Several tests were designed to determine the correct error variances for the GEM-T1 gravitational solution which was derived exclusively from satellite tracking data. The basic method employs both wholly independent and dependent subset data solutions and produces a full field coefficient by coefficient estimate of the model uncertainties. The GEM-T1 errors were further analyzed using a method based upon eigenvalue-eigenvector analysis which calibrates the entire covariance matrix. Dependent satellite and independent altimetric and surface gravity data sets, as well as independent satellite deep resonance information, confirm essentially the same error assessment

Chinn, D. S.

Klosko, S. M.

Lerch, F. J.

Marsh, J. G.

Patel, G. B.

Pavlis, E. C.

Wagner, C. A.

English

Several tests were designed to determine the correct error variances for the Goddard Earth Model (GEM)-T1 gravitational solution which was derived exclusively from satellite tracking data. The basic method employs both wholly independent and dependent subset data solutions and produces a full field coefficient estimate of the model uncertainties. The GEM-T1 errors were further analyzed using a method based upon eigenvalue-eigenvector analysis which calibrates the entire covariance matrix. Dependent satellite and independent altimetric and surface gravity data sets, as well as independent satellite deep resonance information, confirm essentially the same error assessment. These calibrations (utilizing each of the major data subsets within the solution) yield very stable calibration factors which vary by approximately 10 percent over the range of tests employed. Measurements of gravity anomalies obtained from altimetry were also used directly as observations to show that GEM-T1 is calibrated. The mathematical representation of the covariance error in the presence of unmodeled systematic error effects in the data is analyzed and an optimum weighting technique is developed for these conditions. This technique yields an internal self-calibration of the error model, a process which GEM-T1 is shown to approximate

NASA Technical Reports Server

An improved error assessment for the GEM-T1 gravitational model

AN IMPROVED ERROR ASSESSMENT FOR THE GEM-TI_
GRAVITATIONAL MODEL
F.J. Lerch and J.G. Marsh
NASA/GSFC, Greenbelt, MD
S.M. Klosko and E.C. Pavlis
EG&G WASC, Lanham, MD
G.B. Patel and D.S. Chinn
STX, Lanham, MD
C.A. Wagner
NGS, Rockville, MD
N90-20518
ABSTRACT
Several tests have been designed to determine the correct error
variances for the GEM-TI gravitational solution which was derived
exclusively from satellite tracking data. The basic method employs both
wholly independent and dependent subset data solutions and produces a full
field coefficient by coefficient estimate of the model uncertainties. The
GEM-TI errors have been further analyzed using a method based upon
eigenvalue-eigenvector analysis which calibrates the entire covariance
matrix. Dependent satellite and independent alt_metric and surface gravity
data sets, as well as independent satellite deep resonance information,
confirm essentially the same error assessment.
OVERVIEW
The principal calibration technique (Lerch, 1985) is based upon the
comparison of solutions (independent or dependent) which analyzes the
consistency of the coefficient differences and the error estimates between
the solutions as described in Table I.
Calibrations utilizing each of the major data subsets within the
solution yield very stable calibration factors which vary by approximately
10% over the range of tests employed. Measurements of gravity anomalies
obtained from altimetry were also used directly as observations to show
that GEM-TI is calibrated. Based upon these calibrated error estimates,
GEM-TI is a significantly improved solution which to degree and order 8 is
twice as accurate as earlier satellite derived models. By being complete
to degree and order 36, GEM-TI is much larger than earlier gravitational
solutions calculated from direct satellite tracking and has significantly
reduced aliasing effects that were present in previous models. The
mathematical representation of the covariance error in the presence of
unmodeled systematic error effects in the data is analyzed and an optimum
weighting technique is developed for these conditions. This technique
yields an internal self-calibration of the error model, a process which
GEM-TI is shown to approximate. This geopotential field with calibrated
error estimates, predicts 25 cm (Table 2) for the radial RMS uncertainty of
the TOPEX orbit. The TOPEX Mission has a requirement for 10 cm radial
orbital modeling which is needed to support the oceanographic applications
of a high quality spaceborne altimeter.
RESULTS
Taking full advantage of the "super-computing" environment available
at NASA/Goddard Space Flight Center, many solutions have been compared
providing a completeness of field testing heretofore impossible within
earlier computing environments. The results show a model remarkably
consistent in stability for the calibration of its errors. With the
exception of a few known and understood high order resonance terms (and the
limitations of the high altitude Lageos satellite providing data suitable
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https://ntrs.nasa.gov/search.jsp?R=19900011202 2020-03-19T22:42:44+00:00Z
for the calibration of a full 36x36 field), the calibrations show a
stability in error assessment at the 10% level for each of the major data
subsets employed in this evaluation. The published coefficient
uncertainties for GEM-TI and its error covariance matrix are herein found
to be reasonably well calibrated and reliable. For example, the average
calibration factor (k) for GEM-TI using nine major sets of data in Table 3
(excluding the anomalous result for LAGEOS data) gave k=0.99 (± .08) for
the coefficient calibration and k=0.95 (± .09) for the eigenvector
calibration. This is a gratifying result, particularly, since formal least
squares error formulae based on random variables were employed with
compensating downwe_ghting factors to account for more general formulae
involving error sources with unknown systematic effects. The mathematical
validity of the error estimation techique for the gravity model was studied
extensively and an optimal weighting technique w_th internal self-
calibration of the error model was developed.
The authors would like to thank C.K. Shum, John Ries and Byron Tapley
of the Center of Space Research, University of Texas, for their thoughtful
reading and constructive comments in the preparation of this report.
George Rosborough of the University of Colorado likewise made valuable
comments. The authors would like to also thank Dick Rapp and Nick Pavlis
of Ohio State University for their assistance in processing and helping us
successfully utilize the surface gravimetry.
REFERENCES
Lerch, F.J., Error Spectrum of Goddard Satellite Models for the Gravity
Field, Geodynamics Branch Annual Report-1984, NASA TM86223, August
1985.
Lerch, F.J., Marsh, J.G., Klosko, S.M., Pavlis, E.C., Patel, G.B., Chinn,
D.S., and Wagner, C.A., An Improved Error Assessment for the GEM-TI
Gravitational Model, NASA TM.... , in print, November 1988.
Lerch, F.J., Klosko, S.M., Wagner, C.A. and Patel, G.B., On the Accuracy of
Recent Goddard Gravity Models, J. Geophys. Res., 90, (B11), pp. 9312-
9334, 1985.
Marsh, J.G., eta]., A New Gravitational Model for the Earth from Sate]litu
Tracking Data: GEM-T], J. Geoph_s. Res. 93, B6, 6169-6215, 1988.
2O
TABI,E ]
FORMULAE FOR ERROR CALIBRATION
TWO FIELDS Y 8, P
F " Cl.,,,, .St.,., o "s ( coeff, errors )
I
RMS_ ( AF) = Y.
m=o 21 + !
I
0 2 0-2 "_
[ I (Ct,,,) + (SI._ ) ]01 = y. 'm=o 214 1
el = E(RMS_)2
= oI + _ when F is independent of
= o_ - _ when datain F C
CALIBRATION FACTORS
RMSI
e I
for degree 1
RM&.,_
el.rn
for individual coeff, pair
21
TABLE 2
Radial Orbital Errors (RMS)
for Three Day Arc Lengths
Using Calibrated Covariance Matrices
Geopotential Radial
Model RMS Error (cm)
GEM-L2 6.5
GEM-T1 25
GEM-TI +
Surface Gravimetry + 17
Altimetry
TABLE 3
I.
II FROM GEM-T ! FIELD ASSESSMENTS
1'
COEFFICIENT
CALIBRATION
• (GEM-T1) vs (GEM-TI minus DATA SUBSET)
4-LASERS (GEOS 1.2.3. BE-C)
STARLETTE LASER
05CAR * SEASAT DOPPLER
OPTICAL (I I SATS)
LAGEGS LASER
1.06
1.10
I 09
004
1.45
I GEM-T1 vs GEM-TI + SURFACE GRAVITY 095
• GEM-TI vs GEM-T! • SURFACE GRAVITY
SEASAT ALTIMETRY 094
• GEM-TI vs SURFACE GRAVITY ,* SEASAT
ALTIM 0.99
• GEM-TI minus LAGEOS vs. LAGEOS *
SURFACE GRAVITY • SEASAT ALTIMETRY 095
• GEM-T I vs. GEM-T I • Lumped Resonance
Data
I O0
RMS WEIGHTED
PROJECTED
EIGENVECTOR
CALIBRATION
FACTOR ONTO
GEM-T__L
094
0 99
107
089
I 59
092
OB9
090
OBB
1 06
• GEM-TI with I0 times the Data Wetght vl;
GEM-TI minus 4-LASERS with
I0 times the Date Weight
2.75 245
22


An improved error assessment for the GEM-T1 gravitational model

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