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Outline

Radiomagnetotelluric 2D forward and inversemodelling with displacement currents

Thomas Kalscheuer1 Laust B. Pedersen2

Weerachai Siripunvaraporn3

1Department of Earth SciencesETH Zürich, Switzerland

2Department of Earth SciencesUppsala University, Sweden

3Department of Physics, Faculty of ScienceMahidol University, Thailand

3/9/2009

Kalscheuer et al. Radiomagnetotelluric 2D modelling

Outline

Introduction

Goal: Inclusion of displacement currents in a 2D inversescheme of radiomagnetotelluric (RMT) data withsimplifications:

using vertically incident plane waves andassuming constant dielectric permittivity insubsurface.

Method: Modification of routines for forward and sensitivitycomputations of an existing 2D inverse code (REBOCC)that utilizes finite-difference approach.


Outline

Outline

1 TheoryRMT Field SetupGoverning equations

2 Synthetic examples1D forward modelling exampleEffect of oblique incidence2D forward modelling example2D inverse modelling example

3 Field example from Ävrö, Sweden

4 Conclusions


TheorySynthetic examples

Field example from Ävrö, SwedenConclusions

RMT Field SetupGoverning equations

RMT Field Setup I

Surface measurement of electric and magnetic field components.

Frequency range typically 10 to 300 kHz.

Primary signal from remote radio transmitters.Hence, plane-wave assumption.





RMT Field Setup II

Ηz Ηx

Ηy

air

Earthstrike parallelto x-direction

structure with anomalous electrical properties

z

xy

Εx

Εy

datalogger

Figure: RMT field setup.





Governing equations

Governing equations:

In frequency domain:

∇× E = −(iωµ0)H Faraday′s law (1)

∇× H = (σ + iωǫ) E Ampere′s law (2)

Conduction currents: σE.Displacement currents: iωǫE.

So far quasi-static assumption, i.e. displacement currentsnegligible: ωǫ << σ.

Assume ǫr = 5 and ρ = 1/σ = 10000 Ωm. Then, conduction anddisplacement currents are equally strong at f = 360 kHz.





Responses on 2D Earth for vertical incidence only (!):Impedance tensor Z:

[

Ex

Ey

]

=

[

0 Zxy

Zyx 0

] [

Hx

Hy

]

TE − modeTM − mode

(3)

givingapparent resistivities ρ

ija = 1

ωµ0|Zij |

2 and

phases φij = arg(

Zij)

.Vertical magnetic transfer function (VMT) B:

Hz = B · Hy . (4)

Given error level of 2 % on elements of Z, a homogeneoushalf-space and vertically incident plane waves, effect ofdisplacement currents on φ is above the error level at e.g.f = 15 kHz and ρ = 10000 Ωm or f = 170 kHz and ρ = 1000 Ωmfor ǫr = 5.




1D forward modelling exampleEffect of oblique incidence2D forward modelling example2D inverse modelling example

1D forward modelling example

8200

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8800

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9200

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9800

10000

10200

0 25 50 75 100 125 150 175 200 225 250

ρ a (

Ωm

)

frequency (kHz)

apparent resistivity

1D w/o. disp. curr.1D w. disp. curr.

2D FDA TM 27.5

30

32.5

35

37.5

40

42.5

45

47.5

0 25 50 75 100 125 150 175 200 225 250φ

(°)

frequency (kHz)

phase

1D w/o. disp. curr.1D w. disp. curr.

2D FDA TM

Figure: Analytic 1D solution and 2D FDA solution of apparent resistivity andphase for the TM-mode on the surface of a homogeneous half-space withρ = 10000 Ωm and ǫr = 5.





Effect of oblique incidence I

As only vertically incident plane waves are considered, effect ofoblique incidence needs to be estimated for typical 1D examples.

Typical depth section of later field example:

ρ1=600Ωm, εr=6

ρ3=600Ωm, εr=6

ρ2=30000Ωm, εr=6

h1=25m

h2=75m

x

z

yθi

Figure: 1D model; θi = 0 for vertical incidence.





Effect of oblique incidence II

0.984

0.988

0.988

0.9920.992

0.99

60.996

0.996

1.00

0

1.0001.0001.000

1.004

0

10

20

30

40

50

60

70

80

90

angl

e (°

)

104 105

frequency (Hz)

(a) relative amplitude

−0.40

−0.30

−0.30

−0.20−0.20 −0

.10

−0.10

0.00

0.00

0.100.10

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0

10

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90

angl

e (°

)

104 105

frequency (Hz)

(b) phase difference ()

Figure: Relative amplitude and phase difference of TM-mode impedancew.r.t. case of normal incidence (angle= 0) at frequencies between 10 and300 kHz.





2D forward modelling example

020406080

100120140

z (m

)

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800

y (m)

200 500 1000 2000 5000 10000 20000 50000ρ(Ω m)

Figure: Simple 2D model with a conductive block of ρ = 1000 Ωm in ahalf-space with a resistivity of ρ = 10000 Ωm and ǫr = 5 throughout. Receiverpositions are indicated by black triangles.





2000

4000

6000

8000

10000

12000

14000

ρ a(Ω

m)

0 200 400 600 800y (m)

(a) ρa of TM-mode

20

30

40

50

60

70

φ(°)

0 200 400 600 800y (m)

250.00 kHz100.00 kHz10.00 kHz

rebocc w. disp.

250.00 kHz100.00 kHz10.00 kHz

rebocc w/o. disp.

(b) φ of TM-mode

Figure: Comparison of FDA responses of block model with displacementcurrents (symbols) with FDA solution computed in quasi-static approximation(dotted lines with symbols).





−0.2

−0.1

0.0

0.1

0.2

Re(

B)

0.0 200.0 400.0 600.0 800.0y (m)

(1)

(2)(6)

(7)

(c) Re(B) of VMT

−0.2

−0.1

0.0

0.1

0.2

Im(B

)0.0 200.0 400.0 600.0 800.0

y (m)

250.00 kHz100.00 kHz10.00 kHz

rebocc w. disp.

250.00 kHz100.00 kHz10.00 kHz

rebocc w/o. disp.

(d) Im(B) of VMT

Figure: – continued





−5.0·10−5+0.0·10+0+5.0·10−5+1.0·10−4+1.5·10−4+2.0·10−4+2.5·10−4+3.0·10−4+3.5·10−4+4.0·10−4+4.5·10−4

Re(

j tot/E

x0)

(A/V

m)

0 100 200 300 400 500 600 700 800

y (m)

0

50

100

150

200

250

300

z (m

)

(a) Re (jx) with displacement currents

−5.0·10−5+0.0·10+0+5.0·10−5+1.0·10−4+1.5·10−4+2.0·10−4+2.5·10−4+3.0·10−4+3.5·10−4+4.0·10−4+4.5·10−4

Re(

j con

d/E

x0)

(A/V

m)

0 100 200 300 400 500 600 700 800

y (m)

0

50

100

150

200

250

300

z (m

)

(b) Re (jx) of quasi-static case

Figure: Real part of current density jx of the TE-mode at f = 250 kHz for thegeneral case with displacement currents (left) and the quasi-static case(right).





air

Earth

z

xy

current systemlaterallyhomogeneous

current systemlaterallyhomogeneous

Hh Hh

Hb

(1) (2) (3) (4) (5) (6) (7)

conductiveblock

−0.20

−0.10

0.00

0.10

0.20

0 100 200 300 400 500 600 700 800

Re(

B)

y (m)

(1) (2)(3)

(4) (5)(6) (7)

Figure: Real part of current density jx with real part of the magnetic field H(top) and the real part of the VMT response Re (B) (bottom).





2D inverse modelling example

Simple model with block in a two-layer host.

Synthetic data for TE- and TM-mode were computed andcontaminated with 5% Gaussian noise on the impedances.

Results from inversions with and without displacement currentsare compared.





0

50

100

150

200

250

300

z (m

)

0 50 100 150 200 250 300 350 400

y (m)

200

500

1000

2000

5000

10000

20000

50000ρ(Ω

m)

Figure: Model with a buried elongated block of a resistivity of 1000 Ωm in aresistive layer of 10000 Ωm underlain by a half-space of 500 Ωm and ǫr = 5throughout. Receiver positions are indicated by black triangles.





0

50

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300

z (m

)

0 50 100 150 200 250 300 350 400

y (m)

200

500

1000

2000

5000

10000

20000

50000

ρ(Ω m

)

Figure: 2D REBOCC inversion result of synthetic data from block model,displacement currents were allowed for; RMS = 1.04.





0

50

100

150

200

250

300

z (m

)

0 50 100 150 200 250 300 350 400

y (m)

200

500

1000

2000

5000

10000

20000

50000

ρ(Ω m

)

Figure: 2D REBOCC inversion result of synthetic data from block model,displacement currents were not allowed for; RMS = 1.95.




Field example from Ävrö, Sweden

RMT profile: 96 Rx and frequencies from 14 to 226 kHz.

Linde and Pedersen [2004] restricted data for inversion to lowerfrequencies up to 56 kHz.

Only determinant inversion due to 3D effects at ends of profile.

Comparison of models from inversions in quasi-staticapproximation and with displacement currents (with ǫr = 6throughout) for both data set restricted to lower frequencies andfull data set.

Comparison of models with seismic reflectors C and D by Juhlinand Palm [1999] and a normal-resistivity log of borehole KAV01by Gentzschein et al. [1987].




0

50

100

150

200

250

300

350

400

z (m

)

0 100 200 300 400 500 600 700 800 900

y (m)

CD

100 200 500 1000 2000 5000 10000 20000 50000 100000ρ(Ω m)

Figure: Model QL for low-frequency data set w/o. disp. curr.; RMS = 1.56.




0

50

100

150

200

250

300

350

400

z (m

)

0 100 200 300 400 500 600 700 800 900

y (m)

CD

KAV01

100 200 500 1000 2000 5000 10000 20000 50000 100000ρ(Ω m)

Figure: Model DL for low-frequency data set w. disp. curr.; RMS = 2.03.




0

50

100

150

200

250

300

350

400

z (m

)

0 100 200 300 400 500 600 700 800 900

y (m)

100 200 500 1000 2000 5000 10000 20000 50000 100000ρ(Ω m)

Figure: Model QF for full set of frequencies w/o. disp. curr.; RMS = 3.16.




0

50

100

150

200

250

300

350

400

z (m

)

0 100 200 300 400 500 600 700 800 900

y (m)

CD

KAV01

100 200 500 1000 2000 5000 10000 20000 50000 100000ρ(Ω m)

Figure: Model DF for full set of frequencies w. disp. curr.; RMS = 2.60.




Conclusions

1 If displacement currents are present,apparent resistivity and phase responses are smaller than inquasi-static approximation,tipper responses can show more distinct sign reversals, andquasi-static inverse models are prone to artefactual structures.

2 If displacement currents are accounted for, field example showsbetter agreement with

seismic reflectors anda normal-resistivity borehole log.

3 Even if restricted to lower frequencies quasi-static inversion mightgive strongly distorted models.


References

References I

B. Gentzschein, G. Nilsson, and L. Steinberg. Preliminaryinvestigations of fracture zones at Ävrö - Results from investigationsperformed July 1986 - May 1987. SKB Progress Report 25-87-16,SKB, 1987.

C. Juhlin and H. Palm. 3-D structure below Ävrö island fromhigh-resolution reflection seismic studies, southeastern Sweden.Geophysics, 64(3):662–667, 1999.

N. Linde and L. B. Pedersen. Characterization of a fractured graniteusing radio magnetotelluric (RMT) data. Geophysics, 69(5):1155–1165, 2004.


AppendixFit of model QFFit of model DF

A dark chapter ...

... with figures that should not be shown to a critical audience ...



Fit of model QF I

104

105

freq

uenc

y (H

z)

0 100 200 300 400 500 600 700 800 900profile (m)

−11

−9

−7

−5

−3

−1

1

3

5

7

9

11

rel. err. ρapp (%

/100)

Figure: Data fit of model QF to apparent resistivity of determinant.



Fit of model QF II

104

105

Per

iod

(s)

0 100 200 300 400 500 600 700 800 900Profile (m)

−11

−9

−7

−5

−3

−1

1

3

5

7

9

11

rel. err. φ (%/100)

Figure: Data fit of model QF to phase of determinant.



Fit of model DF I

Model DF fits

apparent resistivity rather well at all frequencies, some 3D effectsare visible at the beginning and end of profile and

phase quite badly at high frequencies, might be an effect of localsources (nuclear power plant), i.e. strong near-field effects inphase.



Fit of model DF II

104

105

freq

uenc

y (H

z)

0 100 200 300 400 500 600 700 800 900profile (m)

−11

−9

−7

−5

−3

−1

1

3

5

7

9

11

rel. err. ρapp (%

/100)

Figure: Data fit of model DF to apparent resistivity of determinant.



Fit of model DF III

104

105

Per

iod

(s)

0 100 200 300 400 500 600 700 800 900Profile (m)

−11

−9

−7

−5

−3

−1

1

3

5

7

9

11

rel. err. φ (%/100)

Figure: Data fit of model DF to phase of determinant.
