GOCE data analysis: the space-wise approach and the first space-wise gravity field model

ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)

GOCE data analysis:GOCE data analysis:

the space-wise approach andthe space-wise approach and

the first space-wise gravity field modelthe first space-wise gravity field model

C.C. Tscherning, M. Veicherts

University of Copenhagen

F. Migliaccio, M. Reguzzoni, F. Sansò

Politecnico di Milano


COLLOCATIONCOLLOCATION

The space-wise approach

data

The main idea behind the space-wise approach is to estimate the spherical harmonic coefficients of the geo-potential model by exploiting the spatial correlation of the Earth gravitational field.

model coeffs

time dependent noise covariances (spectra)

spatial dependent signal covariance

[ E

2 ]

[degrees]0 1 2 3 4 5-10

-5

0

5

10

15

20

25



A unique collocation solution is computationally unfeasible due to the huge amount of data downloaded from the GOCE satellite.

A two-step collocation solution is implemented.

datalocal

gridding

localgridding

harmonicanalysis

harmonicanalysis

model coeffs

space-wise solver

spherical harmonics

mY

S

mm dYfa

T

),(),(4

1

spherical grid with local patches



Wiener filter

Wiener filter

datalocal

gridding

localgridding

harmonicanalysis

harmonicanalysis-

prior model

prior model

model coeffs

space-wise solver

In order to implement the local gridding:

- a prior model is used to reduce the spatial correlation of the signal

- a Wiener orbital filter is used to reduce the highly time correlated noise of the gradiometer

+

[Hz]10-6 10-4 10-2 100

10-4

10-2

100



Wiener filter

Wiener filter

datalocal

gridding

localgridding

harmonicanalysis

harmonicanalysis

along tracksynthesis


Wiener filter and GRF/LORF corrections


+

prior model

The procedure is iterated to:

- recover the signal frequencies cancelled by the Wiener orbital filter

- improve the rotation from gradiometer to local orbital reference frame

model coeffs

space-wise solver

-

prior model

+



Wiener filter

Wiener filter

datalocal

gridding

localgridding

harmonicanalysis

harmonicanalysis

prior model

Intermediate results that can be used for local applications:

- filtered data (potential and gravity gradients) along the orbit

- grid values at mean satellite altitude

model coeffs

space-wise solver

filtered data gridded data

+ -

prior model

+






The processed data

common mode accelerations

satellite attitude quaternions

gravity gradients

reduced dynamic orbits (for geo-locating gravity gradients)

kinematic orbits with their error estimates(for low-degree gravity field recovery)

• from the gradiometer:

• from the GPS receiver:

• external information such as Sun and Moon ephemerides or ocean tides for modelling tidal effects

Output data spherical harmonic coefficients and their error covariance matrix

GOCE quick look as prior model

other geopotential models as reference and to compute signal degree variances

• geopotential models:

Inp

ut

dat

a (f

rom

Nov

embe

r 20

09 to

mid

Jan

uary

201

0)


The data processing

• The data analysis basically consisted of three steps:

• Data preprocessing: outlier detection, data gap filling, unexpected behaviours tagging, etc.

• SST solution: to estimate the low degrees of the field (that are then removed from the SGG data)

• SST+SGG solution: to estimate the final model in termsof spherical harmonic expansion

• Error estimates are computed by Monte Carlo methods.In particular, few samples are used to control the evolution of the solution, while the final error covariance matrix is based on a larger set of samples.


Preprocessing philosophy

empirical cov. function

empirical cov. function

mean

collocation

time

Outliers and data gaps are replaced with values estimated by collocation.

The idea is to preserve the stochastic characteristics of the observations

time


Preprocessing example

• An example of data gap filling applied to the difference between kinematic and reduced dynamic orbits.

Cubic spline interpolation around the data gap to recover the long period behaviour

Collocation interpolation inside the gap to recover the stochastic behaviour of the signal


space-wise solver

SST solution philosophy

The energy conservation approach requires to:

• detect outliers and data gaps in the kinematic orbits;

• derive velocities from positions by least-squares interpolation;

• calibrate biases in the common mode accelerations;

• correct potential estimates from non-gravitational and tidal effects.

energy conservation

energy conservation

collocation gridding

collocation gridding

numerical integration

numerical integration

SST data SST model+

prior model

prior model


Estimated potential along track

Absolute differences w.r.t. EGM08

Predicted error standard deviation

1.704 m2/s2

predicted error rms (from MC)

1.523 m2/s2

empirical error rms(w.r.t. EGM08)

Non-stationary noise covariance is used in the gridding


Error calibration of the estimated potential

Error spectrum w.r.t. EGM08Predicted error spectrum

If we do not remove spikes we get this error pattern on the grid

low frequency zoom

• some periodical behaviours are not modelled (the highest with 2 cpr period)

• at very low frequency, the predicted spectrum is lower than the empirical one

Error calibration introducing prior information (EGM2008)

Remarks:

2 cpr period


Choice of prior model

full signal degree variances (estimated from EGM08)

quick-look predicted error degree variances

predicted residual degree variances if a rescaled quick-look model is used

Used for signal covariance modelling in the gridding

rescaled signal degree variances scale factor = 0.975

• A degraded version of the GOCE quick-look is used as prior model to reduce the influence on the final solution

Predicted (residual) degree variances

QUICK-LOOK

DEGRADEDQUICK-LOOK


Estimated potential on the grid

Estimated signal [m2/s2]

-83° < < 83°

empirical (w.r.t. EGM08) error rms

0.041 m2/s2

0.106 m2/s2

-90° < < 90°

latitude interval predicted error rms

0.026 m2/s2

0.135 m2/s2

Predicted error [m2/s2]


Estimated SST model

EGM08 degree variances

ITG-GRACE predicted error degree variances

Error degree variances w.r.t. ITG-GRACE

SST model predicted error degree variances

Above degree 60 the estimated model is the (degraded) quick look model, as corrections are negligible

Gravity gradients are needed for further improvements

GRACE

SST GOCE

Error degree variances


SST+SGG solution philosophy

Data gridding

FFT-1

complementaryWiener filter

Data synthesis along orbit

Wiener filter

FFT

+

FFT

LORF/GRFcorrection

test

Harmonic analysis

Spa

ce-wise solver

Final modelSST SGG

+

Energy conservation

Harmonic analysis

Data gridding

Space-wise solver

- -

Low degree model


Estimation error along the orbit

T

[m2/s2]TXX

[mE]

TXZ

[mE]

TYY

[mE]

TZZ

[mE]

0.091 2.4 4.4 4.6 6.0

Error rms of the Wiener filtered observations along the orbitEmpirical values from differences w.r.t. EGM08

T

[m2/s2]TXX

[mE]

TXZ

[mE]

TYY

[mE]

TZZ

[mE]

0.089 2.5 4.2 4.6 5.9

Error rms of the Wiener filtered observations along the orbitPredicted values from Monte Carlo simulations


Signal covariance modelling

m

m22 ˆˆ 22 ˆ)12(ˆ m

mmed

residual signal variances after removing SST-model

zoo

m

variances of degree 30log10 scale

Approximate degree variances are used for collocation

Single coefficient variances are used for error modelling by Monte Carlo


Estimation error on the grid

|| < 83°

empirical error rms

0.020 m2/s2

0.048 m2/s2

|| < 90°


0.016 m2/s2

0.026 m2/s2

|| < 83°

empirical error rms

2.64 mE

3.92 mE|| < 90°


1.44 mE

1.71 mE

empirical error computed w.r.t. EGM08

T predicted error [m2/s2] Trr predicted error [mE]


Estimated space-wise model

EGM08 degree variances

Error degree variances w.r.t. EGM08

Error degree variances w.r.t. ITG-GRACE

Predicted error degree variances

Differences w.r.t. EGM08 (d/o 150) = 8.4 cm Differences w.r.t. ITG-GRACE (d/o 150) = 5.1 cm

GOCE

GOCE vs EGM08

GOCE vs GRACE

Error degree variances

Geoid error [cm] Geoid error [cm]



Log10scale

Predicted error variances of the GOCE space-wise model



Assuming a mission length of 18 months, (9 sets of two months + some refinement)

one can expect an improvement of factor 3

in terms of accuracy, with the same spatial error distribution

10.86 cm

Predicted gravity anomaly error

3.03 mgal

Predicted geoid error

for || < 83° and up to d/o 200

Geoid error [cm] Predicted cumulative geoid error [cm]


Conclusions and future work

• The analysis of the first two months of GOCE data shows that the space-wise approach is able to provide good results.

• The main characteristic of the space-wise solution is to be a solution fully computed by collocation, with its pros and cons.

Furthermore, intermediate results such as filtered data along track and grid values at satellite level can be used for local applications.

• At medium-high degrees the solution is driven by GOCE data, while at very low degrees a dependence from prior models can be seen. This dependence will be removed in the next solutions.

• A new solution will be computed for a longer data period, that implies to optimally combine grids at mean satellite altitude based on different data subsets.

Documents

GOCE data analysis: the space-wise approach and the first space-wise gravity field model