential
(sin
is
the
Legendre
_ _nm
constant
geocentric _ is
¢)
nm
gravitational
latitude,
the
_
the
of
cos
satellite of The
the
149
by:
ml+_
sin
Earth
distance, east
m
(7.1)
(elsewhere
referred
of
_
¢ is
longitude,
the first use
given
rm
satellite
functions
coefficients.
is
the
kind,
and
normalized
the
satellite
_r_n(sin _
and
¢) _
harmonics
are are is
indicated
by
the
unnormalized
functions
=
where
(26+I) (n+m) ! (2-6om)-11/2
6
is the Kronecker om otherwise equals 0.
The and on The
the
ocean
the
simple
in
time. terms
either
cm
parameter
UB
the
is
The
However,
normalized
and
equals
consists
of
body
the
given
above,
or
are
where
to
I
the
of
the
body
the
is
more
is
Earth
based model. of may
coefficients
conventionally amplitudes
contribution
and
expansion
potentials
the
0
potential
modeled
Wahr
these
where
m
tide
harmonic
of
the
when
potential
spherical Both
phase,
helght
tide
response
potentials
and
(7.2) nm
model.
form
tldal
amplitude
upon
layer
standard
tide
the
P
which
elastic
based
density
of
between
to
are the
body
tide
-- _ Af f k2, f
ocean
tide
potential
L3
I ae _r--
potential
-U° --_ f
_ K_ _,q,±
is given
P2m(Sin
is
¢)
similarly
(efB
+
expressed
related
P_q(Sin¢)
150
cos
to
elasticity
_2,f )
(7.3)
as
-- 6+I
C+l_q I _-ae ----
be
expressed
by
cos
a
vary
k2.
The
and
potential.
model
the
of
adopted
dependent
surface
expressed
in
tide
tide
delta,
potential
frequency
ocean
with
tidal
relationship
is
(n-m)!
nm
The
overbar.
± + B_q + ,f) (a_q,f
(7.4)
where indicates
summation
Af
is a body
tide
B ef
is
the
of
the
k2,f'62,f
are
the
which
m
is
constant
angular body
Love
is an
number
amplitude
the
body response
tides,
+ a_q, f
is
ocean
the
constituents
with
associated
associated I for
seml-dlurnal
K_
tidal
f.
constituent
with
f.
constituent
f
tide.
order
period
all
associated
argument
describe
the
over
with
the
and
f
diurnal
phase
respectively
of
the
Earth.
and
is
0
tides,
for and
the 2
long
for
the
tides.
tide
angular
subharmonic
of
constant
argument the
associated
associated
ocean
tide
with
degree
with
the
generated
by
4.
(_,q,_+)
constituent
f.
are
the
monic
Each should
be
of
constituent noted
that
f
amplitude
and
phase
the
tide
generated
is
ocean
associated
with
of
an
the
(_,q,+)
by
subhar-
constituent
unique
f.
frequency.
It
if
k2, f B k 2
_2,f
for
all
the
time
f,
m 62
then
domain
(7.5)
the using
total the
body
tide
potential
potential
151
may
be
simply
computed
in
k2 dael 1311 where
r d is
the
geocentric
gravitational
constant
model
Love
for the
single
band
simple
background
differ
of
was
The argument
the
Doodson
element
rates.
Sun
most
or
of
model
correct
from
the
is
constituent Table
f
7.1
Moon a
variations
terms
for
matching
number.
The
that
uniquely
identifies
(approximate)
Note
is
the
and
_d
frequency
are
concentrated
efficient
which
the
is
the
dependent
to
Love
reference
values.
identified
by
these
same
The a priori
Static
Geopotential
The
models
adopted
principal
Darwinian
frequencies
7.1.2
in a use
a
numbers This
are
frequencies
tidal
symbol based are
for upon
also
the
Doockgon
frequencies each the
corresecliptic
present
in the
effects.
a priori
for
LAGEOS
PGS-1331'
for
Starlette
PGS-S4'
for
SEASAT
GEM-lOB'
for
models
ellipsoid of
the
all
were
parameters new
speed
Models
for
GEM-L2'
gravity
definition
For
background
tide
modern
or
computationally
ocean
These
Sun
Moon.
the
It
and
the
adopted.
tidal
ponding
to
diurnal).
number.
gives
the
numbers,
significantly
procedure
and
(the
vector
(7.6)
other
the
development
are:
satellites
analytically (ae--6378137m, of
GEM-TI
light
152
corrected
to
zero
f-I=298.257),
(c=2.99792458x108m/sec).
and
mean the
pole, adopted
TABLE
Darwinlan
Doodson'
Symbol
Argument Number
s
Period (hr)
7. I
Description
M2
255.555
12.42
Principal
lunar
semidiurnal
S2
273.555
1 2.00
Principal
solar
semldurnal
N2
245.655
12.66
Larger
lunar
elliptic
semi dl urnal K2
275.555
11.97
Lunar/Solar
L2
265.455
12.19
Smaller
KI
165.555
23.93
Lunar/Solar
01
145.555
25.82
Principal
lunar
diurnal
PI
163.555
24.07
Principal
solar
diurnal
Mf
O75.555
1 3.66d
Lunar
fortnightly
Mm
065.455
27.55d
Lunar
monthly
Ssa
O57.555
188.62d
Solar
seml-annual
153
semidlurnal
lunar
elliptic
diurnal
7.1.3
The
a priori
Table upon
the
62, f
7.2
is
Earth
7.1.4
A__prlori
The set
of
where
of
The
amplitudes
Laplace
are
the
and
tidal
oceans
(_f
(1975).
the
characterize
generating
to
- _f
argument
model the
based
Note is
that
free
of
of
the
response
potential.
the
tide
generating
potential
is
a
are
associated and
computed
Such
such
(7.7)
(P))
amplitude
Equations.
tidal
5f(P)
necessary
tide
i.e.,
(1979),
phase from
solutions
models
are
with
constituent
respectively numerical
involve
available
at
high
only
Af(P)
point
P.
of
the
solutions
a
for
f and
computational
a limited
number
constituents.
The
the
Dziewonski
model, fully
Wahr
Models
phases
presently
&
by
heights
tidal
and
Tide
elastic
cos
angular
computed
Gilbert
numbers
the
tide
numbers
of
Tides
-- Af(P)
_f(P)
Given
Ocean
the
Love
non-loadlng
constituent
and
of
this Love
response
mf is
burden
for
the
Model
I066A
These to
_f(P)
the
Model
zero
I066A
Tide
gives
Earth
dissipation.
Body
=
heights
_ £,q,±
global for
the
are
C± _q,f
tidal
expanded
P£q(Sin
heights,
evaluation
of
into
¢)
cos
spherical
± f± (0_q,
the
coefficients
the
potential
154
harmonics
± ) e£q,f
C± _q,f can
by:
be
and
(7.8)
phases
computed.
± E_q,f
TABLE WAHR
NUMBERS
Tidal
Band
Long
LOVE
Period
Diurnal
7.2
Line
All
I077A
k2,f
•299
145555
(O1)
•298
163555
(PI)
•287
165545 165555
.259 (KI)
165565 166554
Semi-Diurnal
FOR
•256 .253
(PSI)
All
.466
.302
155
Observed semidiurnal, I° x I° which
tide diurnal,
global
constituents at any
perturbing
GEOS-3
linear have
effects It
these
must
tide
is
A crude
The
tide
of
tracking
data
frequency.
degree the
terms
orbit
amplitude that
the
of
at
230
an
the
and
tide
the
also
included
coefficients
I ppm
a
incl inat i on.
156
.001
some
of to
satellites,
of
model. the
nominal
orbits
body ocean
represent reasonable
number
of
satellites
as the
function
analysis,
to
be
I cm.
in the Note
eccentricity
perturbation
of third
perturbation
this
the
a
principal
a
assumed
arcsec
be
have
In
in
must
all
produced
was
perturbation of
10%
53
if
I ppm.
for
53
their
and the
order
geopotential
for
on
proportional
of
inclination
decomposition than
tides
These
shows
and
constituents terms
is
due
first
the
about
altitudes,
7.1 in
were
criterion
minor
orbit
These
is
frequencies.
arcsec
tide
ampli-
amplitude
of
improved
evaluated
tide of
an
were
greater
ocean
to
orbits
Figure
ocean
half
analysis
effect
Satellites
tidal
significant
TOPEX
response
tide
.001
have
minor
ocean
inclinations
over
the
qualitative
tidal
can
the
a
scheme
tide
a Kaula-type
associated
contribute
the
for
radially.
tide.
criterion to
the
than
major
eccentricity
equivalent
More
upon
the
from
of
based
of
might
ocean
proposed
potential
histories.
tidal
the
tide
orbital
effects
on
decimeter
a
on
These
perturbation
minor
satellite
having
radial
theory.
presents
computed
data.
total
the
orbit.
was
that
gauge
amplitude,
in
integration
available
the
perturbations
a variety
in
been
an
the
if
data
53
are
estimated
I
of
be modeled
estimate
tide.
models
have
tide
90%
small
exceed
7.1
tracking
no
analysis
probable
raising
Figure whose
This
which
over
constituents
using
sea
constituents
perturbation
modeled.
the
tide
orbit.
also
for
the
tide
bands
deep
a satellite's
shows
major
Schwiderski
although
ocean
11 period
available
on
7.3
orbit
E.W.
However,
effects
the major
long
account
which
Table
the
the
point.
constituents,
to
by
should
for
and
grid
incorporates
tude
models
in
is the
(_I
I
_
_D
o,I
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.,_
m
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I_'i'I'Ia_w£
158
This
analysis
constituents. main
A substantial
tidal
frequencies,
periodicities of
the
the
tides
resonance
in
as
orbital
the
the
of
thus
harmonic on
these
the
complete
sents
only
the
main
order
to
provide
existing error and
which
due
However,
only
expansion
of
the was
the
on
degree
also
satellite
and
The
is
global
The
Cartwright
tide I is
r(¢,l,t)
-
terms
use
the
both
derived matching
the
was
that
of
to
spherical
of
the
a
of
repre-
from
quantitative
the
ocean
in
low
degree
solution
tidal
model
3.
36
of
minor
admittance, tides,
Schwiderski.
harmonics
admittance
concern
terms
simultaneous
concept
some
with
developed
sideband
complete
for
significant
available
perform
a true
in Appendix
terms
attenuation
procedure
The
upon
have
tides
for
in
order
fundamental
the recoverability
given
based
grid
was
for
which
These the
motivated
as are
models
subsequent by
the
study
(1977).
raising
given
is
were
converted
studies.
assess
orbital
_^^
ocean
to
and
the
_^
at
commensurability
can
Our
those
seen
exciting
degree
of
^_
errors
order
to
tides
A
geopotential.
Models
been
their
tidal
as a__prlori
procedure
below.
longitude
of
^^_^_
of a__priori
in
better
terms used
a one
to
the
the The
with
frequencies
their
low
objects.
^_^
models
harmonic
detailed
of Munk
and
the
with
frequencies.
closer to
tidal
not
convolved
the
because
orbiting
and
spherical
The
have
terms
frequencies.
estimates
analysis order
with
tidal
set
are
produce are
on
associated,
motion
to
significant
sideband
terms
tidal
oceanographic
are
sideband
wi _11 _lle _,_A_ wavel^_
most
possibly
nearby
satellites
effects
The
the
Earth
150
these
orbital
the
the
of
with
dominant
effects
distance
on
frequencies.
spherical
than
number
satellite
of
the
more
but
seen
Some
than
potential
is
of
satellite.
with
revealed
potential
at
time
t
and
at
latitude
¢ and
by
_ rs(¢,l,t) 8
= [ g n8 8
159
P2m(Sln
¢)cos[oBt+xs+ml]
(7.9)
where
B designates
equilibrium the
tide
of
gravitational is
terms
the for
degree
tide
AB(¢,_)
r_
for
and
to
the
the
of
degree
perturbing
with
O, 1900.
represented
2
and
effect
B, m is
o B and
associated
January
is
negligible
specifying
this
of
surface of
frequency
constant
epoch
Earth's
2 are
cos[o 8 t +
are
also
potential
input
are
e
B is
X8 -
The
by
g.
order
m.
The
(e.g.
Munk
and
specified.
is
the
set
of
and
the
complex
signal
r B with
given
(7.10)
_8(¢,_)]
amplitude
relating
the
AB g _B
the
the
constituent
admittances hand,
phase
function
in
function
ZB(¢'X)
other
to
_B(¢,X)
--
These
that
= AS(¢,k)
corresponding _
the
for
of
heights
admittance
signal
than
Note
is the
or Moon at
greater
_B(¢,k,t)
The
Sun
constituent
X8
Legendre
response
constituent
qB"
acceleration
Cartwright,1977).
where
the
associated
The
particular
amplitude
ephemerides
P2m
the
phase
respectively.
exponential the
signal
complex
output
by
-J(_B +m_)
(7.11)
P2m
readily
admittance
computed is
known
from for
the
known
constituent
tides.
If,
8, then
on
one
the may
compute
Thus, from
if
: Re[
Re[Z r
reasonable
admittance
the known
tides,
the
unknown
(7.12)
function tides
160
descriptions could
be
could
estimated.
be
obtained
The 1980b), i.e.
major
were
tide
values
tide
obtained
values
of
errors
estimated
long
period
for
distribution only
on
a
available
of
the
a
in
the
one
7.4.
amplitude,
as
equilibrium only
estimated
in
M 2,
rms
is not
not
provide
errors
sense
-
and
for
the
long
us the well
12.8%.
The
geographic
Note
three
each
as
i.e.,
available.
diurnal,
and
did
rms
NSWC's
tides
Nominal
as
an overall
four
are
proportionately
tide
errors
semidiurnal,
NSWC
being
The
harmonic
shown
diurnal
tides.
tape,
grid.
spherical
Also
qS"
period
(1980a,
GOTD-1981
global
their
and
models
NSWC
degree from
Table
long
Schwiderski
standard
semidiurnal
the
are
four
of
estimated
to
the
computed
tide
for
were
relative
errors
_8
the
errors band
determined
form
tabulated
equilibrium
with
are
A 8 and
are
constituent's
the
data,
constituents,
representation,
model
in
for
these
estimated
constituent
that
there
period
tides
available.
From
the
semidiurnal,
diurnal,
frequency
being
that
the
i.e.
within
tidal
reason
that
band.
3 or
band
are
at
the
band
to at
three
cannot
be
assumption
tidal
to
their of
best
there
period
the
a
be
four
is
NSWC
Mm
equilibrium
linearity
to
the
of
band.
161
Mf
tides
values. the
on
is
are to
frequency, the
Earth's
because
global
for
the
practical
This
admittances
of
or
(or
semidiurnal so
the
smaller suggests across
in
frequency
available,
reduce
are
over
illustrated
span
diurnal
tides
of
of
assumes
interpolate
greater
segmented and
also,
procedure
in
period
further
and
a much
than
point
the
range
proced_e
adopted
points
The
band long
was
for
the
function
¢,A
small,
that
The
linear
particular
analyses
so
limited.
band.
only
Also,
more
separate
bands
assumption
each
long
do
period
each
only
in
to
locally
anticipated are
4 compared
the
is
linearity
However,
interpolation.
period
admittance
Proportionally, in
frequency
with
was
from)
7.2.
variation
of
represented
there
extrapolate
chose long
This
nonlinearities
we and
each
surface.
Figure
outset,
range
by
this of
a factor
a conflict the
long
TABLE NSWC
Cumulative
RMS
Tide
Values
7.4
TIDE
Summed
MODELS
to
Degree
30
and
their
RMS
NSWC Equilibrium Tide
Constituent
Amplitude
Tide NSWC
n B (cm)
RMS
(cm)
Errors
Model
Errors (cm Ampl itude
& deg)* Phase
M2
24.2
30.0
3.11
3.72
S2
11.3
12.2
1.28
4.24
M2
4.6
6.5
0.51
4.12
K2
3.1
3.4
0.23
3.13
KI
14.1
10.9
0.94
9.95
01
10.1
7.9
0.57
3.42
PI
4.7
3.5
0.20
4.14
QI
I .9
I .7
0.08
2.41
Mf
4.2
I .0
Mm
2.2
0.8
Ssa
I. 9
I. 6
*From Table Island and private
of Comparison of Deep-Sea Stations
Empirical and Modeled (used and not used),
communications.
162
Ocean E.W.
Tides at 195 Schwiderski,
-180 tidal heights, phases and l°z
O
l °
errors given Schv¢iderski major tide constituents
-90 X
odm_ ttosu:e
_
T
YLsO tia,,
o/for
hI°ok tO x 1o
S
!
I
: :
:
i :
:
_ freqaesu:y
:
fi q, f_, f'3 Long
Period
:
:
:
- fzoqwe_ny
q' f_ f'3 f_ Band
Diurnal
= known
admittances
= predicted
heights,
Band
:
for fp's
& errors
for fp'S
Figure
7.2.
Background
163
:
:
Semi-Diurnal
for fl • f2, f3 frequencies
admittance
phases
:
. freque-_7
fi" f;' f'2'f'3'
Tides
Model
Development.
Band
by for
The of
residuals
the NSWC
squares
tides
where
For
the the
amplitude.
residuals the
will
should
diurnal
be
hypothesis.
The
NSWC
reflect
fitting
each
estimated
the
disagreement
process
A 8 cos
_B
is a or
A 8 sin
in AB(_,X)
and
long
period
tides,
estimated
as
M2,
dominated
Shelf
and
These
specific
hypothesis
were
to
admittance
concept,
be
are
which
local
more
The
residuals Schwiderski
to
a
of
tide
the
the
the
models
also
to
reflect
{1980a,
as
our
such
solar
and
linear
justifiable
data,
due
as
to
the
radiation
any
Clearly,
the
was
nonlinearities
1980b)
tide.
error
in areas
response
error
equilibrium
nonlinearities
replace
physically
that
12.8%
with
global
particular to
represents
nonlinearities
will
of
6_
is
differential
adopted
a
we
dealing
by
the
resonance.
modeling
errors
for
be
_B
(1.13)
error
we
leas_
errors:
the
same
present.
physical
data
process
weighting
Because
Patagonian
Earth's
fitting
: (6a 2 + A_6_2) -I
proportionally tide
this
specified
6a represents
in #B(_,X).
the
regression
to the
weight
the
with
linear
according
from
systematic
if arc
nonlinear
in
a
nonlinear
based
model
on
the
would
be
computed
by
essential.
The
global
Schwiderski The
and
models
sense, true
are
we
Figure
all
7.5
general lower that error,
not
of
As the
area part
of
the error greater
and
numerical
qualitatively
shows
magnitude). dominate
our
have
for
amplitudes
the
Figure
than
same,
local
20%,
which
shows
the
less
of
also
the
the
western
The
regions to
7.4.
in a global is
upper
part
of
in
M2
(vector
Patagonian
Shelf
differences
relative
correspond
164
the
and
This
residuals as
7.3
tide.
The
significant
20%.
generally
important
such
and
as
that,
tides.
percentage
than
tide
indicates
diurnal
Islands
M2
in Figures
this
areas
are
the
compared
amplitude
There
is typically
are
and
Marquesas 7.5
for
mismodeled
global
expected,
the
the
semidiurnal
residuals. of
model
seriously
the
phases
in
Atlantic. error, of
The
indicating
high
amphidrome
the
relative locations
OE poor _u+_cn'_
Schwiderski .
.
M 2 Tide
-
Amplitude
in Cm
i
J
--q
L_r.
°_._ Ig.
-.
o."_ jp-,,. _,., • -95. _
.
C',\
,. ,_. __'
•
.ate."
f ,,m o
.
4.
D.
313.
113.
I_.
Im.
I_JID.
Interpolated
14_.
M 2 Tide -
I10.
Amplitude
IN_.
in Cm. • _""I
/ .'q. _d
Figure
7.3.
Comparison
of
165
M2
Tide
Amplitudes.
Schwiderski M 2 Tide -
Interpolated
Phases in Deg.
M 2 Tide - Phases in Deg.
I,
I.
Q.
Figure
7.4.
Comparison
166
of M 2 Tide
Phases.
OF POOR Amplitude
of Error In Cm
Relative Error in Percent
Figure
7.5.
Error
in Interpolated
167
M 2 Tide.
Q'bA,L.ITY
where the
the
amplitude
results
we
semidiurnal
there
as
for
error,
and
M2
has
still
has
rms
on
the
an
tides
in
is
typical
the
of
diurnal
the
and
weighted equilibrium
tide
tides
show
and
difficulties
frequency
range
within
major
tides
our
regions
These
three
that
large
band.
substantial
a
available
statistical
in
cm
shown
of
01
and
bands
factor rms
in
was M 2,
shows
approach
in
of
two
power
this
procedure,
of
less
conclusion
the
in
entire
residual
amplitudes
are
the
linear
estimated
has were
this
band.
this
band,
7
out
be
cm 5%.
these
tide
fit
in
model
The
fits are
rms
twice
seen
in
admiteach
of
the 30
in
01 worst
cm, the
yet long
not
adequately
residual
amplitude
However,
our
Table
from
the
NSWC.
M2,
a total
of
disagrees
in
amplitude.
only
different
rms
of
tides
weighted
each
the
error
of
rms
can
global
quantity.
is
quite
that
for
from
this
the
NSWC
As
rms
than
amplitude
amplitude.
computed
in
that
that
summary
the
the
disagreement
the
of
global
approximately
confirm
tide
period
tidal
fit
by
by
rms
long
period
diurnal
input
order
the
indicates
exception
error
with
Mf
of
the
example
which
the
of
a weighted
band
modeled
This
alternative.
The
disagree
period
all
small.
100%,
only
presents
NSWC
case,
are
7.5
With
the
the
practical
semidiurnal
with
long
given
tides.
tance.
over
the
there
is no
NSWC
and
comparisons
with
Table
is
for
unexpected
However,
the
obtained
relative
difficulties not
is
bands.
Similar of high
variability
12.8%
7.4,
the
the
of
the
the
NSWC
equilibrium
Mm tide
amplitudes.
The provide adjusted
the in
standard factors order
distribution. the is
long not
deviations
period
by to
The band
inconsistent
which map
the
the
semidiurnal is with
off the
of NSWC
weighted and
by
unit
a
rms
weight amplitude
residuals
diurnal
factor
semidiurnal
168
of
given
bands 2.5. and
in
errors
into are Thus
diurnal
need
the
unit
near the
Table
7.5 to
normal
unity, linear
data,
but
be
but model it
is
•o
o r,.,
I_, 0
r._
_
B
c_
0
0
0
0 "cI
oO
0
I._
04
oD
o'_
o_
t-*-
c_
c_
_
c_
_
O_
OJ
0
0
0
_
L_ 0
CX,I _--"
O_ 0
',,.0 0
d
d
c_ c_
0
N
r
or.,_
c_
_o
oJ
c;
c_ c;
o_
_
on
e_ N
m E-,
_
0
cc_ c_
C-,v
•
0
•
-,m'*
.,-4
[]
_-_
o_
o_
_
_--G
0
169
inconsistent estimates than are
for for
40%
of
only
the
these
the
about
twice
associated
corrected
to
computing of unit
DRAG
AND
SOLAR
spacecraft
Formulation
the
: - 2 CD
the
these
we
on
errors
matrix
are
error
still
period
tides,
and
tides.
have
also
computed have
a
point
is
associated
with
at
less
tides,
which
error
RADIATION
which are
the
of
been simply
each
point
PRESSURE
are
the
of
concern
forces
of
in
modeling
atmospheric
drag
the and
the Models
acceleration
due
to atmospheric
drag
is
131 v O D Vr
satellite
satellite,
M
of
the
atmosphere,
v
to
the
atmosphere
and
1971Jacchia;
fits
long in
projected
pressure.
In GEODYN,
is
The
covariance
orbit
the
Mathematical
of
the
rms
error
based
variance.
the
these
unknown
tides
of
7.2.1
CD
these
for
nominal
the
forces
radiation
where
However,
amplitude
non-conservative
solar
AD
correct.
assuming
frequency.
ATMOSPHERIC
evolution
are
tides
estimated
by propagating
The
area
to
attain
the desired
7.2
the
errors
to
obtained
tides
period
equilibrium
In addition the
long
r
drag is the
is
the
Vr
the atmosphere
is
(7.14)
r
coefficient, mass
of
velocity its
the
170
of The
to
is
the
satellite,
vector
modulus.
is presumed
A
rotate
the
cross-sectional
PD
is the
satellite
atmospere with
the
density relative
model Earth.
is
the
The
acceleration
due
to
solar
radiation
pressure
is given
by
(7.15)
PS 7
"A--{
s
where by
_
is
the
the
body
ficient,
A
of
vicinity
sun
in AU,
are the
and
r s is
of
these
adjustment
accommodates shape. because often
in
the
to
capability
plecewise within
C ," CO
the
+ C
the
distance
R s is
the
the
either
time,
drag
coefficients
interval,
the
the
is
or
over
the
the
However,
spacecraft
efforts,
Atmospheric
Drag
Almost
of
are
GEODYN effects
has using
intervals,
according
and,
to
(7.16)
(t-t O )
present
but,
coefficients
time
the Sun.
accommodated,
pressure
vary
to
in
coefficient
variations.
solar
can
toward
with
specified
coefficient
satellite
a sphere.
drag
drag
coef-
pressure
pressure
multiple
satellite
pressure
pointing
similarly
observed
the
from
associated are
the
radiation
radiation
model with
solar
satellite
error
of
radiation
vector
and/or
model
accomodate
satellite
unit
assume
varies
to model
time
is the
density
discontinuous
each
Ps
the
atmosphere
required
before,
drag
the
shadowing
the
models
of
for
is
geocentric
the
much
the
as
the
accounting
CR
Earth,
of
Errors
factor
Earth,
M of
Both
For
the
and
the
the
eclipse
we
are
only
using
this
used
in
capability
with
the
drag
modeling.
7.2.2
cantly shape
all
perturbed of
the
by
Model
the
satellites
drag.
spacecraft
Testing
Given and
in
that the
there
are
atmospheric
171
our
analyses
model density
are
errors model,
signifi-
in both the
the
major
question
to
be
minimize
the
answered
atmospheric
parameterization
over
(c)
solution
for
C D values
as
per
was
it has
area
the arc
BE-C
in the
arc,
specified
performed
modeling
The
the
so
as
solutions.
once
of
drag
to The
arc
or
time
intervals,
length.
using
capabilities
-- specifically
1971).
both
satellite
the
GEODYN
a
selection
and
the
65
and
was
used
I
71Jacchia
as
the
basis
investigation.
BE-C
orbit
to determine
ranging
the
has
laser
example
were
stabilized,
San
Smith
used
difficult
in
received Andreas
which
et
the
orbit
cross-section
each
surface
1975.
Given
that
data,
tests
of
BE-C
al.,
area also
attention and
distances
Of
the
GEM-TI,
set
cross-sectional
with
a
model has
intercomparisons
and
analysis
observing
using
perigee for
BE-C height
BE-C
a reasonably error of
real
tracking
directly.
172
be
track data
of
designed errors and
of
the
magnetically
has
940km.
set
one
area
developed
strong
could
along
was
of
satellites
also of
analysis
laser
It was
in-plane
its
California
presented
its
from
within
BE-C
problems.
revolution.
the
of
modeling
modeling
the
station
of
of
drag
drag
sites
deal
Experiment
1977).
orbital
{e=0.O257)
good
Fault
creation
caused
over
a
intersite
atmospheric
significantly eccentric
length
over
the
orbital
adjusted
conveniently
models
1965,
the
CD,
the
over
most
to
for
day)
variable
density
(Jacchia,
parameter,
several
contributions
most
were:
C D adjusted
The
which
investigated
a C D and
this
(see
within
scale
once
parameterize
error
(b)
atmospheric
for
to
a constant
investigation
models
best
(a)
software, of
how
drag
options
(i.e.
This
was
to
vary
a
somewhat
A
variable
by
Safren,
laser
ranging
using sensed
resulting
orbit at
the
orbits
Several as
to
five-day
represent
example, passes
a
well
was
passes
the
geographic
7.2.2.1
On of
The
800201)
((a)
ways-with
a to
and
models
every
shown
7.6.
in Table
Table
7.6
differences
for
cases,
the
orbit,
less
in all are
drag
due
to
estimate
of
minimized where
minimization
has
data
the
of
and
gravity
long
number a
period
of
degrees
minimum
unless
of
RMS
7.6
of
78
only
30
shows
the
arcs.
cross and
epoch.
The
solution
of
drag
strong
show There
effects
signals.
freedom
devoted
evidence
the
along
track
RMS
differences force
this
scaling the
greatest
to drag was
173
some
result
Therefore,
it scale
present
7.7.
In
component
of
were
same
trajectories
error
as
are
Therefore,
concern
this
tha _ ovec-
aliasing
desirable
parameters indicating
an
to be
agreement,
in an was
being
the
construed
drag
as
component
parameters.
is also
could
be
were
always
in the
can
1965
intervals
Figure
models
differences
of
was
trajectory in
which
day)
Jacchia
day
shown
effects drag
5
the two
trajectories
as
non-drag
of
_,_e_
per
the
resulting
track
was
once
Both
tested
Each
(c)
and
arcs.
along
The
models
,'_o-_
respective
track
was
comparisons.
these
perturbed
error.
the
of their
effective.
parameterization
day
differences
density been
For
total
having
these
so
BE-C.
parameters
12hrs.
comparisons
sets
same
model
arc
of
drag
_o,,e
five
All
and
modeling
through
different
the
these
radial
drag drag
of
the
of
of
every
summarizes
the
a
Table
in each
trajectory
adjusted
over
with
weaker
chosen
on
having
selected.
_,,_]ined
minute
The
were
available
790417)
found
of
predominantly
0.6m.
orbits
also
employed.
each
with
than
(c_
each
were
of
solution
series
through
intercompared
all
and a
converge
tracking
arcs
Results
coefficient
utilized
These
a somewhat
the data
representation
approaches
hand
was
of
of
(epoch
the other
through
selected.
spectrum
Comparison
preliminarily
were
arc
distribution
Orbit
1971
full
tracked
used.
(epoch
arcs
)f drag that
be held a
need
the to for
NASA G - 6 5 -
6565 Figure 7.6. 174
BE-C.
Table
BE-C
TEST
ARCS:
5 DAY
ARC
DRAG
MODELING
7 6
ORBITAL
LENGTHS 790417
No.
COMPARISONS
800201
of passes: W. USA E. USA S. Am. Hawaii TOTAL
51 2_ 3 0 78
ORBIT
790417:
COMPARISONS: J?l
RMS
2_ 1 0 5 30
ALONG
TRACK
DIFFERENCES
(m)
CD+CDOT J71CD/DAY
J71CD/DAY
3.0
J71CD/12H
3._
0.8
J65
1.6
_.3
4.7 _ .1 4.6
J71
CD/12H J65
CD+CDOT
CD+CDOT J65
J65
CDIDAY
3.0
1.2
1.2
J65
CD/12H
3.6
1.9
1.4
800201:
ORBIT
COMPARISONS:
RHS ALONG
TRACK
CD/DAY
1.1
DIFFERENCES
(m)
J71CD+CDOT J71CD/DAY J71
CD/DAY
9.3 J71CD/12H
J71CD/12H
9.2
1.5
J65
1.4
8.7
8.5 10.7 10.4
J65 CD+CDOT
CD+CDOT J65
J65
CDIDAY
11.5
2.7
2.9
J65
CD/12H
11.2
3.1
2.5
RHS
Cross
Track
and
Radlal
Dl££erences
t75
are
all
less
1.5
than
0.6 m
CD/DAY
J65 CD/DAY
VS. J71 CD/DAY
4
m
o
,
•
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•
•
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7
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120.
RADIAL DIFFERENCES CROSS TRACK DIFFERENCES ALONG TRACK DIFFERENCES •tory
1-, ......... c._
176
'o'E'
(_
_ lr_,,
A,, ,.r.t,
additional
drag
clearly
had
producing which than
the
most
the as
model
CD/12hr
area
available
model
This
second
errors
satellite's
motion
causes with
the
apparent
timing
parameterizations
The associated for
to
are
the drag
the
based
upon
is
each in
pass
its
the
which
son
evaluation. errors
are
variable
a station
was area
analyzed.
the
of
the
types
of
is
arc
of
drag
assumed
through
177
some
of
if
one
the
was
not
most
of
the
the
calculated
either
early
the
7.8
when
the
different
errors.
timing
errors
provided could
parameterlzatlon,
or
so-called
presents
drag
the
when
apparent
parameterization that,
was
the
in
811012)
of
be
constant
As
Figure
spectra
to
using
are
minimization
than
constant
found
appear
(epoch
a
modeling
error
to
worse
cross-sectional
data.
an
different
approach
modeling
analysis
tracking
5 day
for
It
tests.
observations--these
in a
employed
CD/day
proceeded
plane,
at
actual
seen
of
orbital time
the
analysis
better
improvement
these
variable
Errors
various
model
the
no
and
results of
using
results
improvement
Timing
intercompari with
data
agreement
calculated
Apparent
errors" errors
As
error
yielded
clear
the
values.
acquisition
respect "timing
eliminate
the
of
area
in is
basis
significant
approach
seen
apparent
basis
II,
of
arc, no
invoking
utilized.
drag
calculated
was
trajectories
No
was
GEODYN
Evaluation
timing
with
area.
showed
second
the
representations
reducing
parameterlzation, on
2hr
representation
tracking
there
tested
cross-sectional
7.2.2.2
late
compared
in
satellite
were
for
This
weaker
Since
drag
drag/1
trajectories
the
desirable
orbits
surface
variable
orbit
most
and
satellite
between for
and
performance
strongest
obtained.
the
These area
track
drag/day
orbits.
the
Even
were
in
adopted
simillar
along
RMS
The
overall
with
models.
m
seen
best
arcs
2m RMS
density 3.1
the
the
on
parameters.
the
completely then
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251
ERRORS
x
109
GEM-L2 6EM-TI 18 ULE 16
14
12
I0
6
2
0
DEGREE Figure
10.2.
RMS
of Coefficient
252
Error
Per
Degree.
uncertainty the
for
zonal
based
and
upon
lost
in by
scrutiny these of
of
of
high
their
reasonable orbital
Figure
field
power.
8.7, this
to a
magnitude
degree
terms
expected and tracking
power
desirable
they
of
to
are
model
be
not is
amount
Nevertheless, the
the
calibrated
well
resolved.
However,
information
valuable
based
upon
this
coefficient
as
their
estimated
error;
253
contain
of
to
data.
terms
poorly
have
of
error
for
constrained
terms
in
unjustified,
been
for
those
100%
these
have
result
(except
nearly
collectively
degree
significant
in
shown
While
a lower
approach.
the
terms
are
individually,
the
taking
degree
orders)
expected
signal,
truncation
highest
resonant
their
valuable
shown
the
as already is
preliminary
uncertainties, no
and
more this
resolved
than is by
100%
both dynamic
a
10.1
THE
GEM-TI
GRAVITY
Mean
free
from
the
and
Moritz,
_max Ags--
X _=2
CALIBRATION
ANOMALY
air
spherical
OF A SATELLITE
MODEL'S
ERRORS
USING
DATA
gravity
anomalies
harmonics
of
a
(on
the
gravity
geoid)
field
as
can
be
follows
calculated (Heiskanen
1967):
_ X m=O
a Y(_-I)B_(_)
_m(Sin_)
[_mCOS
m_+S_mSin
mA]
(10.1)
where
ae
is
the
mean
value
is
the
earth's
is
the
radius
of
equatorial
semi-major
to
the
gravity.
axis.
surface
of
the
best
fitting
earth
ellipsoid.
_m(Sin_)
is for
the
normalized
geocentric
is the
B_
fully
is
latitude
geographic
Pellinen'
Katsambalos, which
Legendre
function
4-
longitude,
s
smoothing
1979)
Ag s is
anomaly
associated
corresponding
averaged
over.
factor to
(described the
(Note:
block B£=I
size for
in over point
values)
and
C£m, S£m
are
the
normalized
with
the
terms
only)
spherical
reference subtracted.
254
ellipsoid
harmonics zonal
of
the
potential
field (even
If < >
global
E
average
value
statistically
i
E
Error
S
in
expected
Ag s from
value
and
coefficient
commission
errors
then &max E
_- OModel 2 (Ag s)
_
_ 9,=2
_ _2(_-I)2 m--O
2 ) B_
o2(_m,_m
(10.2)
where
E
is
the
expected
error
in
the
gravity
anomalies
based
S
upon
the
estimated
a2(_m,_m
) is the
Section have
been
covariance.
we
have
the
(both
best
estimated
directly
without
essential
by
is
quantity
and
expected
value
of
this
If
one
also
can
also
the
model.
surface
data
the
harmonic
one field
and
by
the
and
measures.
Kaula
commission)
in
a
between
the
is the expected reliable
the
the
information omission
rms difference
255
and
GEM-TI,
and
within
showed
its
which how
field
can
surface
and
global on
data.
errors
(truncation) of
of
commission the
the
two
be
The
variance
measured
the
gravity
information. global
a
undergone
altimetry
global
surface
which
has
harmonic
the
of computed
model
(1966)
independent
for
for
factors
errors
this
gravimetry
estimate
computing
scaling
coefficient
and
-_ C_m,S_m.
solution
calibration
covarlance
statistic
and
balanced
with
coefficients
coefficients
weights
difference
has
of
potential
refined
harmonics
computed
of
data
the
comparison
the
pair
potential
independent
omission
satellite
a well
present
available
forming
statistic
yield
to
as
of
the
realistic
wish
employed
errors
to
the
of the
describes
has
We
upon
data
8.2
which
in
variance
determined
solution
based
errors
the The
error in
the
error
in
data
sets.
In mean
terms
square
of
gravity
commission
anomalies
errors
are
as
developed
estimated
for
by
a
Kaula
given
(1966)
blocksize
as:
E < 2s> - < g2s> -
where
the calculated
Ag s = Agtrue
value
(for
Ag s
and
the measured
Ag
Ag
= Agtrue
omission
errors
2 E =
To
is
blocksize
size;
The
(I0.3)
harmonics
given
the
noise
further
uncertainties
in
the
same
block
Ag.
as:
[ - ]
-
coefficient from
fact or
surface k
in
the
equation:
EST
etrue(Ags)
= E
= k
eGEM-TI
where
k is
to
be
determined
from
this
256
analysis.
(10.5)
Unfortunately, degree
fields
(Eq.
10.3)
give
reliable
is
for
large
gravity
and
is unreliable
simple
these
for
with
global
altimetry.
This
calibration
anomalies
field.
Table
The
sets
used
from
two
is
10.1
the
But
high
the
results
error
models
of
the
error
appears
derived
high
this
low
to
especially
values
to
for
commission
technique
marine
sensitive
presents
for
degree
including most
omission
estimate
terms.
complete
comparisons
order
this
for
results
anomalies
in from
degree
calibration
and for
GEM-TI.
were
obtained
anomalies (or
were
obtained
geophysically
anomalies in
data
the
from form
SEASAT
uncertainties
total
: _
mean
10.1 for
B_2
were
in
Altimeter
These
gravity
anomalies
commission
surface
They
means.
used.
estimated
_ _£ I_ _=2 36 m:O
(!98!).
areal
area
geopotential
Terrestrial
Rapp,
also
satellite
5 ° anomalies
gravimetric !°x! ° observed
derived
anomalies
computed
error
for
models
from
were
the
GEM-TI
is given
gravity used
original
based
on
the
terms
of
by:
(£-I )2o2(_£m ' S_m )_I I12 = 4.5
mgals
:
Bg
is
Table the
were
in Figure
aGE M T1(Ags)
where
from
5 ° equal
The
our
sources.
predicted)
of
I° values.
to calibrate
Pellinen's
10.1
additional
alone, when
shows
that
estimation
2.
have
GEM-L2
chosen
discrepancy (Lerch
our
is also
in the
This
presents
factor,
altimetry
We
smoothing
was et
k. O_m
al.,
results
This have
field more
not
of
been
it seems model
found and
to
the
that
calibrations based to we
uncertainties
this
estimate extent
it appears
257
5 ° anomalies.
estimated
conservative
1985b)
for
calibration,
utilized,
of our the
the
operator
to
on
in
surface
within
4%.
have
been
by
nearly
of
field
in occur
the in
gravity However,
conservative a factor
of
uncertainty.
calibration GEM-TI
due
of to
its
solution
for
constraint is
to
give
can
be
power
be
a
of
seen
gravity
to have
10.3
degrees much
blocks
comparison
predecessor.
purposes,
improvement
the
original
anomalies and to
a
done
in
(1981)
and
for
It
is
and
this
altimetric
gravity
alluded
to earlier,
new
As what
insensitive
to
are
the
with
data
longer
with
lower
the set
10.3.1 at
the
see are
future
gravity
wavelength
as
contains
GEM-TI
a new
here
of
as for
SEASAT
set
been
Note
and
of alti-
compared
GEM-TI
with
GEOS-3
altimetry
is
it
progress
too,
is
in
both
calibration
data
subject global and
incorporate activities.
sets
field.
both
good
unique
to
to
gravity
quite
to an
changes
gravity
well
model.
have
anomaly
GEM-L2
comparison.
converging
necessary
agrees
the
which
fields,
that
does
shown
SEASAT
show
different
3163,
to us
We
we
model
nearly
in this the
recent
undulations.
than
these
our
into
258
with
models
comparison
of to
these
and
while
analyses
making
GEM
PGS
(1985)
that
encouraging
answer.
This
I°xl ° estimates
Figure
performs
best
A
testing
absolute
the
available
recent
point
(Kaula's)
field
surface
GEM-TI
found
made
altimeter
We
of
10.3.1.
independent
improvement. modeling
the
which
power.
"satellite-only"
anomalies
most
a
Rapp's
In
data.
PGS-T2'
Figure
use
sea
for
performs
gravity
altimetry
information
version
recently
ocean
underscores
source
gravity
has
lower
more
power
here
SEASAT
GEM-TI
expected, over
Rapp
based was
as
with
values.
altimetric
a
used
in
we
GEM-TI
altimeter is
and
Richard
as
the
its
also,
GEM-TI
Note
lowered
errors
from
anomaly
is
data,
metrically
gravity
utilized
Again
these
Seasat
altimeter the
from
the
which
of
obtained
with
which
GEM-lOB
agreement
anomalies
relatively
errors.
directly.
truncation.
better
model
more
residual
of
the
have
field
the
underestimated
have
estimator a
where
terms
and
error
favors
(10.3)
shows
5°x5 °
computed
which
Equation
These
solution
The
answer
gravity
formed
the
collocation
models
oceanic
terms.
unrealistic.
in
is seen
have
the
biased
Figure GEM
degree
within
known
will
higher
are
Figure
some10.4
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o
r_ •_ 0'_ n+. 0
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