ESA living planet symposium 2010 ESA living planet symposium 28 June – 2 July 2010, Bergen, Norway GOCE data analysis: realization of the invariants approach

ESA living planet symposium 2010

ESA living planet symposium28 June – 2 July 2010, Bergen, Norway

GOCE data analysis:realization of the invariants approach

in a high performance computing environment

Oliver Baur, Nico Sneeuw, Jianqing Cai, Matthias Roth

Institute of GeodesyUniversity of Stuttgart


Outline

GOCE gradiometry

Invariants representation

Linearization

Synthesis of unobserved GGs

Stochastic model

High performance computing

Summary


2V

V2

... observation tensor

... gravitational tensor

... centrifugal tensor

... Euler tensor

separation of centrifugal and Euler effects

(star tracker, gradiometer)

GOCE gradiometry

0, 332211

332313

232212

131211

VVV

VVV

VVV

VVV

VijV

gravitational gradients

satellite


z

y

x

0

p1,x

p1,z

p2,x

p1,yp2,y

p2,z

high-sensitive axes accuracy: 10-12 m s-2

low-sensitive axes accuracy: 10-9 m s-2

ES

A-A

OES

Med

iala

b

high-accurate determination of 13332211 ,,, VVVV

GOCE gradiometry


M2e0

*0standard approach:

observations: Vij

gradiometer orientation essential (tensor transformation)

alternative approach:

observations: J = J{V} = J{Vij}

no orientation information required

rotational invariants

gradiometer frame (GRF)

model frame (LNOF)

}0|,,{ *G3

G2

G1 eee

}0|,,{ M3

M2

M1 eee

M3e

M1e

G2e

G3e

G1e



system I

Vtr 1 J

22 tr VJ

33 tr VJ

system II

Vtr 1 I

222 tr -)tr (5,0 VVI

Vdet 3 I

system III

1

2

3

Waring formula

Newton-Girard formula

characteristic equation

0322

13 III

0) det()( 3 IVp


rotational invariant

complete system (minimal basis) consists of three independentinvariants

}{J}{J MG VV


invariants system of a symmetric, trace-free second-order tensor in 3

2

12332

13222

23112313123322113

223

213

212

233

222

2112

1

2

2

1

0

VVVVVVVVVVVVI

VVVVVVI

I

analysis of I1 provides the trivial solution ( constraints) non-linear gravity field functionals gravitational gradients products mixing of gravitational gradients




reference gravitational gradients in the GRF

reference invariants


pros and cons of the gravitational gradients tensor invariants approach


pros

scalar-valued functionals

independent of the gradiometer orientation in space

independent of the orientation accuracy

independent of reference frame rotations / parameterization

cons

non-linear observables

gravitational gradients required with compatible accuracy (full tensor gradiometry)

huge computational costs, iterative parameter estimation

stochastic model of invariants



pros

scalar-valued functionals

independent of the gradiometer orientation in space

independent of the orientation accuracy

independent of reference frame rotations / parameterization

cons

non-linear observables

gravitational gradients required with compatible accuracy (full tensor gradiometry)

huge computational costs, iterative parameter estimation

stochastic model of invariants

pros and cons of the gravitational gradients tensor invariants approach

linearization

high performance computing

error propagation

synthesis of inaccurate gravitational gradients


L

l

l

mlmlmij

s,c;r,,fr,,V0 0

)()(

22112 jiji VVI

gravitational gradients:

invariants:

3322113 jijiji VVVI

linearization (calculation of perturbations): Vij= Uij+Tij

)(

)(2 2

232313131212

333322221111ref222

ijT

TUTUTU

TUTUTUIII

additional effort (per iteration): synthesis of Uij up to LL ref

Linearization


dependence on maximum resolution Lref of reference field

GGM signal

DE-RMS values relative to GGM:

analysis V33

DE-RMS values relative to V33 :

1st it. analysis I2, Lref = 0 (GRS80)


1st it. analysis I2, Lref = 200 (OSU86F)

2nd it. analysis I2

3rd it. analysis I2

degree

Linearization

V33

GGM signal


degree

Linearization

GGM signal


analysis V33





2nd it. analysis I2

3rd it. analysis I2

V33 I2 , Lref = 0

GGM signal



degree

Linearization

GGM signal


analysis V33





2nd it. analysis I2

3rd it. analysis I2

V33 I2 , Lref = 0

I2 , Lref = 2

GGM signal



degree

Linearization

GGM signal


analysis V33





2nd it. analysis I2

3rd it. analysis I2

V33 I2 , Lref = 0

I2 , Lref = 2

I2 , Lref = 200

GGM signal



degree

Linearization

GGM signal


analysis V33





2nd it. analysis I2

3rd it. analysis I2

V33 I2 , Lref = 0

I2 , Lref = 2

I2 , Lref = 200

2nd iteration

GGM signal



degree

V33 I2 , Lref = 0

I2 , Lref = 2

I2 , Lref = 200

Linearization

GGM signal


analysis V33





2nd it. analysis I2

3rd it. analysis I2

2nd, 3rd iteration

GGM signal


conclusions small linearization error fast convergence numerically efficient insensitive towards linearization field


invariants representation requires the GGs with compatible accuracy (full tensor gradiometry)

GOCE: V12 and V23 highly reduced in accuracy

synthetic calculation of inaccurate GGs (forward modeling)

avoid a priori information to leak into gravity field estimate

minor influence

additional effort (per iteration): synthesis of V12 and V23


3322112312 ,,, VVVVV



2

1

223

213

212

233

222

2112

VVV

VVVI



2

1

223

213

212

233

222

2112

VVV

VVVI


degree

V33

I2 , full tensor

GGM signal

GGM signal


analysis V33


2nd it. analysis I2

1st it. analyse I2 , Syn. 1. It.: -

2. it. analyse I2 , Syn. 1. It.: -

1. it. analyse I2 , Syn. 1. It.: OSU86F

2. it. analyse I2 , Syn. 1. It.: OSU86F


impact of GGs synthesis on the estimation of geopotential parameters


GGM signal


analysis V33


2nd it. analysis I2

1st it. analysis I2 , syn. 1st it.: -

2nd it. analysis I2 , syn. 1st it.: -

1. It. Analyse I2 , Syn. 1. It.: OSU86F

2. It. Analyse I2 , Syn. 1. It.: OSU86F

degree

V33

I2 , full tensor

GGM signal

I2 , syn 1. It.: -

I2 , syn. 1. It.: -


impact of GGs synthesis on the estimation of geopotential parameters

conclusions full tensor

reconstructed no a priori

information needed no impact on overall convergence numerically efficient


Stochastic model

linearized invariant

3362352241331221112 TcTcTcTcTcTcI

336

235

224

133

122

111

2

2

2

Uc

Uc

Uc

Uc

Uc

Uc

linearized overall functional model

stochastic model of gravitational gradients

)(.

)(

)()()(

)(

33

12

3311121111

Tsym

T

,TT,TTT

D

D

CCD

TD

correlations neglected

nnnnn T

T

c

c

T

T

c

c

I

I

33

133

6

16

11

111

1

11

2

12

0

0

0

0


Stochastic model

)(0

)(

0)(

)(

33

12

11

T

T

T

D

D

D

TD

linearized invariant

3362352241331221112 TcTcTcTcTcTcI

336

235

224

133

122

111

2

2

2

Uc

Uc

Uc

Uc

Uc

Uc

linearized overall functional model

stochastic model of gravitational gradients

nnnnn T

T

c

c

T

T

c

c

I

I

33

133

6

16

11

111

1

11

2

12

0

0

0

0


Stochastic model

error propagation

TVV

TVV

VTVij

I

Tijij

)(...)()(

)()(

16

16

11

112

1

33331111

FJFJFJFJD

FFD

matrices of linear factors (Jacobian)

6,...,1,diag 1 kcc nkkk J

conclusion invariants variance-covariance matrix by products between

the (diagonal) matrices of linear factors andthe inverse gravitational gradients filter matrices


“brute-force” normal equations system “inversion”

splitting the computational effort on several CPUs

parallelization using OpenMP, MPI or OpenMP+MPI

computation platforms provided by the High Performance Computing Centre Stuttgart (HLRS)

NEC SX-9 (array processor)

12 nodes, 192 CPUs

TPP: 19.2 TFlops




shared memory systems

parallelization via OpenMP

block-wise design matrix assembly

successive normal equations system assembly

algebraic operations by BLAS routines

normal equations system “inversion” by Cholesky decomposition

LAPACK routines



distributed memory systems

parallelization via MPI

block-wise design matrix assembly

successive normal equations system assembly

block-cyclic data distribution

algebraic operations by PBLAS routines

normal equations system “inversion” by Cholesky decomposition

ScaLAPACK routines



# CPUs 1 4 8

design matrix assembly (%)

0.8 0.8 0.8

NES assmebly (%) 93.8 92.1 88.8

NES inversion (%) 1.5 3.4 6.8

speed-up (-) 1 3.8 7.2

good runtime scaling on shared memory architectures limited to ~8 CPUs

normal equations system assembly most time-consuming part

hybrid implementation established recently

design matrixassembly

NESassembly

totalruntime

conclusion parallelization performed successfully


the invariants approach is an independent alternative for SGG analysis to more conventional methods

independent of reference frame rotations and the gradiometer frame orientation in space

efficient linearization: small linearization error, fast convergence

full tensor gradiometry reconstruction by synthesis of unobserved gravitational gradients

approaches for the stochastic model handling of invariants

algorithms successfully implemented on high performance computing platforms

analysis of simulated GOCE data demonstrates the invariants approach to be a viable method for gravity field recovery

application on real data

Summary


invariants system of a symmetric second-order tensors in 3

V

VV

V

det 2

tr )tr (2

1

tr

21233

21322

223112313123322113

22223

213

2123322331122112

3322111

VVVVVVVVVVVVI

VVVVVVVVVI

VVVI




reference gravitational gradients in the GRF reference invariants



reference gravitational gradients in the GRF reference invariants


Linearization

2

1

223

213

212

233

222

2112

VVV

VVVI


linearized observation equation

approximate solution x0

calculation of V12 and V23

calculation of invariants

solution of the linear system of equations

new estimate iii xxx ˆˆˆ 1

final solution:

i = i+1

truncation?yes

no

ix̂


Documents

ESA living planet symposium 2010 ESA living planet symposium 28 June – 2 July 2010, Bergen, Norway GOCE data analysis: realization of the invariants approach