in review J. Geod 2007
figures to this manuscript are in extra file
Improvement in the radial accuracy of satellite orbits from
long term altimetry: information from covariance projections
\
J. Klokočník 1, J. Kostelecký 2,3, C. A. Wagner 4
1 CEDR - Astronomical Institute, Academy of Sciences of the Czech Republic,
CZ-251 65 Ondrejov Observatory, Czech Republic, jklokocn@asu.cas.cz
2 CEDR - Research Institute for Geodesy, Topography and Cartography,
CZ-250 66 Zdiby 98, Czech Republic, kost@fsv.cvut.cz
3 Department of Advanced Geodesy, Czech Technical University,
CZ-166 29 Praha 6, Thakurova 7, Czech Republic, kost@fsv.cvut.cz
4 Laboratory for Satellite Altimetry, NOAA
1335 East_West Highway, Silver Spring, MD, USA, carl.wagner@noaa.gov
Rosborough’s theory (1986), ammended by our newer works, describes Earth static gravity induced radial orbit
error as a function of latitude and longitude. Using this theory we show on many examples the improvement in
the accuracy of Earth gravity models, from the early 1980s (order of tens of meters), to the present (order of
centimeters and less). The early models, with higher correlations between potential coefficients, show strong
variations of the error in longitude as well as latitude, compared to the more recent fields. Currently the static
gravity errors in the best of the Earth models are believed to be below the systematic environmental errors in the
long-term altimetry.
Keywords
orbits of Earth artificial satellites, gravity field of the Earth, radial orbit error,
Rosborough theory, single- and dual-satellite crossovers, evolution of radial error
Introduction, motivation of this study
Since the first satellite altimeter (on Skylab, 1973), the Earth's sea level has been monitored from space by
subtracting the measured altimeter height there from the satellite's height above the Earth. Precise and reliable
products of satellite (radar) altimetry (with measurements in the nadir-radial direction) depend (among others) on
the accuracy with which we know the radial orbit component of the satellites equipped by the altimeters. If one
wants to study a signal with size 10 cm, say detailed marine geoid or sea surface topography and its time changes
or tides, one cannot have the radial orbit error larger than ±10cm, but smaller. The radial orbit error depends on
gravitational and non-gravitational effects. Until very recently the major error source in the referred sea level
measurement has been the error in the satelllite's distance from the Earth (it's orbit radius) due to uncertainty in
the Earth's static geopotential (e.g., its harmonic coefficients of s/c Earth gravity field models); it was several
meters at the begining of satellite altimetry; the precison of altimetry measurements themselves was around one
meter.
Because the geopotential-caused orbit error depends strongly on the satellite's height as well as inclination [high
orbits such as Topex/Poseidon (1992+) and Jason (1996+) having smaller errors than lower orbits such as Geosat
(1985+), GFO (1996+), ERS(1992+) and Envisat (2000+)], this error source has varied widely over many
altimeter missions ranging from tens of meters for field models in the missions of the 1970s and 1980s to
perhaps less than one 1 cm today with the ultra precise geopotential models from CHAMP (2000+) and
GRACE(2002+). The improvement since begining of satellite altimetry till now is enormous and fascinating.
Prior to the mid 1980s the only theory available for computing the geopotential-orbit error was as a time series
(eg., Wagner, 1983), inconvenient for mapping its structure on the world's ocean. Rosborough (1986) and
Engelis (1987) made a major advance by transforming this time series to stationary geographic coordinates,
valid for all the near circular orbits of the past and current altimeter missions.
In this paper we show the evolution of the static gravity field error since nearly the beginning of era of satellite
altimetry till the present time. We focus on covariance projections from the gravity models to the radial errors.
First we recall Rosborough's theory (see section Theory), then we review conditions for objective evaluation of
the radial error (see Validation), then we shorly review historical models (see Historical models - evolution),
and finally we show the radial (crossover) error for the recent models (Recent models).
This “review” can be considered as a first step to a more general description of evolution of the radial accuracy
of satellite altimetry orbits. Second step (Wagner et al, 2007, in preparation) will deal with up-to-date actual long
term altimetry observations (e.g., from ENVISAT) computed with the aid of different gravity models, from
historical to recent.
Theory
Theory used here follows the Rosborough’s work (1986) and our older works (e.g., Klokočník et al 1995, 2000a,
Klokočník and Wagner 1999) and concerns the radial orbit component only, because the altimetry measurements
are given in this direction. More general approach, including the along- and cross-track components is e.g. in
Rosborough and Taply (1987). Radial displacements are only weakly measured in the normal geodetic tracking
instruments (like SLR), thus satellite altimetry is very suitable for testing gravity field models (e.g., Klokočník
et al 1998, 2000b, 2002, Wagner and Klokočník 1994, Wagner et al 1997).
The radial orbit perturbation Δr due solely to the static Earth gravity field, represented by geopotential harmonic
coefficients Cl,m, Sl,m (l degree, m order, fully normalized values), is given by the following equation
(Rosborough, 1986), valid for effectively circular orbit
l max
l
l
~ C C cos m  S sin m  
r   Dlmp
lmp
lm
lm
(1)
l  2 m0 p 0
l max
l
l
~ S C cos m  S sin m 
  Dlmp
lmp
lm
lm
l 2 m0 p 0
where
Dlmp depends on the inclination and semimajor axis of the satellite orbit,
~ C ( S ) are functions of inclination and latitude,

lmp
+/-
refer to the ascending/descending tracks.
C (S )
It is useful to define the “influence functions” Qlm
:
l
(2)
~ C(S ) .
C(S )
Qlm
  Dlmp
lmp
p 0
Rosborough (1986) introduced the mean regional part of Δr:
l max
(3)
l
C
Clm cos m  Slm sin m 
   Qlm
l  2 m0
called by him the geographically correlated part of Δr and the variable portion around
the geographically correlated mean
l max
(4)
l
S
Clm cos m  Slm sin m  ,
   Qlm
l  2 m 0
so then
r     .
The error of (1) can be written as follows (e.g., Klokočník et al., 1994)
l max
(5)
 2x  
l max
l1
l2
  Q
l1  2 l 2  2 m1  0 m 2  0
where
T
1
LT1 k 12 L 2 Q 2
 QlCi mi .1  d 
,
Q  S
 Ql m .1  d  
 i i

x
i
 1  d cos mi  t 1  d sin mi  
 ,
Lxi  




1

d

sin
m


t
1

d

cos
m

i
i 



 COVAR Cl1m1 , Cl 2 m2
k12  
 COVAR Sl1m1 , Cl 2 m2
x  r ,  ,  ;


i = 1, 2;
d  0, d  0, t  1
for  : d  0, d  1,
for  : d  1, d  0, t  1 .
for r :
The single-satellite crossover difference is defined as
(6)
x1  r asc  r desc ,


COVAR Cl1m1 , Sl 2 m2
COVAR Sl1m1 , Sl 2 m2

 ,
thus
x            2 .
The error of thiss crossover difference, the single-satellite crossover error, SSC, reads
 x  2.  .
(7)
The dual-satellite crossover difference is the difference defined between two orbits over the same geographic
location (e.g. Shum et al., 1990). The dual-satellite crossovers (DSC) are important, since they permit a
comparison of ocean surfaces over decades and study of relative coordinate frame shifts (e.g., Klokočník et al.,
1995; Wagner et al., 1977; Klokočník et al., 2000).
There are just four types of DSCs:
X 12AD   1   2   1   2 
X 12DA   1   2   1   2 
(8)
X 12AA   1   2   1   2 
X 12DD   1   2   1   2 
where the lower indices „1“ and „2“ belong to the first and to the second orbit of the satellite’s pair (e.g.
X12AD  r1A  r2D ). We immediately see that the crossover altimetry quantities SSCs or DSCs and the
radial error have different sensitivity to the gravity field parameters.
x12AD , the dual-satellite crossover error, is explicitly
For example the error of

2
AD
x12
l max

l max
   COVARC
l1 m1

, Cl 2 m2 
dQ cos m    Q sin m   . dQ
 COVARS , S 
dQ sin m    Q cos m   . dQ
 COVARC , S 
dQ cos m    Q sin m   . dQ
 COVARC , S 
dQ sin m    Q cos m   . dQ
C
l1 m1
1
C
l 2 m2
cos m2   QlS2 m2 sin m2 
S
l1 m1
1
C
l 2 m2
sin m2   QlS2 m2 cos m2 
S
l1 m1
1
C
l 2 m2
sin m2   QlS2 m2 cos m2 
S
l1 m1
1
C
l 2 m2
cos m2   QlS2 m2 sin m2
l 2 m2
1
l1 m1
C
l1 m1
C
l1 m1

l 2 m2
1


l 2 m2
1
l1 m1

S
l1 m1
1
l1 m1
where
l2
l1  2 l 2  2 m1  0 m 2  0
C
l1 m1
(9)
l1
 
 Q
dQlC1m1  QlCi mi 1  QlCi mi 2 ,
S
l1m1

 
 QlSi mi 1  QlSi mi

2

.
Validation
These are the conditions for objective computation and evaluation of the radial static gravity error:
(1) all COVARiance terms, not only the VARiance terms, of the given variance-covariance matrix of the tested
gravity field model must be used. If the covariance terms are ignored, and only standard deviations of Clm, Slm
are used, then totaly false estimate of the radial error may be obtained.
Figs. 1 a, b with EGM96 SSC and Figs 2 a, b for the historical JGM 3 show illustrative examples. For more
examples see, e.g., Klokočník et al (1994). The radial error computed only with the VAR is much larger than the
“correct error” accounting for all COVAR terms in our case and it has a “zonal character” (dependence of the
error on longitude is depressed).
The error computed only with VAR is usually larger than with COVAR for orbits either used in a model or near
to one used because for these the correlations between geopotential terms are generally strong after they are used
to 'fit' the tracking data in these orbits. This is just the opposite for orbits far from those used in a given model,
for example orbits of low inclination where the COVAR estimates are generally larger than the VAR estimates.
With VAR-only and because sine and cosine geopotential coefficients tend to be assessed equally strongly over
all missions, the Rosborough transformation shows that the VAR-only estimate will always be very close to
zonal in character. The fact that the COVAR estimates of recent models are also zonal in character is due as
much to the better (lower) correlation of terms in these than in the older models. This is interesting because the
CHAMP and GRACE models are from one satellite (compared to the previous models). But the tracking in
them is nearly global, uniform (pole to pole), and dense, in stark contrast to the scattered tracking on mostly nonpolar orbits in the older models.
(2) a cut of the covariance matrix at certain maximum degree Lmax of the harmonics cannot be at “too low”
degree. The choice of an appropriate limit depends on several factors. First, the matrix (provided by the authors
of the gravity model) should be to as high degree and order as possible (today usually to l,m =90). Computations
with such huge matrices are complicated and very time consuming. So it is logically to ask how low Lmax is still
acceptable to avoid significant degradation of the estimated radial error.
Figs. 3 a,b,c,d,e show -- for the given Earth model (EGM96) and selected orbit (ERS type) – how the error
estimate degrades with decreasing Lmax. The difference between Lmax =70 and 90 is negligible (not shown here),
the differences to 50, 30, 20, 10 are higher and higher, so the cut below about Lmax = 50 is not acceptable (for
the lower orbits of ERS type). For other examples see (Klokočník et al, 1994).
It also depends very much on the orbit of the mission: obviously the higher orbits of TOPEX/Poseidon or
JASON have a great advantage. Figs. 4 a,b,c show the SSC error with EGM 96 to Lmax = 50 for orbits of
ERS1,2, ENVISAT, for Geosat/GFO and T/P&JASON.
(3) A cut of orbit perturbations. A delicate problem is where to cut the perturbations in the Langrange planetary
equation (Theory, eqs. 2-4, for more details see Rosborough, 1986). Long-periodic perturbations for the
particular orbit may destroy the whole “picture” of the radial error. Generally, the periods of the orbit
perturbations and arc lengths used for the orbit computations should be comparable. It would give us a cut at
about 4 days for all tested satellites excluding T/P and JASON where the limit would be 10 days. Empirically
chosen most appropriate orbit perturbation cut is 1.3 days for the ERS-type orbit and 4 days for Geosat-type orbit
(Klokočník et al, 2000). The effects of long-period perturbations are not shown here generally because they are
not seen directly in satellite altimetry since the orbits which produce the altimetric sea level heights are reinitialized every 1-4 days (10 days for T/P) from tracking data, which effectively absorbs these perturbations.
More about filtering of orbit perturbations see in (Klokočník et al, 1995).
Figs. 5 a,b,c,d show the changes of the SSC error with decreasing length of the cut. For more examples see
(Klokočník et al, 1994). Our example is for the SSC errors in the ERS 1 orbit with the cut 25 days, 4 days, 0.6
day and 3 hours. We clearly see a resonant structure of the error when the arc is long. We can watch an evident
loss of gravity information for too short cuts.
(4) number of valid digits in the covariance-matrices must be sufficient to ensure objective computation of the
radial error (complicated procedure with milions of mathematical operations and rounding errors).
For the older models, having higher radial errors, using only 3 valid digits of the matrices (provided to us by the
authors of the models) was not sufficient in some cases. Thus, we asked for 4-5 valid digits for the older GRIM5
and EGM96 models (to correct our previous computations) as well as for all recent gravity models.
Historical models - evolution
Great progress in the Earth gravity field modelling between about 1988 and 1998, from the model GEM T2 via
JGM 2 and 3 to EGM 96 (see References), is shown on “historical series” in Figs. 6, 7, 8 a,b,c,d, for these 4
models with full covariance matrices to (50,50), 4 or 10 day cut of orbit perturbations, for GEOSAT, ERS and
T/P, respectively. The progress was enormous (see the RMS values on left hand sides of the figures), from a
meter/decimeter for the lower/higher orbits using GEM T2 to few centimeters/a centimeter with the calibrated
covariances of EGM 96. (Figures 6-8 are reproduced from Klokočník et al, 1998). The along-track signiture of
the radial error was clearly visible for T/P with GEM T2 or for ERS with JGM 2 (for more similar examples see,
e.g., Klokočník et al, 1994). The variability of the radial error with latitude and longitude decreased significantly
from time of GEM T2 to EGM 96 (and is still smaller for the most recent models). For the individual gravity
field models, description of the data and methodology, see References.
Recent models
We have tested various recent gravity field models. Figs. 9-13 show the series of the radial orbit errors and their
components or the SSC errors (see Theory) for EIGEN 1S, TUGRAZ04, EIGEN GRACE 02S, and EIGEN
GL04C models (see References). We used their full COVAR to Lmax = 70, the orbit perturbations cut 1.3 day for
ERS and 4 days for Geosat-type orbits. For TUGRAZ04 and EIGEN GRACE 02S, we have available only their
formal covariance matrices so their comparison with previous models with scaled/calibrated matrices must be
very careful. Nevertheless, we can prove very low variability of the radial or SSC error geographically in all
recent gravity models. This is due to a dense, uniform, precise and global data exclusive or dominating in these
models from CHAMP and/or GRACE.
From GRIM5S1/C1 (the last pre-CHAMP gravity model from GFZ/CNES), EIGEN 1S (already with CHAMP
data added to GRIM 5S1 base) to the very recent models like EIGEN GL04S/C, the radial error is smaller and
smaller, its variability with latitude and longitude becomes negligible. It reflects quality and very good
geographic distributions of observations of these missions and thus of their orbits. The fact that the COVAR
estimates of recent models are also zonal in character like the VAR-only (see test above) is due to lower
correlation of terms in these new Earth models than in the older models. This is interesting because the CHAMP
and GRACE models are from one satellite, but the tracking in them is nearly global and uniform (as mentioned above).
Figs. 14 a, b show example of the DSC errors, with historical JGM 3 model and for ERS/Topex combinations
AA and DD (see Theory). It is a typical example: while the signal has the same magnitude for AA, AD, DA,
and DD and it is as in the SSCs, the geographic distribution (namely in longitude) of the error for the different
types of DSC differs (is shifted in longitude), and differs also between the DSCs and SSCs. For more examples
and more about various applications of the DSC see Klokočník et al, (1995, 2000) or Wagner et al (1997).
Conclusion
The radial orbit error for altimetry missions due to the static Earth gravity field error (expresed by the variancecovariance matrix of the harmonic geopotential coefficients (Stokes parameters) of the tested models) has been
studied with Rosborough’s and Klokocnik’s theory. Numerical examples show evolution since the historical
models, like GEM T2 or JGM 2/3 till now, to EIGEN–GRACE 02S or EIGEN GL 04C: the error has decreased
from meters to centimeters and its variations with latitude and longitude are now very small. Now we
approached the situation when the radial orbit error is not dominated by the static gravity induced error, but the
environmental (non-gravitational) errors became most important. This study can be considered as a first step to
a more general description of evolution of the radial accuracy of satellite altimetry orbits. Second step (Wagner
et al, 2007, in preparation) will deal with up-to-date actual long term altimetry observations computed with the
aid of different gravity models, from historical to recent.
Acknowlegment
We thank Grant Agency of the Academy of Sciences of the Czech Republic for the grant # A3407 and Ministry
for Education of the Czech Republic for grant LC 506 (CEDR).
References:
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