Ocean tides from satellite altimetry and GRACE
ABSTRACT Satellite altimetry and GRACE observations carry both the signature of ocean tides and have in general complementary potential to resolve tidal constituents. It is therefore straightforward to perform a combined estimation of a global ocean tide model based on these two data sources. The present paper develops and applies a three step procedure for generating such a combined ocean tide model. First, the processing of multi-mission altimetry data is described along with the harmonic analysis applied to derive initially a pure empirical ocean tide model. Then the capability of GRACE to sense particular tidal constituents is elaborated and an approach to estimate tidal constituents from GRACE is outlined. In a third step a combination strategy with optimal stochastic data treatment is developed and applied to the altimetry-only tide model EOT08a and four years of GRACE observations, leading to the combined model EOT08ag. The differential contributions of GRACE to EOT08ag remain small and are mainly concentrated to the Arctic Ocean, an area with little or poor altimetry data. In comparison with other tide models, EOT08ag is validated by K-band range residuals, the impact on gravity field modelling and on precise orbit determination and by variance reduction of crossover differences. None of these comparison exhibits a significant improvement over the altimetry-only tide model except for a few areas above 60°N. Overall the improvements of the combination remain small and appear to stay below the current GRACE baseline accuracy.
Ocean tides from satellite altimetry and GRACE
W. Boscha, Savcenko, R.a, Mayer-G¨ urr, T.b, Flechtner, F.c, Dahle, C.c, Daras, I.c
aDeutsches Geod¨ atisches Forschungsinstitut (DGFI), Munich, Germany
bInstitute of Geodesy and Geoinformation, Bonn University, Bonn, Germany
cGeoForschungsZentrum Potsdam (GFZ), Potsdam, Germany
Text of abstract
Keywords: Altimetry, GRACE, EOT08a, EOT08ag, EOT10a, Ocean tides
Precise knowledge of ocean tides is indispensible for practi-
cal application like ship routing, the prediction of tidal heights
and currents, the protection of the coastal ecosystem, for study-
ing the dissipation of tidal energy, but also for the quantifica-
tion of short-term mass variations in the Earth system, an ap-
plication relevant in the context of the dedicated gravity field
missions like GRACE and GOCE. The numerical treatment of
ocean tides is based on Laplace shallow water equation as de-
scribed for example by Hendershott (1977). Beside the gravita-
tional forces of Moon and Sun, the land/ocean distribution and
the sea floor topography must be taken into account to solve the
differential equation. A challenge in the numerical treatment of
tides is a proper accounting of dissipation, solid Earth effects,
and modelling of baroclinic tides. Assimilating tidal constants
from coastal or shallow water tide gauges have been performed
to improve the accuracy of tidal prediction. A review on data
assimilation may be found in Egbert and Bennett (1996).
In this paper we focus on pure empirical analysis of ocean
tides as can be realized, for example, at an individual tide gauge
by harmonic analysis of a sufficiently long record. On global
scale the empirical analysis became feasible after satellite al-
timetry developed towards an operational technique and pro-
vided a few years time series of sea surface height measure-
ments with a precision of a few centimetres (Mazzega, 1985).
Consequently, after TOPEX completed a few years of operation
there was a great deal of new ocean tide models (Shum et al.,
Since then, the altimeter observation scenario has been ex-
tremely improved. Today, there are nearly two decades of re-
peated sea level height measurements, performed over com-
plementary ground track pattern of up to six different mis-
sions. There is significant progress in the precision of satel-
lite ephemerides which are today exclusively based on GRACE
gravity field models. The calibration of radiometer sensors has
been improved as well as models used to correct for environ-
Email address: email@example.com (W. Bosch)
mental effects (e.g. Scharroo et al., 2004). This altogether jus-
tifies a pure empirical approach, as demonstrated, for exam-
ple, with the global ocean tide model EOT08a (Savcenko and
Bosch, 2008). The comparison with other recent global ocean
tide models demonstrate that the results of a pure empirical
procedure are similar in performance as alternative approaches
with or without assimilation (Ray et al., 2010).
However, some principle deficiencies of satellite altimetry
remain: The sampling is not really global: above the latitude
coverage of TOPEX, there are only GFO suffering from a poor
orbit determination, and the ESA missions ERS1/2 as well as
ENVISAT, which fail to contribute to solar tides due to their
sun-synchronous orbit configuration. Above all the southern
ocean has considerable large areas with seasonal sea ice cover-
age, making a reliable time series analysis nearly impossible.
Although alias problem for ocean tides are also present in
the GRACE data (see section 3.1) it has been demonstrated that
GRACE observation carries the signature of residual ocean tide
signals for selected constituents, (Bosch et al., 2009). A quali-
tative comparison of global ocean tide models by analysing the
K-Band range data of GRACE was performed by Ray et al.
(2009). Egbert et al. (2009) assimilate localized mass anoma-
lies derived from GRACE ranging measurements into a hydro-
dynamic model to improve tidal solutions around Antarctica for
the M2, S2, and O1 constituents. At the 2009 GRACE Science
Team Meeting (Austin, Texas) a few presentations addressed
again the potential to estimate ocean tides from time series of
GRACE gravity fields. Han et al. (2009) points out that di-
urnal tides along the ascending and descending GRACE orbits
are roughly opposite in phase and therefore mostly cancel in
monthly gravity field solutions. They promote a separate treat-
ment of ascending and descending tracks. Kilett et al. (2009)
solve for amplitudes and phases of major solar and lunar tides
in the Arctic Ocean by inverting five years of GRACE inter-
satellite acceleration data using the mascon-approach.
It is therefore self-evident to combine satellite altimetry and
GRACE data for a common estimation of a global ocean tide
model. The primary goal in this paper is to describe the proce-
dure and first results for estimating a precise global ocean tide
Preprint submitted to Journal of GeodynamicsOctober 1, 2010
model by combining observations from GRACE and satellite
altimetry in order to provide the most precise global descrip-
tion of the short-term tidal mass variations, a new ocean tide
reference model for GRACE gravity field modelling, for satel-
lite altimetry, and for studying the impact on Earth rotation,
loading effects and other applications.
The present paper will first describe the processing of the
altimeter data and the harmonic analysis performed for an
altimetry-derived ocean tide model, EOT08a. In section 3 the
tidal aliasing for the GRACE mission is explained, the sensi-
tivity is investigated and the procedure is described, how in-
dividual tidal constituents can be estimated from GRACE ob-
servations. Then, section 4 outlines the approach to combine
the altimetry-derived tide model with residual ocean tide sig-
nals derived from GRACE. The results of such a combination,
the model EOT08ag, is introduced and validated by crossover
statistics and analysis of GRACE range rate residuals. A final
discussion and outlook reviews the combination procedure and
indicates possible improvements for follow-on combined ocean
2. Altimetry data processing
2.1. Processing strategy and data pre-processing
Most of the classical pulse-limited radar altimeter systems
provide sea surface height measurements along nominal ground
track profiles, which are repeated after a fixed number of days.
Sampling a high frequent ocean tide signal with dominant pe-
riods of 12 and 24 hours only every few days causes the well-
known alias effects (see e.g. general discussion by Parke et al.
(1987) and Smith (1999)): the high frequent signal becomes
visible only after an alias period, much longer than the sam-
pling period of the altimeter system. Sun-synchronous mis-
sions like (ERS and ENVISAT) pass the same location always
at the same local time and cannot at all sense solar tides (they
have an infinite alias period). In order to solve and mitigate
the alias situation we take advantage of multi-mission altime-
try with at least two, sometimes even five missions operating
simultaneously with complementary ground track pattern. This
multi-mission scenario exist since September 1992 (see 1). The
common ground track patterns of all altimeter satellites jus-
tify to perform – independent of the ground tracks of individ-
ual missions – the empirical analysis on a regular equidistant
grid and to analyse all data inside a cap with spherical radius
rmax around the grid nodes. This however is only possible
if the data of all missions is updated, harmonized, and cross-
calibrated. Updating implies to apply the most recent orbits
and best known mission specific corrections, e.g. for the on
board microwave radiometers or the sea state biases. In ad-
dition the data is harmonized by applying identical geophysical
correction models e.g. for the inverted barometer effect, the ref-
erence tide model, and ionospheric prediction models for single
frequency radar systems. Harmonization minimizes the risk to
map systematic differences of correction models to the solve-
for parameters of the ocean tide model. The cross-calibration is
realized by means of a common least-square analysis of cross-
over differences performed between all altimeter systems op-
erating simultaneously (Bosch and Savcenko, 2007). The esti-
mated radial correction capture relative rang biases and other
geographically correlated errors. In summary, the laborious
pre-processing results in widely consistent multi-mission data
holdings, which justify treating all observations as if they were
taken by a single altimeter mission.
2.2. Harmonic analysis of tidal constituents
The data of all missions of table 1 were pre-processed and
de-tided by applying the ocean tide corrections derived from
FES2004, a hydrodynamic model subsequently taken as refer-
ence. The harmonic analysis is then straight-forward and con-
sists in estimating at every grid node residual tides from the sea
surface heights hk
hk(t) + vk=
i=1fi(t)[cicos(ωit + ui(t)) + sisin(ωit + ui(t))]
where vkare residuals, m is a mean value and ciand siare co-
sine and sine-coefficients to be solved for individual tidal con-
stituents. The fiand uiare nodal corrections for amplitudes
and phases of the astronomical arguments wi respectively. In
addition, cosine and sine-coeffcients Ajand Bjfor annual and
semi-annual periods Tjwith Ωj= 2 ∗ π/Tjare solved for. The
analysis is performed on every node of a 15’x15’ geographi-
cal grid using cap sizes with rmax= 1.5◦in shallow water and
rmax= 4.5◦in open ocean. The observations are weighted by
a Gauss function taking its half weight width at 0.3 · rmax. The
EOT08a solution (Savcenko and Bosch, 2008) solved for the
constituents M2, S2, N2, K2, 2N2, O1, K1, P1, Q1, and M4.
The residual amplitudes of the most dominant constituent M2
is shown in Fig.1. Shortly after the edition of EOT08a, a re-
placement of the dynamic atmospheric correction (DAC)” for
the ocean response to atmospheric pressure variations became
available (AVISO ?) and led to significantly improved estimate
of the residual S2 amplitudes (cf. Fig.1).
For some more reasons a modified solution, EOT10a, was
also generated. EOT10a accounts for correlations among sub-
sequent altimeter observation along the same pass and provides
a more realistic error estimate. A variance-component estimate
was appliedto achievea realisticrelative weighting betweenthe
data from different missions. To rebut the suspicion that any
tidal signal is mapped to the radial error estimates, the cross-
calibration results were not applied for EOT10a. Instead a mis-
sion specific offset was estimated at each grid node. Correla-
tion analyses (Savcenko and Bosch, 2008) showed that the tidal
constituents are basically uncorrelated. This justified to per-
form harmonic analyses for individual constituents with homo-
geneous but varying cap size (rmax= 1.5◦, 3◦, 4.5◦). EOT10a is
a patchwork of tables from individual solution selected accord-
ing to the results of the validations described below. Compared
to EOT08a, EOT10a provides additional tables for Mf, Mm,
Ajcos(Ωjt) + Bjsin(Ωjt)
04.1992 – 04.1996
09.1992 – 10.2005
04.1995 – 04.2003
01.2000 – 09.2008
09.2002 – ...
10.2003 – ...
07.2008 – ...
Table 1: Satellite altimetry - mission overview
0 cm 1 cm 2 cm 3 cm4 cm 5 cm
Figure 1: Residual tidal signal wrt. FES2004 of M2 and S2 (replacement grid)
2.3. Results and Validation
The residual amplitudes of EOT08a and EOT10a can hit sig-
nificant magnitudes. For S2 and M2 the amplitude assume up to
15 cm over extended areas in shallow water like the North-West
European shelf or the Yellow Sea (see Figure 2). Residual am-
plitudes of the same magnitude are also found in the Indonesian
Waters and on the Patagonian Shelf (both not shown here).
A widely used measure for the quality of global ocean tide
models is the comparison with tidal constants. Table 2 com-
piles the RMS differences between global tide models and tidal
constants of the ST102p set of tide gauges (Ray) and a set of
WOCE sites (Ref.?). For nearly all comparisons the empir-
ical models GOT4.7(Ray 200?), EOT08a, and EOT10a (this
paper) show smaller RMS values than the hydrodynamic ref-
erence model FES2004 although some of the tide gauge data
have been assimilated to FES2004 (Lyard et al., 2006).
For the ST102p tide gauges the EOT10a RMS values are
always slightly lower than for EOT08a. The most significant
improvements are achieved for the S2 constituent. However
the comparisons at selected tide gauges are globally not abso-
lute representative. An exhaustive quality assessment requires
much more comparisons. For the validation of EOT08a, statis-
tics of crossover differences have been performed in addition
(Savcenko and Bosch, 2008) and show for shallow water areas
a significant gain in variance, if the residual tide correction of
EOT08a were applied. Also the impact on GRACE range resid-
uals have been studied for EOT08a. As these comprehensive
quality assessments for EOT10a are not yet fully completed, it
was decided to use EOT08a with an updated table for the S2
constituent for the combination with GRACE.
3. GRACE data processing
3.1. Tidal aliasing
The determination of ocean tides from GRACE data requires
as a first step the investigation which tidal constituents can be
estimated together with the monthly gravity field variations.
GRACE does not permanently monitor every point of the ocean
surface due to the orbit configuration. This insufficient sam-
pling rate results in the high frequent tidal signal being mapped
120˚125˚130˚120˚ 125˚130˚ 120˚125˚ 130˚120˚125˚130˚120˚ 125˚130˚
0 cm0 cm
2 cm2 cm
4 cm4 cm
6 cm6 cm
8 cm8 cm
10 cm10 cm
350˚355˚ 0˚5˚10˚350˚ 355˚ 0˚5˚10˚350˚355˚0˚ 5˚10˚350˚355˚0˚5˚10˚350˚355˚0˚5˚10˚
0 cm0 cm
2 cm2 cm
4 cm4 cm
Figure 2: Residual tidal signal wrt. FES2004 of the some partial tides in Yellow Sea
ST102 (96-102 TGs)
WOCE (158 TG)
Table 2: RMS differences [cm] of tidal constants at tide gauges of the ST102p data set and of WOCE. Smallest values are indicated in bold
into lower frequencies (aliasing). In order to be able to separate
the tides from the monthly variations, the aliasing periods of
the tidal signal into the GRACE observations have to be shorter
than one month.
As GRACE has no precise repeat orbit, it is not possible
to calculate the aliasing frequencies of the different tidal con-
stituents using closed formulas, but an analysis of the real
GRACE orbits is required.Therefore, the over flights by
GRACE during the time span from 2003 till 2007 are com-
pared to the oscillations of the major tides.
results are illustrated for a test area in the North Atlantic
([20◦W − 35◦W] × [30◦N − 40◦N]). The tidal oscillations are
indicated by the blue lines, the red dots mark the time when
GRACE passes over this area.
The left part of Fig. 3 shows tidal constituents with sub-
monthly alias periods such as M2, O1, N2 and Q1. These con-
In Fig. 3 the
stituents can be obtained from GRACE and will therefore be
estimated together with the monthly gravity field variations in
the following investigations. The right part of the figure illus-
trates constituents with aliasing periods longer than one month.
Therefore GRACE is not capable to solve for those tidal con-
stituents. For S2 and P1, for example, the tidal signals are not
clearly separable from the semi-annual signal GRACE is ex-
periencing. For K1, GRACE still needs more time in orbit to
completely sample the signal.
3.2. Sensitivity Analysis
If we would solve the full spectrum of all relevant con-
stituents up to a certain degree and order (e.g. 20 or 30) the
number of additional parameters (besides the gravity field co-
efficients) would increase dramatically. Therefore, a sensitiv-
ity study based on the two ocean tide models FES2004 and
Figure 3: GRACE alias periods of different tidal constituents. The over flights
by GRACE are compared to the oscillations of the major tides in a test area in
the North Atlantic ([20◦W −35◦W]×[30◦N −40◦N]). The partial tides can be
separated into those featuring an alias period of less than one month (left) and
those with a longer alias period (right).
GOT4.7 has been performed in order to reduce the ocean tide
unknowns to a reasonable smaller number. The tide models
are given in a spherical harmonic representation of amplitudes
lm,s) and phases (ε±
lm,scan be retrieved by (Dow, 1988):
lm,s), from which the coefficients C±
lm,scos ˆ ε±
lm,ssin ˆ ε±
such as for M2.
According to the Schwiderski notation, the instantaneous
ocean tide ζζ(λ,φ,t) can be expressed in terms of amplitude
ξζ(λ,φ,t) and phase lag ξζ(λ,φ,t) as:
lm,s− χ and χ = 0 for a semidiurnal tides
ζζ= ξζcos(θζ+ χζ− δζ)(3)
where θζis the Doodson argument and χζdepends on the tide
(Dow, 1988). Expression 3 can therefore be expanded to:
From equation (2) we solve for alm,s, blm,s, clm,s, dlm,scoef-
ficients for both models. These coefficients are then used in
equation (5) Dow (1988) to compute the disturbing potential
difference between the two ocean tide models at a near-GRACE
orbital level of 500 km.
where ρwis the density of the sea water.
2l+1, kl,sand are the load Love numbers.
In the next step, a harmonic analysis of the disturbing poten-
tial difference with respect to the reference sphere of a radius
equal to the semi-major axis of the Earth was performed, which
resulted to the back tracing of the alm,s, blm,s, clm,s, dlm,scoef-
ficients. From those coefficients the C±
ference of the two models after the analysis.
Fig. 4 (top) shows the sensitivity study result for M2 retro-
grade and pro-grade amplitudes up to degree and order 20 with
a threshold of 0.01 mm (result of various tests). Sensitive co-
efficients are marked not dark blue. This threshold has a max-
imum impact of 0.01 mm for the accumulated pro-grade and
0.02 mm for the accumulated retro-grade coefficients (Fig.4
bottom) when taking into account all or only sensitive pro- and
retrograde coefficients depicted in Fig. 4 (top). This maximum
error is about a factor of 10 below the current GRACE error
level (Flechtner, personnel communication) and can therefore
safely be used to decrease the number of solved-for ocean tide
coefficients. For M2, the number of sensitive coefficients could
consequently be reduced from 436 to 99 or to 23%. For N2 and
O1 we have received similar results for the coefficient reduc-
tion and accumulated errors (figures not shown): A N2 thresh-
old of 0.005 mm resulted in 67 sensitive coefficients (reduction
to 15%) and a threshold of 0.004 mm for O1 resulted in 76
sensitive coefficients (reduction to 17%).
lm,sand finally the
lm,sare retrieved back, but this time representing the dif-
3.3. Estimation of tidal constituents from GRACE data
In the following, GRACE observations are used to improve
existing ocean tide models. Here only those constituents with
short aliasing periods are used that were identified to be sen-
sitive in the investigations of section 3.1. More than 4 years
(09/2002 04/2007) of GRACE observations have been used in
the calculations applying the same short arc approach and the
same background models (astronomical tide of sun, moon and
planets, Earth tides, atmospheric and ocean dealiasing prod-
uct (AOD1B), ...) which have been used in the calculation
of the ITG-Grace03s gravity field model, (Mayer-G¨ urr et al.,
2010). As the ocean tide model FES2004 was applied as back-
ground model, the estimated constituents are the residuals to
the FES2004 model. For each partial tide, both the sine and
the cosine part are represented by a complete set of potential
coefficients up to degree and order n = 20 (for M2 n = 25).
tide signals but all gravity field effects from other mass trans-
ports. Therefore, monthly mean gravity field solutions have
been co-estimated together with the ocean tide parameters in
order to separate the ocean tide effect from other mass varia-
tions e.g. caused by hydrology. Thus it is reasonable not to
include the constituents with long aliasing periods into the anal-
ysis as, for example, S2 would be mapped completely into the
monthly solutions and could not be distinguished from other
For each month of GRACE data the observation equations
can be formulated according to
lG= AGδx + ?
Figure 4: Sensitivity study result for M2 retro-grade and pro-grade amplitudes
up to degree and order 20 with a threshold of 0.01 mm. Sensitive coefficients
are marked not dark blue. Bottom figure shows corresponding accumulated
amplitudes taking into account all (blue (pro-grade) and purple (retro-grade))
or only sensitive coefficients from top (red (pro-grade) and green (retro-grade)).
Figure 5: Residual amplitudes of different ocean tide constituents as estimated
from GRACE data compared to the FES2004 model.
where lGrepresents the GRACE K-band range observations re-
duced by the influence of the background models mentioned
above. δx are the unkown parameters in terms of spherical
harmonics and AGstands for the functional model which es-
tablishes the relationhsip between observations and unknowns.
The variance-covariance matrix of the observations Σllis given
by the inverse weight matrix P−1
variance factor σ2
Gmultiplied by an unknown
G. The GRACE normal equations are then
nG= NGδxwithNG= AT
From the monthly normal equations the monthly gravity field
parameters will be eliminated, therefore in the following the δx
only contain the residual ocean tide paramters. Afterwards all
monthly normal equations of the given time span are accumu-
lated to one complete system of normal equations.
The improvements are shown in terms of amplitudes of wa-
ter heights in the Fig. 5. Large residual tidal signals can be
observed in the coastal areas for each of the estimated con-
stituents. Similar patterns can be identified when comparing
the EOT08a model to the FES2004 (not shown here).
4. Combination strategy
As a next step, a combined empirical ocean tide model will
be estimated using both GRACE and the given altimetry model
according to Section 2. Altimetry models are customarily pa-
rameterized in terms of gridded values, whereas the GRACE
solutions are given as spherical harmonic expansion. It was
decided to carry out the combination on the basis of spherical
harmonics, because it is expected that GRACE can only con-
tribute to the larger spatial scales and it is thus possible to re-
strict the combination to these scales by limiting the spherical
harmonic expansion to a moderat upper degree. The smaller
scales are only taken from the gridded altimetry model as will
be described below.
of the respective covariance matrices. As this matrix has not yet
been available in case of the altimetry model, an approximate
model has to be derived. In a first approximation the gridded
and to be uncorrelated. As the model values are not directly
used but transformed into a series of spherical harmonics the
error behavior must be propagated to the Stokes coefficients.
The transformation is performed according to
where h is the tidal height from the EOT08a model and ρwis
the density of sea water. The Love numbers k?
potential of the loading deformation. Taking the property of or-
thogonality of the spherical harmonics into account the Stokes
coefficients are also uncorrelated and the variances are propor-
tional to the following factor:
1 + k?
2n + 1
naccount for the
?1 + k?
2n + 1
This simple approximation also reveals that altimetry is less ac-
curate in the lower degrees than it is in the higher degrees in
terms of gravitational potential. In case of the GRACE solu-
tions the situation is exactly opposite, here the lower degrees
have higher accuracies than the higher degrees. This shows
the potential of the combination of the two measurement tech-
The altimetry model is introduced as pseudo-observations xA
in terms of spherical harmonics. The coefficients are reduced
by the reference ocean tide model x0(FES2004) to be consis-
tent with the GRACE processing model. This results in the
following observation equations for the altimetric part
xA− x0= Iδx + ?
The variance-covariance matrix is taken from (9). The com-
bination of the GRACE observations and the altimetry pseudo
observations results then in the following system of combined
Gbetween the GRACE normal
The relative weighting σ2
equations and the Altimeter normal equations are unknown be-
forehand and estimated iteratively together with the combined
solutions by means of variance component estimation (VCE,
(Koch and Kusche, 2001)). From the combined model, gridded
values can again be synthesized e.g. for oceanographic applica-
tions. As the combined model in terms of spherical harmonics
detail structures must be taken from the gridde EOT08a model.
These details are given as the difference between the original
EOT08a model and the EOT08a model in terms of spherical
harmonics evaluated at each grid point. Fig. 6 visualizes the
different steps in the combination scheme. The results of the
combined analysis are displayed in Figure 7. The plots show
only the contribution of GRACE to the combined model in
in spherical harmonics
in spherical harmonics
Figure 6: Combination schema of GRACE data and altimetry data.
Figure 7: Amplitudes of the contribution of GRACE to the combined model
terms of amplitudes for different tidal constituents. Compared
to the GRACE-only estimation (differences to FES2004) shown
in Figure 5, the signal is significantly smaller, which indicates
the better agreement of GRACE with the EOT08a model. Fur-
thermore the solution is smoother, as GRACE contributes less
to the higher spatial resolution. Improvements can be observed
especially in the polar regions, which appear to be reasonable
as there is no data coverage by the altimetry satellites in these
We investigated the influence of different background ocean
tide models (FES2004, EOT08a, EOT08ag and EOT10a) on
a monthly gravity field solution till d/o 120 for April 2008
using GFZs EPOS (Earth Parameter and Orbit System) soft-
ware and draft EIGEN-GRACE06S release 5 (RL05) process-
ing standards and background models (update of RL04, Flecht-
ner et al. (2010)). All tests were identical except that the ocean
tide model has been exchanged.
A first indication which of these four models is (absolutely)
(KBRR) residuals which are calculated during the gravity field
Nr. of Arcs
KRR residuals [um/sec]
Figure 8: Daily post-fit K-band Range rate (KRR) residuals (m/s) for fields
using all the different models.
adjustment. Fig. 8 (update number) shows the daily RMS
for all four cases. In principle all KBRR residual curves are
very close to each other, but the combined EOT08ag model
has slightly smaller residuals (especially for arcs 1520) than
the others. Also, the results for sub-daily (regional) compar-
isons may look different and have to further investigated. We
found that the global impact of changing the background ocean
tide models in a monthly gravity field is of minor importance.
The formal geoid errors of the monthly solutions are almost
the same in each case and their degree variance differences are
also very close (Fig. 9) and still below the draft GFZ RL05
calibrated GRACE error level (factor of 11 above the GRACE
baseline). In particular, the degree variance difference between
the field using the EOT08ag and the field using the EOT08a
model is in the level of the baseline of GRACE, which means
that the impact of GRACE to the combination with the altime-
try model is presently on the noise level and cannot be sensed
by the monthly gravity field estimation. As for the KBRR resid-
uals, the regional impact may be more significant and has to be
The indication from the KBRR test, that the combined
EOT08ag model improves monthly gravity field solutions, is
more encouraged by the results in Table 6. Here, the RMS of
geoid heights of monthly gravity fields derived from different
ocean tide models and filters are shown. Both EOT models im-
prove the FES results over the oceans by about 12% in case a
Gauss filter is applied. The unfiltered results, which show the
internal quality of a monthly solution, also improve by 10% for
nmax= 40 (500km) with an additional gain of 1% when using
the EOT08ag model.
The four ocean tide models have also been tested in POD
(precise orbit determination) of various geodetic satellites and
various observation types (SLR, PRARE, Doris). The results
(see Table 4) show that generally a) FES2004 provides the
largest RMS values, b) EOT08ag is slightly better than EOT08a
Spherical Harmonic Degree n
Geoid Height [mm]
10 20 3040 50607080 90100 110 120
GRACE Baseline @ 500km
calibrated RL05 error
EOT10a formal error
EOT10a - EOT08a
EOT10a - EOT08ag
EOT10a - FES2004
EOT08ag - EOT08a
Figure 9: Degree variance differences of monthly gravity field using different
background ocean tide models as well as the formal error of the gravity field
and c) EOT10 gives smallest RMS values.
6. Outlook (all, 1Page)
This work was funded by the Deutsche Forschungsgemein-
schaft (DFG), Bonn, Germany, under the grants BO1228/5-2,
FL592/2-2, and IL 17/9-1
We would like to thank the German Space Operations Center
(GSOC) of the German Aerospace Center (DLR) for providing
continuously and nearly 100% of the raw telemetry data of the
twin GRACE satellites.
Dynamic atmospheric Corrections are produced by CLS
Space Oceanography Division using the Mog2D model from
Legos and distributed by Aviso, with support from Cnes
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Table 3: RMS values of geoid heights for a monthly gravity field using different models. Results are shown for the solution after applying the Gaussian 500km filter,
for the unfiltered solution till d/o 120 and for the unfiltered solution truncated to d/o 40.
#arcs#arc FES2004 EOT08aEOT08agEOT10a
Table 4: SLR[cm], PRARE Range (PRA[cm]), PRARE Doppler(PDO) [mm/s] andDoris (DOR) [mm/s] RMSvalues for various geodeticsatellites and observation
Tide EOT08aEOT08agFES2004 GOT4.7
Table 5: RMS differences [cm] of tidal constants at tide gauges of the ST102p,
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Table 6: (from Ilias) Validation of tide models. The lower RMS value indicates the imrovement of solutions of gravity fields
Figure 10: Variance reduction of sea surface heights at the crossover points
of ENVISAT, GFO, and Jason-1 altimeter missions for the year 2004. The
crossover differences were computed if the maximal time difference between
ascending and descending passes doesn’t exceed the corresponding mission
reapeat periods of 35, 17, and 10 days
Figure 11: Differences of K-band residuals if the FES2004 or EOT08a used
Figure 12: Differences of K-band residuals if the EOT08a or EOT08ag used
−1.0 −0.8 −0.6 −0.4 −0.2 0.00.20.40.6 0.81.0
min/max/wrms [mm]: −0.89/0.99/0.39
Figure 13: Differences in geoid heights if EOT08a instead of FES2004 used
−1.0 −0.8 −0.6 −0.4 −0.2 0.00.20.40.60.81.0
min/max/wrms [mm]: −1.00/0.95/0.41
Figure 14: Differences in geoid heights if EOT08ag instead of FES2004 used
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