1. July 5, 2007, IUGG2007, Perugia, Italy 1 Determination of ground conductivity and system parameters for optimal modeling of geomagnetically induced current flow in technological systems A.Pulkkinen 1, R. Pirjola 2, and A. Viljanen 2 1 UMBC/GEST at NASA/GSFC 2 Finnish Meteorological Institute

2. July 5, 2007, IUGG2007, Perugia, Italy 2 Outline Motivation Two central preliminary results and formulation of the optimization problem. Determination of optimal system parameters Determination of optimal ground model. Example: application to real GIC data. Conclusions Old stuff New stuff

3. July 5, 2007, IUGG2007, Perugia, Italy 3 Motivation Measured GIC contains information about the subsurface geological structures. GIC impacts the performance of long technological conductor systems at the surface of Earth. GIC of both scientific and societal interest: optimal modeling capability of the phenomenon desired.

4. July 5, 2007, IUGG2007, Perugia, Italy 4 Two preliminary results GIC at a site can be modeled to a good approximation as: Where System parameters Surface impedance which depends on the ground conductivity Quantities in spectral domain (1) (2)

5. July 5, 2007, IUGG2007, Perugia, Italy 5 Formulation of the problem Let us combine Eqs. (1) and (2) to express GIC in spectral domain: From measured GIC and, determine optimal a, b and (3) The main objective of this work

6. July 5, 2007, IUGG2007, Perugia, Italy 6 Determination of optimal system parameters From Eq. (3) we obtain by multiplication From these we can eliminate to obtain: (4) (5)

7. July 5, 2007, IUGG2007, Perugia, Italy 7 Determination of optimal system parameters Using the cross-correlation theorem these can be expressed in temporal domain as: (6) (7) Only the ratio can be solved (this won’t affect the GIC modeling)

8. July 5, 2007, IUGG2007, Perugia, Italy 8 Determination of optimal system parameters If the functional form of the correlations is identical, these simplify to: (8) (9)

9. July 5, 2007, IUGG2007, Perugia, Italy 9 Determination of optimal system parameters (10) (11)

10. July 5, 2007, IUGG2007, Perugia, Italy 10 Determination of optimal system parameters Note the identity This means that the formulation above holds identically also for the time derivative of the magnetic field, which will be used below.

11. July 5, 2007, IUGG2007, Perugia, Italy 11 Determination of optimal ground model; surface impedance The surface impedance carries information about the subsurface conductivity structure. From Eq. (3) we obtain formally Note that because we divide the data set into segments, we have an overdetermined system in Eq. (12) and thus, for example, least-squares solution is needed. (12) We need to know these

12. July 5, 2007, IUGG2007, Perugia, Italy 12 Determination of optimal ground model; surface impedance For example due to non-Gaussian errors, a stable solution of Eq. (12) will require some “tricks”. First, express the problem in Eq. (12) as Index over datapoints We want to solve this by minimizing some function of 

13. July 5, 2007, IUGG2007, Perugia, Italy 13 Determination of optimal ground model; surface impedance Then, we use Robust M-estimator familiar from MT studies, which minimizes the loss function and where This gives less weight to the “outliers”

14. July 5, 2007, IUGG2007, Perugia, Italy 14 Determination of optimal ground model; surface impedance The Robust M-estimator is equivalent with iterative weighed least-squares where weights in matrix are given by And the solution for the surface impedace is (13) (14)

15. July 5, 2007, IUGG2007, Perugia, Italy 15 Determination of optimal ground model; inversion Due to ill-posed nature of the problem, constrained optimization has to be used. We use Occam’s (razor) inversion minimizing functional: Gradient of the conductivity “Measured” impedance Forward model (slide 6) (15) Lagrange multiplier (smoothness)

16. July 5, 2007, IUGG2007, Perugia, Italy 16 Determination of optimal ground model; inversion Minimization of Eq. (15) leads to iterative solution: Where Jacobian of the forward model (16)

17. July 5, 2007, IUGG2007, Perugia, Italy 17 And that’s it! So, the objective was to look at Eq. (3): These from correlating GIC and horizontal B (or dB/dt) by using Eq. (10). This from GIC, a, b and B using the robust scheme of Eqs. (13)-(14). And finally this from by using the Occam’s inversion of Eq. (16).

18. July 5, 2007, IUGG2007, Perugia, Italy 18 Example: sites Magnetic field GIC

19. July 5, 2007, IUGG2007, Perugia, Italy 19 Example: data for Oct 24 - Nov 1, 2003 Set a = -70 Akm/V and correlate GIC and dB/dt to obtain b = 40 Akm/V.

20. July 5, 2007, IUGG2007, Perugia, Italy 20 Example: surface impedance

21. July 5, 2007, IUGG2007, Perugia, Italy 21 Example: conductivity model

22. July 5, 2007, IUGG2007, Perugia, Italy 22 Validation Model GIC using the derived system parameters and the conductivity model for out-of-sample Bastille storm period of July 15-16, 2000. Compare to measured GIC.

23. July 5, 2007, IUGG2007, Perugia, Italy 23 Validation Black - measured

24. July 5, 2007, IUGG2007, Perugia, Italy 24 Validation - error distribution “Old” model

25. July 5, 2007, IUGG2007, Perugia, Italy 25 Conclusions New procedures for determining the optimal system parameters and the optimal ground model were introduced. The optimal modeling procedure reproduced measured GIC very accurately. The derived ground conductivity model can be used for geological interpretations.

26. July 5, 2007, IUGG2007, Perugia, Italy 26 Conclusions Pulkkinen A., R. Pirjola, and A. Viljanen, “Determination of ground conductivity and system parameters for optimal modeling of geomagnetically induced current flow in technological systems”, Submitted to Earth, Planets and Space, 2007. See also Session JAS004 poster no. 6142: “Long Period Magnetotellurics (MT) using Geomagnetically Induced Currents in Scottish Power Network” by A. McKay, A. Pulkkinen and A. Thomson.