1. 1 Modeling of the Geomagnetic Field at the Core Surface Bryan Grob Institute for Geophysics, ETH Zurich

2. 2 Departement/Institut/Gruppe Outline 1.Introduction and Aims 2.Data - the U.S. Maury Collection 3.Methodology 4.Results and Conclusions

3. 3 Departement/Institut/Gruppe 1. Introduction and aims

4. 4 Departement/Institut/Gruppe 1.1 Introduction  Modeling of the geomagnetic field at the CMB from historical data expansion of magnetic potential in spherical harmonics Solve resulting non-linear inversion problem  New data set used in geomagnetic modeling U.S. Maury Collection Analysis of new data set

5. 5 Departement/Institut/Gruppe 1.2 Aims  Deduction of field morphology at CMB in form of maps of  Detailed analysis of the U.S. Maury Collection

6. 6 Departement/Institut/Gruppe 2. Data - U.S. Maury Collection Woodruff et al., 2005

7. 7 Departement/Institut/Gruppe 2. Data - U.S. Maury Collection 2.1 Biographical sketch Wikipedia, 2009  Matthew Fontaine Maury (1806-1873)  Birth in Virginia, death in Lexington  Became midshipman at 19 => started studies of the sea and navigation  At the age of 33 => stagecoach accident => severe damage of knee and hip => from now on unable to undertake further sea voyages  1842: officer-in-charge of ‘Depot of Charts and Instruments‘ => intensive studies of old logbooks and charts => Brillant idea!!

8. 8 Departement/Institut/Gruppe 2.1 Biographical sketch Wikipedia, 2009  1843: publication of first wind and current charts (Wind and Current Chart of the North Atlantic, Sailing Directions and Physical Geography of the Seas and Its Meteorology)  1853: first International Hydrographic Conference in Brussels  ‚Pathfinder of the Seas‘

9. 9 Departement/Institut/Gruppe 2.2 The brilliant idea/data provenance  His idea: having a great fleet of volunteers, collecting data on their voyages  His strategy: distribute new logbooks for free against hand in of completely filled out ‘abstract log‘ sheet or in Maury‘s own words: ‘ You are expected in conformity with the agreement as per the foregoing receipt to send to the Observatory the abstract [logs] of every voyage you may make until the charts are completed. Vessels that fail to return abstract [logs] will forfeit their claims to the charts‘

10. 10 Departement/Institut/Gruppe 2.2 The brilliant idea/data provenance  His method: ABSTRACT LOG  Definition of ‘abstract log‘: added extra log sheet prior to actual log pages

11. 11 Departement/Institut/Gruppe 2.2 The brilliant idea Woodruff et al., 2005

12. 12 Departement/Institut/Gruppe 2.2 The brilliant idea/data provenance  Abstract logs contains information about: - date, hour - latitude, longitude - currents, pressure - temperature (air & water) - form/directoion of clouds - duration of fog/rain/hail/snow - magnetic variation (= declination (D)) oceanographic parameters => TOTAL: 78,409 D observations!

13. 13 Departement/Institut/Gruppe 2.3 Data analysis Organisation:

14. 14 Departement/Institut/Gruppe 2.3 Data analysis  Spatial distribution of complete Maury Collection (MC): outliers (total: 42 of 78,409) => 0.5 ‰ return route Inbound route

15. 15 Departement/Institut/Gruppe 2.3 Data analysis  Temporal distribution of complete Maury Collection (MC): Onset American Civil War (1861)

16. 16 Departement/Institut/Gruppe 2.3 Data analysis  Declination reported in two different units: Degrees & minutes (convential) Accuracy: 1/60 = 0.02° Points -32 point compass rose -Accuracy: 1/10 point = 1. 13° 32 points = 360° 1 point = 11.25° Wheeler, 2005

17. 17 Departement/Institut/Gruppe 2.3 Data analysis  Data testing against gufm1 (Jackson et al, 2000): normalized residuals: best estimate (gufm1) prediction error obersvation assigned prediction errors : Errors are assumed gaussian

18. 18 Departement/Institut/Gruppe 2.3 Data analysis  Test for accuracy of ‘point data‘ against gufm1 (Jackson et al, 2000): D [°] D [points] # # ‘point data‘ are retained!

19. 19 Departement/Institut/Gruppe 2.3 Data analysis  Test (complete MC) against gufm1 (Jackson et al, 2000) # (Jackson and Walker, 2000) What about Central Limit Theorem??

20. 20 Departement/Institut/Gruppe 2.3 Data analysis  Verification of Laplace distribution:

21. 21 Departement/Institut/Gruppe 2.3 Data analysis  Spatial residual distribution (MC): Residuals < 10 sigma (# obs 77,065) Residuals < 3 sigma (# obs: 65,662) Residuals > 20 sigma (# obs: 598) Residuals > 50 sigma (# obs: 50)

22. 22 Departement/Institut/Gruppe 2.3 Data analysis  Temporal residual distribution (MC):

23. 23 Departement/Institut/Gruppe 2.4 Final data sets 1820.mod1855.m od  2 criteria: 1)good data coverage 2)as far appart as possible

24. 24 Departement/Institut/Gruppe 2.4 Final data sets Distribution of 1820 final data set Distribution of 1855 final data set

25. 25 Departement/Institut/Gruppe 2.5 Reduction to epoch  Data are reduced to discrete epochs 1820 and 1855

26. 26 Departement/Institut/Gruppe 3. Methodology

27. 27 Departement/Institut/Gruppe 3.1 Boundary conditions/prelimnary assumptions  Magnetic vaccum outside Earth‘s surface  Mantle => insulator  Gaussian assumed errors

28. 28 Departement/Institut/Gruppe 3.2 Core field modeling  Magnetic potential expanded in spherical harmonics:  Spherical harmonic expansion truncated at L=14  Downward continuation of observations => spherical harmonics with large l are amplified  Non-linear inverse problem

29. 29 Departement/Institut/Gruppe 3.2 Non-linear inverse problem  Solved by damped least-squares parameter estimation  Goal: smoothest model for given fit to data  Strategy: find model vector m that minimizes both misfit to data and spatial norm  Non-uniqueness and instabibility resolved by regularization (norm, which measures model complexity)  Associated forward function:

30. 30 Departement/Institut/Gruppe 3.3 Non-linear inverse problem  Dissipation norm (Gubbins, 1975):  Norm in terms of least-squares: => regularization matrix  Errors:

31. 31 Departement/Institut/Gruppe 3.3 Non-linear inverse problem  Combination (=> objective function):  Iterative solution: data kernel matrix =

32. 32 Departement/Institut/Gruppe 4. Results and Conclusions

33. 33 Departement/Institut/Gruppe 4. Results and Conclusions  Effectiveness criterion:  Data subset statistics: Jackson et al, 2000: 1.16 Jackson et al, 2003: 1.97

34. 34 Departement/Institut/Gruppe 4. Results and Conclusions 1820 model red: flux out of core, blue: flux into core; color interval: 100  T

35. 35 Departement/Institut/Gruppe 4. Results and Conclusions 1855 model red: flux out of core, blue: flux into core; color interval: 100  T High intensity, high latitude flux patches Reversed flux patches Active Atlantic Quiet Pacific

36. 36 Departement/Institut/Gruppe 4. Results and Conclusions  Tracing of drifting patches: D1D1 D1D1 D3D3 D3D3 1820 1855 D2D2 D2D2 FeatureDrift rate ---------------------------- D 1 0.46°/a D 2 0.50°/a D 3 0.31°/a

37. 37 Departement/Institut/Gruppe 4. Results and Conclusions  Decay of axial dipole: Decay: 1.4% in 35a Slope: 13nT/a (ref. value: 15.46nT/a)

38. 38 Departement/Institut/Gruppe 4. Results and Conclusions  Prospects for model improvement: Data: 1)increase coverage 2)Correct for altitude Solution finding process: 1)Damping parameter 2)Minimization of L 2 rather than L 1 norm Model type: 1)Usage of time-dependent model