GOCE Gravity Models

Roland Pail

GOCE gravity models

Institute of Astronomical and Physical Geodesy

TU München

Herrsching, 31.05 - 04.06. 2010

GOCE Gravity Models

Contents

• Overview and comparison of models and

methodologies

• Performance evaluation: internal and external validation

• Future prospects

• Global models vs. gradients

GOCE Gravity Models

[ ]∑ ∑∞

= =

+

+⋅

=0 0

1

)sin()cos()(cos),,(n

n

m

nmnmnm

n

mSmCPr

R

R

MGrV λλθλθ

Spherical harmonic series

mGal

Overview & Methods

GOCE Gravity Models

Satellite observations Gravity field parameters

[ ]∑ ∑= =

+

+⋅

=max

0 0

1

)sin()cos()(cos),,(n

n

n

m

nmnmnm

n

mSmCPr

R

R

MGrV λλθλθ

Satellite orbit

i

ix

Va

∂∂

=≈

Gradiometry

ji

2

ijxx

VV

∂∂∂

=

Spherical harmonic coeff.∑ ∑= =

⋅=max

0 0

),,(),,(n

n

n

m

nmnm xrArV λθλθ

User:

• Given: coefficients → comp. of gravity field quantities

• Forward task: summation of terms of the series

Gravity field modelling:

• Given: grav. observations → comp. of coefficients

• Inverse problem

Global gravity field determinationOverview & Methods

GOCE Gravity Models

http://icgem.gfz-potsdam.de/ICGEM/ICGEM.html

Global Gravity Models

• Model coefficients (common format)

• Links to documentation (!)

Overview & Methods

GOCE Gravity Models

GOCE_DIR_R1

GOCE-only and GOCE combination modelsOverview & Methods

GOCE Gravity Models

GOCE

GOCE+GRACE

GOCE+GRACE

GOCE+GRACE+

+CHAMP+Terr.

GOCE

GOCE+GRACE

GOCE+GRACE+

+Terr.

GOCE+GRACE+

+CHAMP

GOCE+GRACE

GOCE-only and GOCE combination models

GOCE Gravity Models

Space-wise method

SPW

• observations distributed

in space

• Large least squares

collocation problem

• observations distributed

along the orbit

• Large least square adjust-

ment problem

Methods of gravity field modeling

Time-wise method

TIM

Direct method

DIR

• observations distributed

along the orbit

• Large least square adjust-

ment problem

• pure GOCE models

• full spectral range of

gradiometry

• combination models

• only measurement

bandwith

Overview & Methods

GOCE Gravity Models

Power spectral density (PSD) of

gradiometer error

MBW

GOCE gradiometer observationsOverview & Methods

GOCE Gravity Models

Gravity Signal (D/O200) and noise of gradiometer component VZZ

Time domainFrequency domain

(Power spectral density)

MBW

MBW @. Measurement bandwidth

GOCE gradients: signal & noiseOverview & Methods

GOCE Gravity Models

Signal (D/O 200) Gradiometer noise

Eötvös

Units

Signal + noise

→ Gradient observations are highly

correlated along the orbit track.

[Eötvös Units]

GOCE gradients: signal & noiseOverview & Methods

GOCE Gravity Models

( ) lll

111ˆ −−− ΣΣ= TT AAAx

TIM

• The full spectral range of the gravity gradients is used.

• Stochastic observation model is introduced for a proper weighting in

the adjustment procedure.

MBW

DIR

• Only gravity gradient signal content within MBW is used.

Differences: TIM vs. DIROverview & Methods

GOCE Gravity Models

This information must come from a different source !!!

(prior information, GRACE, 5)

Effect of disregarding signal outside MBWOverview & Methods

GOCE Gravity Models

Overview & Methods

Gravity anomaly signal Errors (limitation to MBW)

mGal

Effect of disregarding signal outside MBW

GOCE Gravity Models

There is no 1-to-1 correspondence of frequency at SH domain!

• One should not cut/add different gravity models!

• One should not simply replace a certain frequency range by

complementary gravity information !

Further consequencesOverview & Methods

GOCE Gravity Models

Performance evaluation:

internal and external validation

Perferm

ance

GOCE Gravity Models

GOCE

GOCE+GRACE

GOCE+GRACE

GOCE+GRACE+

+CHAMP+Terr.

GOCE

GOCE+GRACE

GOCE+GRACE+

+Terr.

GOCE+GRACE+

+CHAMP

GOCE+GRACE

GOCE-only and GOCE combination models

GOCE Gravity Models

Estim

ated coeffic

ient erro

rsPerfermance

GOCE Gravity Models

Estimated coefficient errors

TIM_R1

TIM_R2

TIM_R3

3

2

Perferm

ance

GOCE Gravity Models

Estimated geoid height errors

Release 1 (D/O 200) Release 2 (D/O 200)

cmRelease 3 (D/O 200)

Full covariance propagation (D/O 200):

• Geoid: 4.6 cm

• Grav. Anom. 1.3 mGal

Perferm

ance

GOCE Gravity Models

TIM models: external validation

-10 -5 0 5 10 mGal

Release 1Release 2Release 3

Comparison with EGM2008: Gravity anomaly differences at

degree/order 200 (100 km spatial wavelength)

→ no systematic effects; correct stochastic modelling

Perferm

ance

GOCE Gravity Models

Grav. anom. differences [mGal] to EGM2008 (D/O 200)Signal: gravity anomalies [mGal] (D/O 200)

South America North America

[mGal] TIM GOCO TIM GOCO

Signal 41.90 19.98

Diff. R1 8.21 8.10 2.39 2.30

Diff. R2 7.67 7.62 1.48 1.46

Diff. R3 7.59 7.54 1.11 1.10

Release 1Release 2Release 3

TIM models: external validationPerferm

ance

GOCE Gravity Models

•Map Source: Pavlis, N. K., Holmes, S., Kenyon, S., Factor, J. 2012. “The Development and Evaluation of the

Earth Gravitational Model 2008 (EGM2008).” Journal of Geophysical Research 117

EGM2008 Data Coverage & Coverage of GPS/Levelling Data

12.0% of Land

42.9% of Land

45.1% of Land

GPS/GPS/GPS/GPS/LEVELLING LEVELLING LEVELLING LEVELLING POINTSPOINTSPOINTSPOINTS

Germany

Saudi-Arabia

Brazil

GPS-Levelling Data as a Tool to Validate Global Gravity Field Models

Japan

External validation with GPS/levelling data

GOCE Gravity Models

RMS of Geoid Differences at GPS-Levelling Points(Omission Error estimated from EGM2008)

Germany

Acknowledgement: GPS levelling data for have been provided for validation purposes by BKG, Frankfurt/Main

External validation with GPS/levelling dataPerferm

ance

GOCE Gravity Models

RMS of Geoid Differences at GPS-Levelling Points(Omission Error estimated from EGM2008)

Japan

Acknowledgement: GPS levelling data for have been provided for validation purposes by the Japanese Geographical Survey Institute

External validation with GPS/levelling dataPerferm

ance

GOCE Gravity Models

RMS of Geoid Differences at GPS-Levelling Points(Omission Error estimated from EGM2008)

Saudi-Arabia

Acknowledgement: GPS levelling data for have been provided for validation purposes by King Abdulaziz City for Science and Technology KACS.

External validation with GPS/levelling dataPerferm

ance

GOCE Gravity Models

RMS of Geoid Differences at GPS-Levelling Points(Omission Error estimated from EGM2008)

Brazil

Acknowledgement: GPS levelling data for have been provided for validation purposes by Brazilian Institute of Geography and Statistics - IBGE, Directorate of

Geosciences - DGC, Coordination of Geodesy - CGED

External validation with GPS/levelling dataPerferm

ance

GOCE Gravity Models

GOCE – achievable accuracies

Geoid height accuracy [cm]

*) Dec. 2012*)

10.0

6.1

4.6

2.9 *)

Perferm

ance

GOCE Gravity Models

Gravity anomaly error [mGal]

*) Dec. 2012*)

2.9

1.8

1.3

0.8 *)

GOCE – achievable accuraciesPerferm

ance

GOCE Gravity Models

GOCE StatusFuture perspectives

• Level 1b data reprocessing

• GOCE orbit lowering

GOCE Gravity Models

PSD of gravity gradient trace

Level 1b data reprocessingFuture perspectives

GOCE Gravity Models

Pail et al. (2012), Stud. geophys. geod.

SGG-only: gravity anom. diff. to GOCO03S (d/o 180)

mGal mGal

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

Level 1b data reprocessing

σσσσ∆∆∆∆g [mGal] Original L1b Reproc. L1b Gain%

SGG-only 2.12 1.45 46

GOCE-only 1.51 1.32 15

GOCE+GRACE 1.25 1.10 14

Future perspectives

GOCE Gravity Models

Signal (D/O 200) Gradiometer noise

Eötvös

Units

[Eötvos Units]

GOCE gradients: signal & noise

ORIGINAL GRADIENTSREPROCESSED GRADIENTS

Signal + noise

Future perspectives

GOCE Gravity Models

GOCE Swan-song

• Lowering of the satellite improves the estimates for the high

degrees, and thus increases spatial resolution.

255 km

• -8.6 km: August 2012 → 1 cycle (62 days)

• -15 km: November 2012 → 1 cycle (67 days)

• -20 km: February 2013 → at least 1 cycle (TBD days)

• April 2013 ???

Future perspectives

GOCE Gravity Models

GOCE: lo

werin

g the altitu

de (1)

Future perspectives

GOCE Gravity Models

GOCE: lo

werin

g the altitu

de (2)

Future perspectives

GOCE Gravity Models

Gravity models vs. measured gradients

Some facts:

• Global gravity models are a globally optimum solution

(= optimum separation of signal and noise).

• We do our best to squeeze out as much information as possible

from the gradients to generate global models.

• With regional models maybe 5-10% added value (???)

but: signal-to-noise separation imposes an x00 % problem!

Global vs. gradients

GOCE Gravity Models

Gravity models vs. measured gradients

Some numbers: accuracy of VZZ gradients:

• full spectrum: ~700 mE (original)

~100 mE (reprocessed)

• MBW: ~3-5 mE → but: not full signal !

Pay attention: wir error measures such as 10 mE/√Hz you can

not directly work with (do not disregard the √Hz !!!)

∫=2

1

)(2

f

f

dffSσ

√√√√S(f)

Global vs. gradients

• synthetized from a global GOCE (+GRACE) model (Release 3)

(full spectrum): ~0.4 mE

GOCE Gravity Models

Resumée

� Currently, we have reached with global GOCE models the mission

goal of 1 mGal gravity anomaly error at 100 km wavelength

(proven by external validation).

� Gravity gradients at orbit altitude synthesized from R3 model have

an accuracy of 0.4 mE (d/o 200).

� Further improvements are expected due to data reprocessing.

� Due to lowering the satellite altitude, the spatial resolution can be

further increased.