1. ESA living planet symposium 2010 ESA living planet symposium 28 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 Geodesy University of Stuttgart

2. ESA living planet symposium 2010 Outline  GOCE gradiometry  Invariants representation  Linearization  Synthesis of unobserved GGs  Stochastic model  High performance computing  Summary

3. ESA living planet symposium 2010... observation tensor... gravitational tensor... centrifugal tensor... Euler tensor separation of centrifugal and Euler effects (star tracker, gradiometer) GOCE gradiometry gravitational gradients satellite

4. ESA living planet symposium 2010 z y x 0 p 1,x p 1,z p 2,x p 1,y p 2,y p 2,z high-sensitive axes accuracy: 10 -12 m s -2 low-sensitive axes accuracy: 10 -9 m s -2 ESA-AOES Medialab high-accurate determination of GOCE gradiometry

5. ESA living planet symposium 2010 standard approach: observations: V ij gradiometer orientation essential (tensor transformation) alternative approach: observations: J = J{V} = J{V ij } no orientation information required rotational invariants gradiometer frame (GRF) model frame (LNOF) Invariants representation

6. ESA living planet symposium 2010 system I system IIsystem III Waring formula Newton-Girard formula characteristic equation Invariants representation  rotational invariant  complete system (minimal basis) consists of three independent invariants

7. ESA living planet symposium 2010 invariants system of a symmetric, trace-free second-order tensor in 3  analysis of I 1 provides the trivial solution ( constraints)  non-linear gravity field functionals  gravitational gradients products  mixing of gravitational gradients Invariants representation

8. ESA living planet symposium 2010 Invariants representation reference gravitational gradients in the GRFreference invariants minor terms main term

9. ESA living planet symposium 2010 pros and cons of the gravitational gradients tensor invariants approach Invariants representation 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

10. ESA living planet symposium 2010 Invariants representation 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

11. ESA living planet symposium 2010 gravitational gradients: invariants: linearization (calculation of perturbations): V ij = U ij +T ij additional effort (per iteration): synthesis of U ij up to Linearization

12. ESA living planet symposium 2010 dependence on maximum resolution L ref of reference field GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 1 st it. analysis I 2, L ref = 0 ( GRS80 ) 1 st it. analysis I 2, L ref = 2 ( GRS80 ) 1 st it. analysis I 2, L ref = 200 ( OSU86F ) 2 nd it. analysis I 2 3 rd it. analysis I 2 degree Linearization V 33 GGM signal

13. ESA living planet symposium 2010 degree Linearization GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 1 st it. analysis I 2, L ref = 0 ( GRS80 ) 1 st it. analysis I 2, L ref = 2 ( GRS80 ) 1 st it. analysis I 2, L ref = 200 ( OSU86F ) 2 nd it. analysis I 2 3 rd it. analysis I 2 V 33 I 2, L ref = 0 GGM signal dependence on maximum resolution L ref of reference field

14. ESA living planet symposium 2010 degree Linearization GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 1 st it. analysis I 2, L ref = 0 ( GRS80 ) 1 st it. analysis I 2, L ref = 2 ( GRS80 ) 1 st it. analysis I 2, L ref = 200 ( OSU86F ) 2 nd it. analysis I 2 3 rd it. analysis I 2 V 33 I 2, L ref = 0 I 2, L ref = 2 GGM signal dependence on maximum resolution L ref of reference field

15. ESA living planet symposium 2010 degree Linearization GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 1 st it. analysis I 2, L ref = 0 ( GRS80 ) 1 st it. analysis I 2, L ref = 2 ( GRS80 ) 1 st it. analysis I 2, L ref = 200 ( OSU86F ) 2 nd it. analysis I 2 3 rd it. analysis I 2 V 33 I 2, L ref = 0 I 2, L ref = 2 I 2, L ref = 200 GGM signal dependence on maximum resolution L ref of reference field

16. ESA living planet symposium 2010 degree Linearization GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 1 st it. analysis I 2, L ref = 0 ( GRS80 ) 1 st it. analysis I 2, L ref = 2 ( GRS80 ) 1 st it. analysis I 2, L ref = 200 ( OSU86F ) 2 nd it. analysis I 2 3 rd it. analysis I 2 V 33 I 2, L ref = 0 I 2, L ref = 2 I 2, L ref = 200 2 nd iteration GGM signal dependence on maximum resolution L ref of reference field

17. ESA living planet symposium 2010 degree V 33 I 2, L ref = 0 I 2, L ref = 2 I 2, L ref = 200 Linearization GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 1 st it. analysis I 2, L ref = 0 ( GRS80 ) 1 st it. analysis I 2, L ref = 2 ( GRS80 ) 1 st it. analysis I 2, L ref = 200 ( OSU86F ) 2 nd it. analysis I 2 3 rd it. analysis I 2 2 nd, 3 rd iteration GGM signal dependence on maximum resolution L ref of reference field conclusions  small linearization error  fast convergence  numerically efficient  insensitive towards linearization field

18. ESA living planet symposium 2010  invariants representation requires the GGs with compatible accuracy (full tensor gradiometry)  GOCE: V 12 and V 23 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 V 12 and V 23 Synthesis of unobserved GGs  3322112312,,,VVVVV

19. ESA living planet symposium 2010 Synthesis of unobserved GGs

20. ESA living planet symposium 2010 Synthesis of unobserved GGs

21. ESA living planet symposium 2010 degree V 33 I 2, full tensor GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 2 nd it. analysis I 2 1 st it. analyse I 2, Syn. 1. It.: - 2. it. analyse I 2, Syn. 1. It.: - 1. it. analyse I 2, Syn. 1. It.: OSU86F 2. it. analyse I 2, Syn. 1. It.: OSU86F Synthesis of unobserved GGs impact of GGs synthesis on the estimation of geopotential parameters

22. ESA living planet symposium 2010 GGM signal DE-RMS values relative to GGM: analysis V 33 DE-RMS values relative to V 33 : 2 nd it. analysis I 2 1 st it. analysis I 2, syn. 1 st it.: - 2 nd it. analysis I 2, syn. 1 st it.: - 1. It. Analyse I 2, Syn. 1. It.: OSU86F 2. It. Analyse I 2, Syn. 1. It.: OSU86F degree V 33 I 2, full tensor GGM signal I 2, syn 1. It.: - I 2, syn. 1. It.: - Synthesis of unobserved GGs 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

23. ESA living planet symposium 2010 Stochastic model linearized invariant linearized overall functional model stochastic model of gravitational gradients correlations neglected

24. ESA living planet symposium 2010 Stochastic model linearized invariant linearized overall functional model stochastic model of gravitational gradients

25. ESA living planet symposium 2010 Stochastic model error propagation matrices of linear factors (Jacobian) conclusion  invariants variance-covariance matrix by products between the (diagonal) matrices of linear factors and the inverse gravitational gradients filter matrices

26. ESA living planet symposium 2010  “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 High performance computing

27. ESA living planet symposium 2010 High performance computing 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

28. ESA living planet symposium 2010 High performance computing 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

29. ESA living planet symposium 2010 High performance computing # CPUs148 design matrix assembly (%)0.8 NES assmebly (%)93.892.188.8 NES inversion (%)1.53.46.8 speed-up (-)13.87.2  good runtime scaling on shared memory architectures limited to ~8 CPUs  normal equations system assembly most time-consuming part  hybrid implementation established recently design matrix assembly NES assembly total runtime conclusion  parallelization performed successfully

30. ESA living planet symposium 2010  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

31. ESA living planet symposium 2010 invariants system of a symmetric second-order tensors in 3 Invariants representation

32. ESA living planet symposium 2010 Invariants representation reference gravitational gradients in the GRFreference invariants

33. ESA living planet symposium 2010 Invariants representation reference gravitational gradients in the GRF reference invariants

34. ESA living planet symposium 2010 Linearization

35. ESA living planet symposium 2010 linearized observation equation approximate solution x 0 calculation of V 12 and V 23 calculation of invariants solution of the linear system of equations new estimate final solution: i = i+1 truncation? yes no Synthesis of unobserved GGs