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Gravimetric Network Adjustments

Gravimetric Network Adjustments. J.-P. Barriot BGI MicroGravimetry School, Lanzarote, 23-28 October, 2005. Why gravimetric network adjustments ?. To determine time drifts of relative gravimeters (Scintrex: 0.5 mgal/day, Lacoste: 0.5 mgal/month)

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Gravimetric Network Adjustments

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  1. Gravimetric Network Adjustments J.-P. Barriot BGI MicroGravimetry School, Lanzarote, 23-28 October, 2005

  2. Why gravimetric network adjustments ? • To determine time drifts of relative gravimeters • (Scintrex: 0.5 mgal/day, Lacoste: 0.5 mgal/month) • To determine calibration factors (F: 10E-6 to 10E-3)

  3. Network Equations: Example of typical form 1 2 Network Equation: Two observations g1 and g2 at different places at times t1 and t2 Network Equation: Difference of these two observations

  4. Network Equations: Difference of the two observations at times t1 and t2 Network Equations: Difference of the two observations at times t1 and t2 (rewriting)

  5. Network Equations: Difference of the two observations at times t1 and t2 (rewriting)

  6. Network Equations: 3 observations with loop 1 and 1’ 2 3 Drift b  0.5 mgal/day Calibration F  10E-3

  7. But if we replace g1 and g2 by g1’ and g2’ in such a way that: Then: We cannot determine g, but g+cte, with cte arbitrary ==> rank deficiency in G

  8. But also: 1 and 1’ 2 3

  9. 1 and 1’ 2 3 ==> interplay between calibration F and time drift b ==> instability

  10. Usual Least-Squares ==>

  11. But: does not exist ! Because we cannot determine g, but g+cte, with cte arbitrary ==> rank deficiency in G Besides: ==> interplay between calibration F and time drift b ==> instability: problem ill-conditioned

  12. Solution 1: Adding Equations, for example: 1 and 1’ 2 3 Sufficient in principle, but ill-conditioning (interplay between F and b) VERY GOOD !

  13. Solution 2: Truncated inverse (singular values decomposition: SVD) Pb: choice of i

  14. Solution 2 (continued): Truncated inverse (singular values decomposition: SVD) Permitted range of the solution around zero, including gravity g, drifts b and calibration factor F • Choice of truncation level i: • norm(solution) not too small, not too big… • AND • - norm (residuals) not too small, not too big… GOOOOD !

  15. Solution 3: Tikonov approximate inverse Regularisation parameter Permitted range of the solution around zero, including gravity g, drifts b and calibration factor F

  16. Solution 3 (continued): Tikonov approximate inverse • Choice of parameter  : • norm(solution) not too small, not too big… • AND • - norm (residuals) not too small, not too big… GOOOOD !

  17. Solution 4: Iterative inverse Iterative process: Example:

  18. Solution 4 (continued): Iterative inverse But: MATHEMATICAL TRICK: REALLY BAAAAD!

  19. A small example, just to illustrate

  20. Network Geometry: station numbering Leg 1 Leg 2 1 2 3 4 8 7 6 5 9 10 11 12

  21. Network Geometry: true gravity in mgal Leg 1 Leg 2 7 10 10 7 7 14 14 7 7 10 10 7

  22. Network Geometry: measurement times along leg 1 in min Leg 1 30 60 90 0/360 210 180 150 120 240 270 300 330

  23. Network Geometry: measurement times along leg 2 in min Leg 2 0/360 150 180 330 30 120 210 300 60 90 240 270

  24. Network: true gravity in mgal along stations mgal station number

  25. Singular values: sv0: no added Eq. sv1: one added Eq. (g(1)=7) sv2: two added Eq. (g(1)=7 and g(6)=14) Log10 Singular value index

  26. Solution with added Eqs.: red crosses: true gravity values blue: no added Eq. cyan: one added Eq. (g(1)=7) green: two added Eq. (g(1)=7 and g(6)=14) mgal station number

  27. Solution with truncated SVD: red: true gravity values blue: trucation 4 green: truncation 8 cyan: truncation 12 mgal station number

  28. Solution with Tikonov: red: true gravity values green: alpha=1E-20 cyan: alpha=1E-3 blue: alpha=1E+3 mgal station number

  29. Solution with Tikonov: choice of regularisation parameter red: norm of the solution green: norm of residuals

  30. Example of iterated solution: red: true gravity values green: iterated solution mgal station number

  31. Determination of calibration factor F and time drifts b1 and b2 TRUE VALUES: F = 0.00112000000000 b1 = 0.0003472222222 b2 = 0.0002777777777 one added eq.: F= -1.00000000061203 b1=0.00000000000023 b2=-0.00000000000004 two added eq.: F=0.00101936318604 b1=0.00035321519057 b2=0.00028507612674 truncated 8: F=-0.00000004414763 b1=0.00000000000000 b2=0.00000000000000 Tikonov: F=-0.00000221038955 b1=0.00019957846307 b2=0.00012944781684 iterated: F=-0.56655269405289 b1=0.00015294430885 b2=0.00012343968305 Time drifts in mgal/min

  32. SUMMARY: Solution 1: Adding Equations: VERY GOOD Solution 2: Truncated Inverse: GOOD Solution 3 Tikonov Inverse: GOOD Solution 4 Iterative Inverse: VERY BAD

  33. THE END Reverend Lejay, founder of BGI

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