The ability of satellite-borne radar altimeter data to measure the global ocean surface with high precision and dense spatial coverage provides a unique tool for the mapping of the Earth's gravity field and its geoid. The altimeter crossover measurements, created by differencing direct altimeter measurements at the subsatellite points where the orbit ground tracks intersect, have the distinct advantage of eliminating geoid error and other nontemporal or long period oceanographic features. In the 1990's, the joint U.S./French TOPEX/POSEIDON mission and the European Space Agency's ERS-1 mission will carry radar altimeter instruments capable of global ocean mapping with high precision. This investigation aims at the development and application of dynamically consistent direct altimeter and altimeter crossover measurement models to the simultaneous mapping of the Earth's gravity field and its geoid, the ocean tides and the quasi-stationary component of the dynamic sea surface topography. Altimeter data collected by SEASAT, GEOS-3, and GEOSAT are used for the investigation

Schutz, B. E.

Shum, C. K.

Tapley, B. D.

English

NASA Technical Reports Server

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8 
ALTIMETER MEASUREMENTS FOR THE DETERMINATION 
OF THE EARTH'S GRAVITY FIELD 
NASA GRANT NAG5-746 
j Annual Research Technical Report 
for the period 
March 18,1986 -March 14,1987 
Submitted to 
Ms. G. Wiseman 
NASA Goddard Space Flight Center 
Mail Code 200 
Greenbelt, MD 20771 
by the 
Center for Space Research 
The University of Texas at Austin 
Austin, Texas 78712 
(512) 471-1356 
Principal Investigator 
Dr. B. D. Tapley 
Co-Principal Investigators: 
Dr. B. E. Schurz 
Dr. C. K. Shun 
April 29,1987 
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ALTIMETER MEASUREMENTS FOR THE DETERMINATION 
OF THE EARTH’S GRAVITY FIELD 
NASA Grant NAG5-746 
INTRODUCTION 
The University of Texas Center for Space Research (UTKSR) research efforts under NASA 
Grant NAG5-746 during the time period from March 18, 1986, to March 14, 1987, have 
concentrated on the following areas: 
Refining altimeter and altimeter cmssover measurement models for precise orbit determination 
and for the solution of the earth’s gravity field, 
Performing experiments using altimeter data for the improvement of precise satellite 
ephemerides and the determination of the earth’s gravity field, 
Analyzing an optimal relative data weighting algorithm to combine various data types in the 
solution of the gravity field. 
0 Examining observability of geopotential coefficients using altimeter crossover measurements 
from two satellites with differcnt inclinations and altitudes. 
Copies of abstracts for some of the publications and conference presentations related to the research 
are attached in the appendices. 
ALTIMETER hlODEL REFINEMENT 
A model of the quasi-stationary sea surface topography (SST) was implemented in the University 
of Tcxas Orbit Processor (UTOPIA). The model uses a surface spherical harmonics representation 
of the Lcvitus modcl of the SST, and it includes the capability to estimate the SST model in terms of 
spherical harmonics cocfficicnts using altimeter data. The model of SST is part of the dynamically 
consistcnt altimctcr mcasurcmcnt modcl which includcs the modeling of the dynamical effect of the 
1 
global ocean surface on the satellite orbit. 
A generalized theory for efficient computation of altimeter crossover measurements for both 
single and dual satellites has been developed. Software refinement is underway to implement dual- 
satellite crossover computation capability. Future work also includes the modeling of dual-satellite 
crossover measurement types in UTOPIA. 
ALTIMETER CROSSOVER EXPERIMENTS 
Orbit determination experiments using altimeter crossover observations were performed. 
Altimeter crossover and ground-based tracking data were used to determine SEASAT 12-day arc 
with orbit epoch beginning at July 7, 1978. When resonant coefficients and 10x10 gravity field 
coefficients were solved using crossoves and tracking data, the SEASAT orbit was computed with a 
better accuracy. A simulation experiment was then performed to confirm the above results. In 
conclusion, the use of crossover measurements significantly improves the radial component of the 
estimated orbit. The improvements in the geopotential gained from processing crossover 
measurements will reduce the geographically correlated orbit emr .  
Observability of zonal geopotential coefficients using dual-satellite crossover measurements was 
examined using linear theory. A variation of inclination and altitude of altimetric satellites was used 
to compute observability indcxcs which reflect the strength of zonal variation using altimeter 
crossover data from two different satellites. Preliminary results indicated that zonals are observable 
using dual-satellite crossovers. 
ALTIMETER EXPERIMENT 
Techniques used to gcnerate the altimeter information equations for the TOPEX gravity field 
solution have been devcloped, and a numerical experiment to determine the gravity field using 
SEASAT altimctcr data was performed. Information equations for the SEASAT altimeter data were 
2 
created for two arcs (12 and 18 days) during the 17-day repeat orbit and one arc (24 days) during the 
3-day repeat orbit. The altimeter data were edited at 10 m and then time sampled at the rate of once 
every 20 seconds. The resulting information equations included the geopotential coefficient 
complete to degree and order 42 and the quasi-stationary sea surface topography (SST) complete to 
degree and order 10. The reference orbits were computed using PTGF1 with altimeter residuals 
greater than 10 meters completely edited. The numerical simulation used 12 days of laser, S-band, 
crossover and altimeter data computed at the same times as the real data for SEASAT 
7/7/78-7/18/78. Only when the altimeter data were included with the other observations was it 
possible to solve for a 20x20 geopotential (no model error). No accurate SST solutions were 
obtained, but this was probably due to a lack of other satellite data to provide separation for the SST 
and the geopotential. Future work will concentrate on filtering the altimetcr data and creating 
altimeter normal points. 
APPLICATION OF ALTIMETER DATA 
A new technique was developed to interpolate unevenly spaced GEOS-3 and SEASAT altimeter 
data to generate the mean sea surface, or marine geoid, with high resolution. Green functions of the 
biharmonic operator, in any number of dimensions, are used for minimum curvature interpolation of 
irregularly spaced data ponts. The interpolating curve (or surface) is a linear combination of Green 
functions ccntcrcd at each data pont. In one (or two) dimensions, this technique is equivalcnt to 
cubic spline (or bicubic spline) interpolation, while in three dimensions, it corresponds to 
multiquadratic interpolation. Although this new tcchnique is relatively slow, it is more flcxible than 
the spline method sincc both slopes and values can be used to find a surface. Moreover, noisy data 
can be fit in a least-squares sense by reducing the number of model parameters. These properties are 
well suited for interpolating irregularly spaced satellite altimeter profiles. The long wavelength 
radial orbit c r m  is supprcsscd by differentiating each profde. The shorter wavelength noise is 
3 
reduced by the least-squares fit to nearby profiles. Using this technique with 0.5 million GEOS-3 
and SEASAT data points, it was found that the marine geoid of the mas covering the Gulf of 
Mexico and the Caribbean are highly correlated with the sea floor topography. This suggests that 
similar applications in more remote areas may reveal new features of the sea floor. Future plans 
include application of altimeter data to produce a high resolution global ocean surface map. 
GEOSAT altimeter data, if available, will be included. 
RELATIVE DATA WEIGHTING 
An optimal relative data weighting algorithm was developed to combine various data types for 
the solution of the earth’s gravity field. The algorithm minimizes the e m r  introduced by the 
observation noise and scales the u priori variances for an optimal solution. Evaluations and 
experiments are underway to examine the ability of the data weighting algorithm for the optimal 
solution of the earth’s gravity field. 
APPLICATION OF ALTIMETER DATA TO THE SOLUTION OF THE GRAVITY FIELD 
SEASAT altimeter data were processed and included in the solution for the TOPEX gravity field, 
PTGFZA. The experimental solution of the PTGF2A gravity field is part of the continuing 
collaborative effort between UT/CSR and NASNGSFC to develop the TOPEX gravity model. The 
purpose, in part, is to assess the roles of both the altimeter crossover and the direct altimeter data in 
their contribution to the solution of thc earth’s gravity field. Preliminary gnvity field information 
cquations wen= generated for both the SEASAT altimeter crossover and direct altimcter data. The 
altimeter information equations were included in the gravity field solution. Assessment of their 
respective contribution to the gravity field is underway. The technique for the generation of normal 
points for direct altimeter data is in development. 
4 
PUBLIC ATIONS/CONFERENCES 
The following is a list of publications and conference presentations pertinent to the research 
activities under NASA Grant NAG5-746: 
Altimeter Crossover Experiments, R. S .  Nerem, C. K. Shum, and J. C. Ries. Presented at the 
Sixth UT/GSFC TOPEX Gravity Model Development Meeting hcld at The University of Texas 
at Austin, April 8-10.1986. 
Altimeter Methods for Gravity Field Determination, R. S .  Nerem, C. K. Shum, and J. C. Ries. 
Presented at the Seventh UT/GSFC TOPEX Gravity Model Development Team Meeting held at 
the Goddard Space Flight Center, November 5-6, 1986. 
Altimeter Crossover Methods for Precision Orbit Determination, C. K. Shum, B. E. Schutz, B. 
D. Tapley, B. H. Zhang, and R. S. Nerem. Presented at the Fall Meeting of the American 
Geophysical Union, San Francisco, California, December 1986. A copy of the abstract is 
included in Appendix A. 
Biharmonic Spline Interpolation of GEOS-3 and SEASAT Altimeter Data, D. T. Sandwell. 
Submitted to EOS for publication. Presented at the Fall Meeting of the American Geophysical 
Union, San Francisco, California, December 1986. A copy of the abstract is included in 
Appendix B. 
Altimeter Crossover Methods for Precision Orbit Determination, C. K. Shurn, B. H. Zhang, B. 
D. Taplcy, and B. E. Schutz. Abstract submitted to the AIAA/AAS Astrodynamics Conference 
to bc hcld August 10-12, 1987, at Kalispell, Montana. A copy of the abstract is included in 
Appcndix C. 
Altimeter Methods for the Determination of the Earth's Gravity Field, C. K. Shum, B. D. Tapley, 
B. E. Schutz, B. H. Zhang, and R. S. Nerem. Abstract submitted to the IUGGDAG 19th General 
Assembly Symposium to be held August 9-22, 1987, in Vancover, Canada. A copy of the 
bstract is included in Appendix D. 
5 
APPENDIX A 
ALTIMETER CROSSOVER METHODS FOR PRECISON 
ORBIT DETERMINATION 
C. K. Sum. B. E. Schuc~ B. D. Tqley, B. H Z m g ,  and R. S. Nerem 
(Center for Space R e s e a  The University of Texis at Austin, Aut& 
Texas 78712) 
The ability of satellite-borne radar altimeter data to masure the global 
ocean surface wirh sub-centimeter xzuracy provides a unique dam set 
to aid precision orbit deurminaion as well Y to map the u r r h ' s  gravity 
Beld and iu geoid. Although satellie 3itimCtry has a distinct ;Idvantage 
in iu global dismbuuon of dat;r considerauon of possible error ulucCu 
when altimeter dau are directly used for orbit decennination and 
geophysical mapping suggaa s e v d  disadvantages. In pyriCulu, long 
wavelength oceanographic features jnd nontunpod ocean ropognphy 
can be absorbcd into the orbit when altimctcr dm ye dinctly ufcd 
Alumctcr cfossovcr &ea, which eliminate the dependence on the 
nontemporal ocean topography, pmvide a vduable dau set in io global 
coverage. In this investigation, techniques to employ lltimcur 
crossover data to perform precision orbit dctaminvion and to c e c o v ~  
geophysical parynncn have kac developed. New tffhniqua of uskg 
dud or multiple satellite cr~ssover dam for precisian orbit defemination 
and geodetic pyyneur mapping have been investigated. 
Aclvrowledgmmr: This research WY supported by NASA under 
ConMcr No. NAG S-746. 
1. 
2 
3. 
4. 
S. 
6. 
7. 
8. 
9. 
1986 Fall Meering 
Dr.cK.shum 
Center for Space Research 
The University of Texas at Austin 
A w t h T X  78712 
(S12) 471-5575 
G 
0 
0% 
Us. Susie Y. Thorn 
Cenrcr for Space Resevch 
The University of Texas at Austin 
Aman.= 78712 
C 
.. 
APPENDIX B 
- .  
BIHARMONIC SPLINE INTERPOLATION OF GEOS-3 
AND SEASAT ALTIIvlETER DATA 
ABSTRACT 
Green functions of the biharmonic operator, in any number of dimensions, are used for minimum 
curvature interpolation of irregularly spaced data ponts. The interpolating curve (or surface) is a 
linear combination of Green functions centered at each data pont. In one (or two) dimensions, this 
technique is equivalent to cubic spline (or bicubic spline) interpolation, while in three dimensions, it 
corresponds to multiquadratic interpolation. Although this new technique is relatively slow. it is 
more flexible than the spline method since both slopes and values can e used to find a surface. 
Moreover, noisy data can be fit in a least-squares sense by reducing the number of model parameters. 
These properties are well suited for interpolating irregularly spaced satellite altimeter profiles. The 
long wavelength radial orbit error is suppressed by differentiating each profile. The shorter 
wavelength noise is reduced by the least-squares fit to nearby profiles. Using this technique with 0.5 
million GEOS-3 and SEASAT data points, it was found that the marine geoid of the Caribbean area 
is highly correlated with the sea floor topography. This suggests that similar applications in more 
remote areas may reveal new features of the sea floor. 
APPENDIX C 
.. 
Altimeter Crossover Methods for 
Precision Orbit Determination 
C. K. Shum, B. H. zhang, B. D. Tapley 
and B. E. Schutz 
Center for Space Research 
The University of Texas at Austin 
Austin. Texas 78712 
The ability of satellite-borne radar altimeter data to measure the global ocean surface 
with sub-centimeter accuracy provides a unique data set to aid precision orbit determination 
as well as to map the earth’s gravity field and its geoid. Although satellite altimetry has a 
distinct advantage in its global distribution of data, consideration of possible error sources 
when altimeter data are used directly for orbit determination and geophysical mapping 
suggests several disadvantages. In particular, long wavelength oceanographic features and 
nontemporal ocean topography can be absorbed into orbit and geodetic parameters when 
altimeter data are used directly. The nontemporal ocean topography, mostly due to error in 
the marine geoid. currently has uncertainty of seved meten and is significant when 
decimeter radial orbit accuracy of certain satellite orbits. such as that of TOPEX. is desired. 
A :schnique which eliminates the dtirneter’s dependence on the nontemporal ocean 
topography is the use of altimeter measurements at the points where the orbit ground tracks 
intersect. These points are referred to as “crossover points.’’ X crossover measurement is 
defined as the difference in the altimeter measurements at the points where the satellite 
gound tracks cross. Since the constant part of each altimeter measurement will be due to 
the marine Seoid and permanent sea surface topography, the crossover or differenced 
altimeter measurements will be a function of the orbit m o r  and the time-varying ocean 
surface topography. The orbit enor will be the major conmbutor to the crossover 
measurement. The orbit error is primarily a long wavelength feature with once!revolution. 
(40.000 km) and twice:revoluuon (20.000 km) components dominating. Although the 
nontemporal portion of the ocem topography can be eliminated at the crossover point. the 
remaming temporal changes. such us Ocean tide. unrnodeled orbit emr.  shon wavelength 
phenomena as well as altimeter time tag error, can still be a b e d  into the ndial orbit error 
on 1 slobai basis. With the exception of the seognphically correlated emor due to 
inaccuracy in [he smh’s  gravity tield. global computation and malysis of singie-satellite 
i:os~OV~r esiduals can provide valuable information about ndial orbit m o r  sources and the 
reduction ot  interesting geophysical panmeters. A new technique has been developed to 
use duril-.;atellite crossover measurements. or crossovers from two satellites carrying 
dtimeter instruments. for precision orbit determination. The dual-satellite crossover 
technique is found to be capable of observing variauon of the emh’s zonal harmonics which 
constitutes part of the Seognphically correlated gravity error. 
The TOPEX/POSEIDON mission. which will be conducted jointly by the National 
Aeronautics and Space Administration (NASA) and France’s Centre Nationale d’Etudes 
Spatiale (CNES), will carry precise radar altimeter instruments in an attempt to measure 
oceanic topography including the mean and variable surface gtostropic currents and tides of 
the world's oceans, and to lay the foundation for a continuing program to provide long-term 
observations of the ocean's circulation and its variability. To meet the objectives of the 
TOPEXPOSEIDON mission, the satellite altitude relative to the center of mass of the earth 
must be determined to an accuracy of 14 centimctcrs or better. In the same time span, the 
European Space Agency (ESA) will launch another altimetric satellite, ERS-1. The 
application of the dual-satellite crossover technique combining ERS-1 and 
TOPEXPOSEIDON altimeter crossover measurements will improve the ERS- 1 orbit. 
Since ERS-1 will cover most of the Ocean surface with its 98" inclination, the use of dual- 
satellite crossover measurements between ERS- 1 and TOPEXPOSEIDON will provide a 
significant means for the mapping of geophysical and oceanographic phenomena of interest. 
In this investigation, single- and dual-satellite altimeter crossover techniques are 
developed for precision orbit determination and the reduction of geophysical parameters. 
Observability of zonal harmonics using dual-satellite crossover measurement was studied. 
Numerical experiments were carried out to examine the role of both single- and dual- 
satellite crossover techniques to the conmbution of precision orbit determination and 
reduction of geophysical parameters. 
Acknowled,qmenr. Funding for this research is provided by NASA, Contract No. 
?JAGS-736. 
APPENDIX D 
ALTIMETER MRHODS FOR THE D W A T I O N  OF THE EARTH'S GRAVITY 
FIELD 
C. K. Shum, The Universiey of Texas at Austin, Center for Space Research, Austin, Texas 
78712 and B. D. Tapley, B. E. Schue, B. H. Zhang and R S. Nemn (same) 
The ability of satellite-borne radar altimeter data to measure the global ocean surface with high 
precision d dense spatid coverage provides a unique tool for the mapping of the earth's 
gnvity field and its geoid. The altimeter crossover measurements, mated by differencing 
direct altimeter measurements at the subsatellite points where the orbit ground tracks intersect 
have the distinct advantage of eliminaang geoid error and other nontemporal or long period 
oceanographic features. In the 1990's. the joint U.SJFench TOPEXPOSEIDON mission and 
the European Space Agency's EM-I mission will cvry radar ala~neter instruments capable of 
global ocean mapping with high precision. This investigation aims at the development and 
application of dynarmcally consiscent direct darneccr and altimeter c m w v e t  measurement 
models to the simultaneous mapping of the earth's gravity field and its geoid, the ocean tides 
and the quasi-staaonaq component of the dynamic sea surface topography. Altimeter data 
collected by SEASAT. GEOS-3 and GEOSAT arc used for the investigation. 
Acknawlcdqmcnr: This msearch is supponcd by the NASA Goddard Space Flight Center, 
Contract No. NAGS-746. 
GSc. Analysis of Satellite .4ltimeay (on1 pmsentaaon) 
IUGG 19th Genenl Assembly, Vancouver. Canada. August 9-22, 1987. 
Convenor: R. Andrrfe. Senior System Engineer, G e n e d  EIectnc Co.. Space Systems Division. 
Box 355543A30. Phladclphia. PA 19101 
IAG President: 51. touis, Is0 Rue de Crcuellc. 75700 Pans. France 
Author's address: (same) 
Telephone: ( 5  12) 47 1-5573 


Altimeter measurements for the determination of the Earth's gravity field

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