july 5, 2007, iugg2007, perugia, italy 1 determination of ground conductivity and system parameters...

July 5, 2007, IUGG2007, July 5, 2007, IUGG2007, Perugia, ItalyPerugia, Italy

1

Determination of ground conductivity and system parameters for optimal

modeling of geomagnetically induced current flow in technological systemsA. Pulkkinen1, R. Pirjola2, and A.

Viljanen2

1 UMBC/GEST at NASA/GSFC2 Finnish Meteorological Institute


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

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

Two preliminary results

• GIC at a site can be modeled to a good approximation as:

• Where

€

GIC = aEx + bEy

€

˜ E x,y = ± ˜ Z ˜ B y,x

μ0

System parameters

Surface impedance which depends on the ground conductivity

Quantities in spectral domain

(1)

(2)


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 €

G˜ I C = a˜ Z (ω,σ )

μ0

˜ B y − b˜ Z (ω,σ )

μ0

˜ B x (3)

€

Bx,y

€

σ

The main objective of this work


6

Determination of optimal system parameters

• From Eq. (3) we obtain by multiplication

• From these we can eliminate to obtain:

€

G˜ I C ˜ B x* = a

˜ Z

μ0

˜ B y ˜ B x* − b

˜ Z

μ0

˜ B x ˜ B x*

€

G˜ I C ˜ B y* = a

˜ Z

μ0

˜ B y ˜ B y* − b

˜ Z

μ0

˜ B x ˜ B y*

(4)

(5)

€

˜ Z


7


• Using the cross-correlation theorem these can be expressed in temporal domain as:

€

c ≡b

a=

˜ B y ˜ B x* − χ ˜ B y

2

˜ B x2

− χ ˜ B x ˜ B y*

(6)

€

χ =G˜ I C ˜ B x

*

G˜ I C ˜ B y* (7)

Only the ratio can be solved (this won’t affect the GIC modeling)


8


• If the functional form of the correlations is identical, these simplify to:

€

c =

< ByBx0

∞

∫ > [τ ]dτ − χ ' < ByBy0

∞

∫ > [τ ]dτ

< BxBx0

∞

∫ > [τ ]dτ − χ ' < BxBy0

∞

∫ > [τ ]dτ(8)

€

χ '=

< GICBx > [τ ]dτ0

∞

∫

< GICBy > [τ ]dτ0

∞

∫(9)


9


€

c =< ByBx > −χ ' '< By

2 >

< Bx2 > −χ ' '< BxBy >

(10)

€

χ ' '=< GICBx >

< GICBy > (11)


10


• 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.

€

˜ B x,y =1

iω

d ˜ B x,y

dt


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. €

˜ Z (ω,σ ) = μ0G˜ I C1

a ˜ B y − b ˜ B x(12)

We need to know these


12


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

€

ηi = G˜ I Ci − a˜ B yμ0

− b˜ B xμ0

⎛

⎝ ⎜

⎞

⎠ ⎟i

˜ Z

Index over datapoints

We want to solve this by minimizing some function of η

€

˜ H i


13


impedance• Then, we use Robust M-estimator

familiar from MT studies, which minimizes the loss function and where

€

ρ(η i) = {η i

2 /2 η i < η 0

η 0 η i −η 02 /2 η i ≥ η 0€

ρ(η i

i

∑ )

This gives less weight to the “outliers”


14


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 €

w(η i) = {1 η i < η 0

η 0 /η i η i ≥ η 0€

W = diag[w(η1),w(η 2),...]

€

˜ Z = ˜ H †W ˜ H ( )−1

˜ H †W G˜ I C( )

(13)

(14)


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:

€

η = ∂σ2

+ μ−1 W ˜ Z −W ˜ Z m 2

Gradient of the conductivity

“Measured” impedance

Forward model (slide 6)

(15)

Lagrange multiplier (smoothness)


16

Determination of optimal ground model; inversion

• Minimization of Eq. (15) leads to iterative solution:

• Where

€

σ n +1 = μ∂T∂ + W∇σ

˜ Z nm ⎛

⎝ ⎜ ⎞

⎠ ⎟T

W∇σ ˜ Z n

m ⎡

⎣ ⎢

⎤

⎦ ⎥

−1

W∇σ ˜ Z n

m ⎛ ⎝ ⎜ ⎞

⎠ ⎟T

W d

€

d = W ˜ Z − ˜ Z nm

+∇σ ˜ Z n

mσ n

⎛ ⎝ ⎜ ⎞

⎠ ⎟

Jacobian of the forward model

(16)


17

And that’s it!

• So, the objective was to look at Eq. (3):

€

G˜ I C = a˜ Z (ω,σ )

μ0

˜ B y − b˜ Z (ω,σ )

μ0

˜ B x

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).

€

˜ Z


18

Example: sites

Magnetic field

GIC


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

Example: surface impedance

€

ρa =˜ Z

2

ωμ0

€

φ=tan−1 Im ˜ Z

Re ˜ Z

⎛

⎝ ⎜

⎞

⎠ ⎟


21

Example: conductivity model


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

ValidationBlack - measured


24

Validation - error distribution

“Old” model


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

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.

july 5, 2007, iugg2007, perugia, italy 1 determination of ground conductivity and system parameters...

Documents

surface of

optimal modeling capability

overdetermined system

surface impedace

surface impedanceexample

derived system parameters

express gic

real gic data