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Journal ArticleDOI

A timewise kinematic method for satellite gradiometry: GOCE simulations

02 Aug 2006-Earth Moon and Planets (Springer Netherlands)-Vol. 97, Iss: 1, pp 37-68
TL;DR: New algorithms for the data processing of a satellite geodesy mission with gradiometer to extract the information on the gravity field coefficients with a realistic estimate of their accuracy are defined to allow a complete simulation of the GOCE mission with affordable computer resources.
Abstract: We have defined new algorithms for the data processing of a satellite geodesy mission with gradiometer (such as the next European mission GOCE) to extract the information on the gravity field coefficients with a realistic estimate of their accuracy. The large scale data processing can be managed by a multistage decomposition. First the spacecraft position is determined, i.e., a kinematic method is normally used. Second we use a new method to perform the necessary digital calibration of the gradiometer. Third we use a multiarc approach to separately solve for the global gravity field parameters. Fourth we use an approximate resonant decomposition, that is we partition in a new way the harmonic coefficients of the gravity field. Thus the normal system is reduced to blocks of manageable size without neglecting significant correlations. Still the normal system is badly conditioned because of the polar gaps in the spatial distribution of the data. We have shown that the principal components of the uncertainty correspond to harmonic anomalies with very small signal in the region where GOCE is flying; these uncertainties cannot be removed by any data processing method. This allows a complete simulation of the GOCE mission with affordable computer resources. We show that it is possible to solve for the harmonic coefficients up to degree 200–220 with signal to error ratio ≥1, taking into account systematic measurement errors. Errors in the spacecraft orbit, as expected from state of the art satellite navigation, do not degrade the solution. Gradiometer calibration is the main problem. By including a systematic error model, we have shown that the results are sensitive to spurious gradiometer signals at frequencies close to the lower limit of the measurement band. If these spurious effects grow as the inverse of the frequency, then the actual error is larger than the formal error only by a factor ≃2, that is the results are not compromised.

Summary (3 min read)

1. Introduction

  • Gravity field and steady-state Ocean Circulation Explorer (GOCE) is an European Space Agency (ESA) mission soon to be launched, with the c© 2005 Kluwer Academic Publishers.
  • The first decomposition of the problem can be obtained from the following considerations.
  • 2. Timewise solution A second basic choice is the organization of the GOCE observations in one of three possible ways.
  • This method is the most convenient in the mission design phase, because it allows to convert requirements in the error spectrum of the gradiometer into error spectrum of the recovered gravity coefficients.
  • The authors show in Section 5 that the polar gaps resulting from the non polar orbit of GOCE introduce an exact symmetry in the functional space of spherical harmonics.

2. Gradiometer Calibration

  • The observations include two independent components of the gravity gradient.
  • Thermal signals are by no means random, they occur with well defined frequencies depending upon the external sources of heat and the control cycles of the thermal stabilization system4.
  • The authors are mostly interested in evaluating the effects of the systematic bias components5.
  • Third, the authors need to stress once again that the calibration model should not constrain the shape of the calibration as a function of time, and should not use information on this shape illegitimately transfered from the simulation step.
  • This selection of base functions does not satisfy the third requirement, because all the base function are periodic of period ∆t and thus have the same value at the two extremes of the arc interval.

3. Multi arc solution

  • The multi arc approach is a form of the least squares method, in which the list of parameters is split in accordance to the partition of the orbit into arcs (Reigber, 1989; Milani and Melchioni, 1989; Milani et al., 1995).
  • This, however, ignores the portion C`g of the normal matrix, forcing Γ`g = 0, and changes the solution ∆g.
  • Thus, for the purpose of these simulations, with the main goal of computing a reliable estimate of the error for high degrees, this approximation is sufficient.
  • The authors can conclude that, if the arc length ∆t, the number of calibration parameters per arc 2K+2, and the minimum degree l are chosen properly, then it is possible to solve separately for the calibration parameters ` and then to solve for the harmonic coefficients g only.

4. Resonant Decomposition

  • This makes difficult to perform this computation without using very expensive hardware.
  • The above results refer to the geopotential, but in fact apply to whatever harmonic function expanded in spherical harmonics.
  • An assessment of the relative size of the off-diagonal blocks was performed in the case of the GPS observables: the off-diagonal block correlations were found to be less than 1% in the most critical cases involving the r = 0 block, less than 0.4% in the other cases .
  • The authors can conclude that, even by using h/k = 31/2, not the most appropriate resonance for the value of n of GOCE, the off-diagonal terms can be neglected and a full second iteration is not necessary6.

5. Consequences of polar gaps

  • It can be checked that the large RMS (and also actual errors) occur in this remainder class only for the zonals m = 0; the other harmonics, with m = 31, 62, . . ., are determined with error/signal ratio well below 1 for all l.
  • To understand why the GOCE results leave undetermined the geoid near the poles, the authors have represented in Figure 6 one of the GOCE observables, the radial component of the gravity gradient of the anomalies λ1 V1 and λ2 V2.
  • The exact symmetry results in a complete degeneracy of the normal matrix, with the number of zero eigenvalues corresponding to the dimensions of the symmetry group.
  • The cap function Φ cannot have a finite spherical harmonics expansion, because it is not an analytic function on the sphere.

6. Error sources

  • The formal error is of course larger than the one of the naive estimate of Figure 2, but only because of the polar gaps.
  • Indeed, for the classes not including low values of m the figures look like the one for r = 1 .
  • To assess the origin of the actual errors exceeding the formal ones, it is important to analyze the specific harmonics where the largest actual/formal error ratio takes place.
  • To test the relevance of this orbit error on the solution for the gravity coefficients the authors have performed a simulation without the gradiometer bias, with the orbit errors as the only systematic error source: as shown in Figure 10, the actual error in the coefficients becomes statistically consistent with the formal error.
  • As a general rule, the formal covariance result is a lower bound for the uncertainty, the actual results can only be worse.

7. Conclusions

  • As a result the authors have been able to perform a full simulation, including data simulation and differential corrections, with PC class computer resources.
  • The authors have improved the theory explaining the remaining uncertainties in the solution for the gravity coefficients, resulting from the polar gaps.
  • This is not a surprise: the calibration of accelerometers is one of the main problems of the new generation satellite geodesy missions (Bruinsma et al., 2003).
  • How to control the situation of the spurious signals in the frequency domain immediately below fm is going to be the main challenge of the GOCE operations and data processing.
  • This work was funded by Agenzia Spaziale Italiana (ASI) under contract I/R/194/02.

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A timewise kinematic method for satellite gradiometry:
GOCE simulations
Andrea Milani
Dept. Mathematics, Pisa University, Largo Pontecorvo 5, 56127 Pisa, Italy
e-mail: milani@dm.unipi.it
Alessandro Rossi
ISTI-CNR, Via Moruzzi 1, 56124 Pisa, Italy
e-mail: Alessandro.Rossi@isti.cnr.it
Daniela Villani
Hyperborea S.c., Via Giuntini 13, 56023 Cascina, Italy
e-mail: d.villani@hyperborea.com
Submitted, April 13, 2005
Abstract. We have defined new algorithms for the data processing of a satellite
geodesy mission with gradiometer (such as the next European mission GOCE) to
extract the information on the gravity field coefficients with a realistic estimate of
their accuracy. The large scale data processing can be managed by a multistage
decomposition. First the spacecraft position is determined, i.e., a kinematic method
is normally used. Second we use a new method to perform the necessary digital
calibration of the gradiometer. Third we use a multiarc approach to separately solve
for the global gravity field parameters. Fourth we use an approximate resonant
decomposition, that is we partition in a new way the harmonic coefficients of the
gravity field. Thus the normal system is reduced to blocks of manageable size with-
out neglecting significant correlations. Still the normal system is badly conditioned
because of the polar gaps in the spatial distribution of the data. We have shown that
the principal components of the uncertainty correspond to harmonic anomalies with
very small signal in the region where GOCE is flying; these uncertainties cannot be
removed by any data processing method. This allows a complete simulation of the
GOCE mission with affordable computer resources. We show that it is possible to
solve for the harmonic coefficients up to degree 200 ÷ 220 with error to signal ratio
1, taking into account systematic measurement errors. Errors in the spacecraft
orbit, as expected from state of the art satellite navigation, do not degrade the
solution. Gradiometer calibration is the main problem. By including a systematic
error model, we have shown that the results are sensitive to spurious gradiometer
signals at frequencies close to the lower limit of the measurement band. If these
spurious effects grow as the inverse of the frequency, then the actual error is larger
than the formal error only by a factor ' 2, that is the results are not compromised.
Keywords: satellite geodesy, geopotential, resonances, accelerometers, GOCE
1. Introduction
Gravity field and steady-state Ocean Circulation Explorer (GOCE) is an
European Space Agency (ESA) mission soon to be launched, with the
c
2005 Kluwer Academic Publishers. Printed in the Netherlands.
gocesim2.tex; 13/04/2005; 9:23; p.1

2 A. Milani et al.
purpose of determining the gravity field of the Earth to unprecedented
accuracy and resolution, by using the data from two main instruments.
A gradiometer measures the second derivatives of the gravitational po-
tential. A GPS receiver measures the spacecraft position by low-high
satellite-to-satellite tracking (ESA, 1999).
To extract the scientifically important results from a mission like
GOCE is a large data processing task. The challenge is not just the
amount of computations, but to find algorithms fully exploiting the
observational information without introducing instabilities and unjus-
tified constraints. Several different methods have been proposed to
decompose the problem into feasible computational steps. The full least
square fit would have a huge normal matrix: we need to find a block
decomposition of the normal matrix with dominant diagonal blocks.
Then the normal matrix can be approximately inverted by ignoring
the off-diagonal blocks. This provides the corrections to the parameters
and also an incomplete estimate of the overall covariance matrix. The
procedure can be iterated to account for the neglected terms.
The selection of this decomposition requires understanding of the
structure of the data and of the physics of the dynamical and measure-
ment processes. This paper discusses both aspects. We have found an
algorithm, based upon a sequence of decompositions, allowing to solve
for the geopotential harmonic coefficients up to a very high degree
and order with modest computational resources. We have tested such
an algorithm in a full scale simulation of the GOCE data processing,
including a data simulation step and a correction step. Our choices are
justified both by mathematical arguments, showing the correlations
and approximate symmetries which need to be controlled, and by the
results of the numerical tests.
1.1. Kinematic solution
The first decomposition of the problem can be obtained from the fol-
lowing considerations. The GPS receivers provide phase measurements
which can be processed to provide the spacecraft positions. To obtain
the best results in this Precision Orbit Determination (POD) stage,
the processing is usually performed with a reduced dynamical method
(e.g., Visser and van den IJssel, 2000): the orbit of the spacecraft is
determined only over a short arc, under a dynamical model including
empirical accelerations to be solved simultaneously. The empirical ac-
celerations absorb both the non gravitational perturbations and the
inaccuracies in the knowledge of the gravitational accelerations. This
process has been shown to be accurate to a few cm in all directions in
the spacecraft position over the entire mission.
gocesim2.tex; 13/04/2005; 9:23; p.2

Satellite gradiometry 3
This has two important implications: first, the data from the on
board accelerometers (components of the gradiometer) do not need
to be used in the POD; second, the GPS data do not provide useful
information on the gravity field, because the empirical acceleration mix
together the gravitational signal with the non gravitational one
1
. It fol-
lows that the normal matrix of the GOCE solution can be approximated
by a matrix with two diagonal blocks: the normal matrix of the POD
and the normal matrix for all the parameters to be solved by fitting
the gradiometer data, including the geopotential coefficients g and the
gradiometer calibrations `. Thus it is possible to solve the full problem
in two steps, the POD and the kinematic solution using the GPS orbit
without corrections. This first step of the problem decomposition has
been adopted by all the authors. We will discuss and test the quality
of the approximation done in neglecting the dependency of the gravity
field solution upon the spacecraft position errors.
1.2. Timewise solution
A second basic choice is the organization of the GOCE observations
in one of three possible ways. In a spacewise solution the data points
are considered by their position in space, in a reference frame rotating
with the solid body of the Earth; in this way the data are a sampling
of the gravity field over a geocentric sphere (more exactly a thin shell:
the orbit is almost circular). This choice has the advantage that most
of the gravitational signal to be determined is also organized spatially,
although there are time dependent signals due to tides and other de-
formations. In a timewise solution the data points are considered as
a discretized time series. This is a less natural way of looking at the
static part of the gravitational signal: indeed, each spherical harmonic
appears as a sum of signals with different frequencies (Kaula, 1966).
The advantage of the timewise methods is in the treatment of the
gradiometer calibrations. In a frequencywise solution the time series of
the gradiometer data are Fourier transformed into the frequency do-
main. Since the effect of each spherical harmonic can be represented by
a Fourier polynomial with good approximation, the fit can be directly
performed in the frequencies space. This method is the most convenient
in the mission design phase, because it allows to convert requirements
in the error spectrum of the gradiometer into error spectrum of the
recovered gravity coefficients.
1
There are other methods of data processing to extract gravity field information
from the GPS data, as used with CHAMP (Reigber et al., 2002), but they cannot
be pushed to the GOCE level of accuracy and resolution.
gocesim2.tex; 13/04/2005; 9:23; p.3

4 A. Milani et al.
These three approaches are equivalent for a perfect distribution of
data. With a spatial distribution uniform on a sphere and a time distri-
bution uniform over an unlimited time span there is a well conditioned
one-to-one correspondence between spherical harmonics, a linear sub-
space of the signal as function of time, and its discrete spectra. However,
such uniformities are impossible in real satellite geodesy missions. The
distinction of the spacewise approach has been well understood since
the early phases of GOCE (Rummel et al., 1993), but the implications
of the difference between the timewise and the frequencywise methods
have been underestimated.
A fundamental problem of all the satellite geodesy missions with
on board accelerometers of whatever type is that all such instruments
provide relative measurements
2
. Thus it is necessary to add a posteriori
calibration parameters to the list of parameters to be determined. The
instrumental calibrations are organized timewise: the accelerometer bi-
ases can be assumed to change smoothly with time, while if they were
organized spatially they would appear almost random.
The calibrations of the individual accelerometer channels can be
combined into calibrations for the gradiometer and calibrations for the
common mode measurements. The latter are obviously strongly corre-
lated with the empirical accelerations solved in the reduced dynamics
POD. The gradiometer calibrations can be determined neither from the
orbit nor from the common mode calibrations.
Several methods have been proposed to remove the effect of the
gradiometer errors, including calibration errors, from the data. In a
frequencywise method digital filters can be used; however, as pointed
out in Albertella et al. (2001), the frequency domains of the calibrations
and of the gravitational signal have some overlap, thus the procedure
has to be much more than a simple linear filtering. Moreover, the
frequencywise methods are very sensitive to data gaps (in time). In
a spacewise method, the calibration error can be partially removed
by averaging/smoothing techniques, but there is no guarantee that
systematic errors would be fully removed. It is clear that solving for
the calibration error is by far simpler within a timewise method.
Our method can be described as timewise kinematic, in that the
orbit is assumed (from the POD) and the predicted gradients are
computed as a time series and directly compared with the individual
observations. The least squares fit is linear; nevertheless it may require
iterations because the normal system of equations is too large to be
2
Absolute a priori calibration of an accelerometer is possible only at moderate
levels of accuracy (such as in inertial guidance systems). In flight calibration is
possible with dedicated devices, but to achieve the level of accuracy envisaged for
the GOCE measurements would be very difficult.
gocesim2.tex; 13/04/2005; 9:23; p.4

Satellite gradiometry 5
solved at once, as discussed in Section 4.3, but the number of necessary
iterations is small.
The separation of the calibrations from the gravity signal is the
second step of the problem decomposition. It is obtained by fitting
the calibrations to a linear combination of slowly varying base func-
tions, separately on each time interval: it is described in Section 2.
This allows the separation of the calibrations as local parameters from
the global ones describing the gravity field, as discussed in Section 3.
A third decomposition step is necessary to allow the full solution to
be computed with limited computing resources. With a method called
resonant decomposition we reorder the normal matrix of the g variables
in such a way that it also has a dominant block diagonal structure: this
is described in Section 4.
A timewise method is insensitive to data gaps organized by time
(e.g., instrument shutdown/communication failures), but all methods
are sensitive to data gaps organized spatially. We show in Section 5 that
the polar gaps resulting from the non polar orbit of GOCE introduce
an exact symmetry in the functional space of spherical harmonics. This
exact symmetry is broken in a finite spherical harmonics expansion,
but still holds as an approximate symmetry. Thus all data processing
methods have to be sensitive to the indetermination resulting from the
polar gaps. We validate the results of our numerical experiments in Sec-
tion 6 by showing that the main indetermination is the one unavoidably
resulting from the polar gaps.
2. Gradiometer Calibration
The observations include two independent components of the grav-
ity gradient. E.g., if they are sampled every 10 s for a measurement
time span of about 8 months
3
, the observations form a vector A with
' 4 × 10
6
components. We have performed a fit to the gradiometer
observations vector A solving for two parameter vectors: the gravity
harmonic coefficients g (e.g., 201
2
coefficients if the field is determined
up to degree and order 200) and the gradiometer calibration parameters
`. The main issue is the dimension of `. E.g., let us suppose a set of 2 cal-
ibration parameters had to be solved for each interval of ' 200 s, since
f
m
= 1/200 Hz is the lower frequency limit of the measurement band,
where the gradiometer data are most accurate (ESA, 1999, Section 8.1):
this would imply ' 2 × 10
5
calibration parameters. This large number
3
This sampling is meant to be the result of a preprocessing stage, in which
digital filtering is used to remove the noise at frequencies higher than the upper
limit f
s
' 1/10 Hz of the measurement band.
gocesim2.tex; 13/04/2005; 9:23; p.5

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Abstract: The removal of low-frequency errors is an important task in Gravity field and steady-state Ocean Circulation Explorer (GOCE) data processing, which suffers from a large number of errors. This paper analyzes the frequency characteristics of gradient data in the time domain and then derives a formula that can represent the relationship between degree and frequency. Moreover, two primary filtering strategies, namely the remove-restore method and the forward-backward filtering technique, are presented. The former can solve low-frequency signal loss during filtering, while the latter can resolve the phase-drift problem. The final result shows that the proposed formula can accurately show the relationship between the maximum cut-off frequency and degree of the spherical harmonic model. The proposed filtering method is proved to be effective in removing low-frequency errors while saving signals in the measurement bandwidth.

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TL;DR: In this article, the boundary value problem with the radial gradient of the gravity can be established on the satellite orbit, and a gravity field model is solved out from the BVP by using actual GOCE data.

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TL;DR: In this article, a new combined error model of the cumulative geoid height is proposed for the future GOCE Follow-On satellite system in respect of the preferred design of the matching measurement accuracy of key payloads comprising the gravity gradient and orbital position and the optimal selection of the orbital altitude of the satellite.
Abstract: Firstly, the new single and combined error models applied to estimate the cumulative geoid height error are efficiently produced by the dominating error sources consisting of the gravity gradient of the satellite-equipped gradiometer and the orbital position of the space-borne GPS/GLONASS receiver using the power spectral principle. At degree 250, the cumulative geoid height error is 1.769 × 10−1 m based on the new combined error model, which preferably accords with a recovery accuracy of 1.760 ×10−1 m from the GOCE-only Earth gravity field model GO_CONS_GCF_2_TIM_R2 released in Germany. Therefore, the new combined error model of the cumulative geoid height is correct and reliable in this study. Secondly, the requirements analysis for the future GOCE Follow-On satellite system is carried out in respect of the preferred design of the matching measurement accuracy of key payloads comprising the gravity gradient and orbital position and the optimal selection of the orbital altitude of the satellite. We recommend the gravity gradient with an accuracy of 10−13−10−15 /s2, the orbital position with a precision of 1-0.1 cm and the orbital altitude of 200-250 km in the future GOCE Follow-On mission.

3 citations

References
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01 Jan 1966
TL;DR: The theory of satellite geodesy writer by Why as discussed by the authors is a best seller book on the planet with excellent value and content is integrated with fascinating words and can be read online or download this book by right here.
Abstract: Have leisure times? Read theory of satellite geodesy writer by Why? A best seller book on the planet with excellent value and content is integrated with fascinating words. Where? Just below, in this website you could check out online. Want download? Obviously available, download them also here. Offered documents are as word, ppt, txt, kindle, pdf, rar, and zip. Whatever our proffesion, theory of satellite geodesy can be excellent resource for reading. Locate the existing documents of word, txt, kindle, ppt, zip, pdf, and rar in this site. You could absolutely read online or download this book by right here. Currently, never miss it. Our goal is always to offer you an assortment of cost-free ebooks too as aid resolve your troubles. We have got a considerable collection of totally free of expense Book for people from every single stroll of life. We have got tried our finest to gather a sizable library of preferred cost-free as well as paid files. GO TO THE TECHNICAL WRITING FOR AN EXPANDED TYPE OF THIS THEORY OF SATELLITE GEODESY, ALONG WITH A CORRECTLY FORMATTED VERSION OF THE INSTANCE MANUAL PAGE ABOVE.

1,355 citations


"A timewise kinematic method for sat..." refers background or methods in this paper

  • ...This is a less natural way of looking at the static part of the gravitational signal: indeed, each spherical harmonic appears as a sum of signals with different frequencies (Kaula, 1966)....

    [...]

  • ...This large number 3 This sampling is meant to be the result of a preprocessing stage, in which digital filtering is used to remove the noise at frequencies higher than the upper limit fs ' 1/10 Hz of the measurement band. gocesim2.tex; 13/04/2005; 9:23; p.5 would be a problem, not only for the…...

    [...]

  • ...When transformed in the orbital elements of the satellite the contribution of a particular harmonic (l,m) reads (Kaula, 1966) Ulm = GMe R⊕ l ∑ p=0 ∞ ∑ q=−∞ ( R⊕ a )l+1 F̄lmp(I)Glpq(e)Alm cos (vlmpq − ψlm) (6) where Me is the Earth mass, R⊕ its radius, a the semimajor axis of the satellite orbit, e…...

    [...]

Book ChapterDOI
01 Jan 1983
TL;DR: The European Space Agency (ESA) as mentioned in this paper is the successor to the European Launcher Development Organization and the European Space Research Organization, which were created under separate conventions concluded in 1962 as European bases for the execution of space activities and programs.
Abstract: This chapter describes the establishment and role of European Space Agency (ESA). The ESA is the successor to the European Launcher Development Organization and the European Space Research Organization, which were created under separate conventions concluded in 1962 as European bases for the execution of space activities and programs. ESA was created by the Convention for the Establishment of a European Space Agency, which was opened for signature in Paris on May 30, 1975. The Director General, who is appointed by the council, acts as ESA's chief executive officer and legal representative. He reports annually to the council, attends its meetings, and may make proposals but has no vote. The administrative structure beneath the Director General consists of function oriented Directorates for Administration, Applications Programmes, Spacecraft Operations, Scientific Programmes, and Space Transportation Systems, as well as the Technical Directorate.

637 citations

Journal ArticleDOI
TL;DR: In this paper, a new long-wavelength global gravity field model, called EIGEN-1S, has been prepared in a joint German-French effort, which is derived solely from analysis of satellite orbit perturbations, independent of oceanic and continental surface gravity data.
Abstract: [1] Using three months of GPS satellite-to-satellite tracking and accelerometer data of the CHAMP satellite mission, a new long-wavelength global gravity field model, called EIGEN-1S, has been prepared in a joint German-French effort. The solution is derived solely from analysis of satellite orbit perturbations, i.e. independent of oceanic and continental surface gravity data. EIGEN-1S results in a geoid with an approximation error of about 20 cm in terms of 5 × 5 degree block mean values, which is an improvement of more than a factor of 2 compared to pre-CHAMP satellite-only gravity field models. This impressive progress is a result of CHAMP's tailored orbit characteristics and dedicated instrumentation, providing continuous tracking and direct on-orbit measurements of non-gravitational satellite accelerations.

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"A timewise kinematic method for sat..." refers methods in this paper

  • ...1 There are other methods of data processing to extract gravity field information from the GPS data, as used with CHAMP (Reigber et al., 2002), but they cannot be pushed to the GOCE level of accuracy and resolution. gocesim2.tex; 13/04/2005; 9:23; p.3 These three approaches are equivalent for a perfect distribution of data....

    [...]

  • ...1 There are other methods of data processing to extract gravity field information from the GPS data, as used with CHAMP (Reigber et al., 2002), but they cannot be pushed to the GOCE level of accuracy and resolution. gocesim2.tex; 13/04/2005; 9:23; p.3 These three approaches are equivalent for a…...

    [...]

  • ...The analysis of the CHAMP and GRACE data should provide more than enough a priori information before the launch of GOCE....

    [...]

Frequently Asked Questions (1)
Q1. What are the contributions in "A timewise kinematic method for satellite gradiometry: goce simulations" ?

The authors have defined new algorithms for the data processing of a satellite geodesy mission with gradiometer ( such as the next European mission GOCE ) to extract the information on the gravity field coefficients with a realistic estimate of their accuracy. Second the authors use a new method to perform the necessary digital calibration of the gradiometer. The authors have shown that the principal components of the uncertainty correspond to harmonic anomalies with very small signal in the region where GOCE is flying ; these uncertainties can not be removed by any data processing method. The authors show that it is possible to solve for the harmonic coefficients up to degree 200 ÷ 220 with error to signal ratio ≥ 1, taking into account systematic measurement errors. By including a systematic error model, the authors have shown that the results are sensitive to spurious gradiometer signals at frequencies close to the lower limit of the measurement band.