Master thesis in interpretation of controlled-sourceradiomagnetotelluric data from Hallandsåsen

Martin HjärtenUppsala universitet

Institutionen för Geovetenskaper - Geofysik

June 4, 2007

Abstract

Controlled Source Tensor Magnetotelluric (CSTMT) ground measurements were exe-cuted on the Hallandsåsen horst where a major tunnel is under construction. The in-strument system EnviroMT are used for this purpose. The major research aspect of thisthesis has been to form an opinion of the effectiveness of the method by comparing theresults from the CSTMT survey with a prior investigation performed with the DC resis-tivity method. Another important part of this thesis has been to compile the basic andfundamental CSTMT and RMT theory, in a way that people outside the EM communityeasily can be introduced to the subject.

When comparing the different inversion models from the CSTMT and DC resistivity sur-veys one can see differences in the depth at which the conductors are resolved. In theCSTMT inversion models (TE+TM) there are two conductors that possibly can reach thedepth of the tunnel in construction. These conductors are not resolved at the deeperstructures in the DC resistivity models. Whether the conductors in the CSTMT inver-sions (TE+TM) truly extend to the depth at which they are modeled, or if they in deeperparts are artificial effects of regularisation in the inversion cannot be said for sure. Ac-counting for the low frequencies utilised in the TE mode, one has very strong argumentsthat the deep conductors seen in the CSTMT model are true.

The TE-mode models have shown to be much less affected by the complex problems ofnear field effects in comparison with the TM-mode models. The evidence of the nearfield effects is very prominent in the TM-mode phase, but in the phase of the TE-modeone can not see any such tendencies. However, one can see a discontinuity in the samepart of three profile lines which shows that the data is disturbed but not nearly as muchas in the TM-mode. The apparent resistivity seems to be over all less affected by the nearfield effects. In the apparent resistivity of the TE-mode, one can not discern any near fieldeffects at all. In the TM-mode, the apparent resistivity shows higher apparent resistivitythan the real apparent resistivity in the near field. To receive more information aboutthe deeper structures, lower controlled source frequencies were allowed in the TE-modethan in the TM-mode inversion models. The RMS in the TE-mode inversions has notbeen deteriorated, which is an another indication that the TE mode is not very disturbedby the near field effects.

The RMT inversion models are shown to be heavily biased in the deeper parts to whichthe RMT data are insensitive and regularization determining the outcome of the inver-sion. One can also see that regularisation is influencing the whole inversion model. Inthe shallow subsurface the inversion models should be same for CSTMT and RMT, butone can see differences in resistivity between the models.

The real induction arrows show features that are not as clearly displayed in either thephase or apparent resistivity. It seems that the real induction arrows are better at de-tecting lateral differences in conductivity in a more resistive media, than the phase andapparent resistivity.

Contents

1 Introduction 4

2 Geo-electromagnetic theory 6

2.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Ampere’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Primary and secondary fields . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Electric properties of rocks and fractures . . . . . . . . . . . . . . . . . . . . 7

2.4 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Plane wave assumption . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.2 The field equation in the frequency domain . . . . . . . . . . . . . . 9

2.5 The 1D-case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5.1 Skin depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.2 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5.3 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 2D-case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6.1 TE-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6.2 TM-mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.3 Impedance and Tipper vector . . . . . . . . . . . . . . . . . . . . . . 14

2.6.4 Strike direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6.5 Strike rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6.6 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Radio Magnetotellurics (RMT) 19

CONTENTS 2

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Transmitter direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 RMT field layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Estimation of the MT transfer functions . . . . . . . . . . . . . . . . . . . . 20

3.5 TSVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Controlled source tensor Magnetotelluric (CSTMT) 23

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Using a source in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 EnviroMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.1 The source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.2 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.3 Electric field sensors and Analogue filter (AF) box . . . . . . . . . . 25

4.3.4 Central processing unit . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Forward modeling and inversion 26

5.1 The forward model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2 Occam Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 REBOCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Interpretation of Magnetotelluric data 31

6.1 Interpretation of the phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.2 Interpretation of apparent resistivity . . . . . . . . . . . . . . . . . . . . . . 32

6.3 Interpretation of the tipper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.4 Near fields effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.4.1 Interpretation of the near fields effects . . . . . . . . . . . . . . . . . 33

6.5 Interpretation of the 2D-inversionmodels TE,TM,TE+TM and the determi-nant DET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Description of the area and Field data 35

7.1 Field measurement in situ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.2 Geological review of the Horst Hallandsåsen . . . . . . . . . . . . . . . . . 35

8 Result 37

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.2 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

CONTENTS 3

8.2.1 TM-mode COMB TSVD, Line1 . . . . . . . . . . . . . . . . . . . . . 38

8.2.2 COMB TSVD, Line3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.2.3 COMB TSVD, Figure Line2 . . . . . . . . . . . . . . . . . . . . . . . 39

8.3 Apparent resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.3.1 Apparent resistivity, Line1 . . . . . . . . . . . . . . . . . . . . . . . . 41

8.4 Real induction arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.4.1 Real induction arrows, Line1 . . . . . . . . . . . . . . . . . . . . . . 42

8.4.2 Real induction arrows, Line3 . . . . . . . . . . . . . . . . . . . . . . 42

8.4.3 Real induction arrows, Line2 . . . . . . . . . . . . . . . . . . . . . . 43

8.5 Strike analysis, Line1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8.6.1 COMB and RMT TSVD TM, Line1 . . . . . . . . . . . . . . . . . . . 46

8.6.2 COMB TSVD TM and TE, Line3 . . . . . . . . . . . . . . . . . . . . 50

8.6.3 COMB TSVD TE+TM and DET, Line2 . . . . . . . . . . . . . . . . . 52

8.7 DC-resistivity survey, Line1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.7.1 DC-resistivity survey, Line3 . . . . . . . . . . . . . . . . . . . . . . . 54

8.7.2 DC-resistivity survey, Line2 . . . . . . . . . . . . . . . . . . . . . . . 55

9 Conclusions 56

10 Acknowledgments 58

Chapter 1

Introduction

Controlled Source Tensor Magnetotellurics (CSTMT) is a frequency domain electromag-netic (EM) method used for ground measurements. The CSTMT method make use ofplane wave electromagnetic signals from remote high powered transmitters in combina-tion with the signals from a mobile controlled source. These signals propagates aboveand below the ground. If a conductor is present a time varying magnetic field inducesan electric field that drive currents within the conductor. The currents will generate theirown secondary EM-fields. A receiver will respond to the resultant of the primary fieldwhich propagates through the air, and the secondary fields propagating by diffusion fromthe conductor. Induction is not a process that happens instantaneously but take timetherefore the measured response will differ both in phase and amplitude relative to theprimary field. These differences between transmitted and received EM fields reveal thepresence of a conductor and gives information abouts it’s geometry and electrical prop-erties. All this is controlled by Maxwell’s equations, the constitutive relations, explainedin the next chapter. Some example of applications for a land based electromagnetic sur-vey is mineral exploration, ground water surveys, landfill surveys, location of geologicalfaults to mention some.

A practical part of this thesis has been to execute field measurements for the sake of locat-ing geological faults in an area where a major tunnel is under construction. The measure-ments took place on Flintalycka situated on the Swedish west cost on the HallandsåsenHorst, and the measurements was performed by my self and B Danielsen (Project assis-tant, Lund University).

The instrument used for the CSTMT survey is the EnviroMT, which is a new controlledsource magnetoutelluric system, still at the prototype stage. The EnviroMT was devel-oped in the frame of a EU project and it’s first test survey was made 1998. The systemis made for ground measurements of the electromagnetic field and components in thefrequency range of 1-250 kHz, and can be operated in two different modes the RadioMagnetoTelluric (RMT) and the Controlled Source TensorMagnetoTelluric (CSTMT). TheRMT method utilizes radio signals from a large number of powerful transmitters cover-ing the band from 14 kHz to 250 kHz. Whilst in the CSTMT mode one operate a mobilesource in the frequency band 1-100 kHz and combining the signals from both the sourceand the distant transmitters.

The target for the investigation is to model the fracture zones to a depth of approximately150 meters, which is the depth to the tunnel in construction. From previous investigation

5

it is known that the area is moderately resistive. This implies that one needs a controlledsource with the usage of it’s lower frequency range to be able to penetrate to the desireddepths.

Themain reasonwhy themeasurements has been executed on Flintalycka is that B.Danielsenhas performed a prior geophysical survey with the traditional DC resistivity method. Tobe able to form an opinion of the effectiveness of the new CSTMT method an importantpart of this thesis is to compare the results from the different methods.

Chapter 2

Geo-electromagnetic theory

2.1 Maxwell’s equations

Experimental evidence show that all electromagnetic phenomena obey four equations,(e.g. Simpson and Bahr, 2005)

2.1.1 Faraday’s law

∇× E = −∂B∂t

(2.1)

A time varying magnetic field produces an electric field. They relate to each other ac-cording to the right hand rule. Put your thumb in the direction of the magnetic inductionvector B, then your coupled fingers show the direction of rotation of the electric field E .

2.1.2 Ampere’s law

∇× H = J +∂D

∂t(2.2)

Here, an electric current and/or a time varying electric field generate amagnetic field.Theyalso relate to each other according to the right hand rule.

2.1.3

∇ · B = 0 (2.3)

The divergence of the magnetic induction is zero. This implies that the magnetic induc-tion B is continuous and that there are no single magnetic poles within the earth.

2.2. PRIMARY AND SECONDARY FIELDS 7

2.1.4

∇ · D = q (2.4)

The divergence of the dielectric displacement equals q, meaning that electric fields beginand end on electric charges.

SI unitsE: electric field (V/m)

B: magnetic induction (Tessla)H: magnetic field intensity (A/m)D: dielectric displacement (Columb/m2)J: electric current density (A/m2).q=electric charge density (C/m3)µ=magnetic permeability (S/m)ǫ=electrical permittivity (F/m)

ǫ = ǫrǫ0 (F/m)ǫ0 = electric permittivity of free space= 8.854* 10−12 (F/m)

ǫr ≈ 10 for common earth material (dimensionless)σ=electrical conductivity (H/m)

ω = angularfrequency = 2πft=time (s)

f=frequency (Hz)

Table 2.1: Electromagnetic entities

2.2 Primary and secondary fields

A transmitter generates a primary electromagnetic field which propagates above and be-low the ground. If a conductor is present a time varying magnetic field induces an elec-tric field according to Faraday’s law eq (2.1). The electric field will then drive a currentwithin the conductor according to Ohm’s law eq (2.7). The currents generates their ownsecondary EM-field according to Ampere’s law eq (2.2). A receiver will respond to theresultant of the primary field which propagates throw the air, and the secondary fieldspropagating by diffusion from the conductor. The measured response will differ in bothphase eq (2.35) and amplitude relatively to the primary field. The differences betweenthe transmitted and received EM fields reveal the presence of a conductor and provideinformation on its geometry and electric properties.

2.3 Electric properties of rocks and fractures

What we really can see in an Magnetotelluric investigation is, to which degree the Earthis able to conduct current. Most of the rock forming minerals have vary high resistivitiesand do not differ much from one bedrock to an other. So the conductivity of a bedrockis mostly controlled by other factors, this can be presence of conductive minerals likegraphite or sulphides. It can also be water filled pores and fractures in the bedrock. The

2.4. CONSTITUTIVE RELATIONS 8

bulk resistivity of fractured bedrock depends mainly on the conductivity of the pore wa-ter, the amount of water-filled fractures and their geometry. Magmatic rocks like granitecan differ from 1 million Ωm to 1000 Ωm depending on the water content, (e.g. Persson,2001).

2.4 Constitutive relations

The linear relation between the five vector fields are expressed in the constitutive rela-tions and follows by

D = ǫE (2.5)

B = µH (2.6)

J = σE. (2.7)

Where µ, ǫ, σ are material parameters and depends only on the properties of the Earth,see table 2.1.

2.4.1 Plane wave assumption

The electromagnetic fields interact with the resistivity structures of the earth. An electro-magnetic wave can be considered to be plane, if the wave length is much smaller thanthe penetration depth. (e.g. Bastani, 2001).If we now assume a plane wave with a surface amplitude E0 and a time dependency ofthe form eiωt, the electric field can be rewritten as,

E = E0eiωt. (2.8)

and the time derivative is,

∂E

∂t= iωE0e

iωt = iωE (2.9)

The same procedure for the B,H andD fields gives,

∂H

∂t= iωH0e

iωt = iωH (2.10)

∂D

∂t= iωD0e

iωt = iωD (2.11)

∂B

∂t= iωB0e

iωt = iωB (2.12)

2.5. THE 1D-CASE 9

Now substitute equation (2.12) with the use of (2.6) into equation (2.1), then (2.7) and(2.11) into equation (2.2), yields

∇× E = −iωµH (2.13)

∇× H = (σ + iωǫ)E. (2.14)

2.4.2 The field equation in the frequency domain

Maxwell’s equations are used to derive the fundamental electromagnetic field equation.In order to do that, first a reminder of a vector identity, for any vector F it follows that∇ × (∇ × F) = (∇ · ∇ · F) − ∇2F. Now take the curl of Faraday’s law, and applythe vector identity ∇ × (∇ × E) = −∇ × (iωB). The left side of the equation is then,∇× (∇× E) = (∇ · ∇ · E) −∇2E. The right side becomes after substituting eq (2.6) andeq (2.14) this, −∇× (iωB) = −iω∇× B = −iωµ(σ + iωǫ)E.

A fundamental assumption in the electromagnetic theory is that there exist no naturalelectric sources in the earth, this implies that the divergence of E can be set to zero. Inmost geophysical investigations σ >> ωǫ which allows us to neglect the displacementcurrents. For high resistivities and frequencies this approximation can not always bedone. When these approximations areused whats left is the electromagnetic field equa-tion, simplifies to

∇2E = iωµσE. (2.15)

Similar manipulation with the magnetic induction B gives,

∇2B = iωµσB. (2.16)

What we end up with are second order differential equations, with the 1D solution of theform (e.g.Boas,1983)

Ex = E1eiωt−qz + E2e

iωt+qz (2.17)

By = B1eiωt−qz +B2e

iωt+qz, (2.18)

here E1, E2, B1, B2 and q are constants.

2.5 The 1D-case

That a model is 1Dmeans that the conductivity varies only in the vertical direction (σ(z)).The coordinate system to be used is an ordinary orthogonal Cartesian coordinate systemwhere z is positive downwards. This means that there can be conductivity variations ithe z-direction, but no variations in the x-y plane. The simplest 1D earth model is a ho-mogeneous half space, where the conductivity is constant along all axes. Such a modelis going to be used here to derive some basic concepts as skin depth and impedance.

2.5. THE 1D-CASE 10

The earth does not generate electromagnetic energy but only dissipates or absorbs it (e.g.Simpson and Bahr, 2005). The electric field solution given by equation (2.17), should belocked upon as a down-going wave E1e

iωt−qz where the electromagnetic energy dissi-pates downwards, and a reflected wave E2e

iωt+qz where the energy dissipates upwards.This condition implies that E2 must be set to zero as there is no reflected wave in onlyone media. That is why in a homogeneous half space this simple case has the solution

Ex = E1eiωt−qz. (2.19)

If we now substitute eq (2.19) into eq (2.15), and account for the fact that the Laplacianonly operates on z, meaning that ∂

2E

∂x2 = ∂2E

∂y2=0. Therefore

∂2Ex∂z2

= q2E1eiωt−qz = q2Ex = iωµσEx, (2.20)

andq =

√

iωµσ. (2.21)

Rewriting eq (2.19) givesEx = E1e

iωt−√iωµσz. (2.22)

Where µ is approximated by µ0 which is the magnetic permeability of free space, andµ0 = 4π10−7ωsm−1.

2.5.1 Skin depth

An important parameter in electromagnetic investigation is the skin depth. The depth ofpenetration can be defined as the depth at which the amplitude is decreased by a factore−1 compared with it’s surface amplitude. Equation (2.22) states that the depth of pene-tration depends on frequency and the conductivity of the medium. Equation (2.22) givesthe amplitude at surface (z=0) as E1e

iωt, multiplied with the factor e−1 gives E1eiωt−1.

Setting this equal to eq (2.22) and solving for z gives the skin depth.

E1eiωt−

√iωµσz = E1e

iωt−1 (2.23)

z =1√iωµσ

(2.24)

Because the skin depth is a real quantity zskin= Re(z)

zskin ≈ 503

√

1

fσ(2.25)

The conductivity is the inverse of resistivity.

σ =1

ρ(2.26)

Rewriting equation (29) in terms of the resistivity gives

zskin ≈ 503

√

ρ

f(2.27)

2.5. THE 1D-CASE 11

Obviously, an EM-wave will penetrate deeper into a medium with higher resistivity atlower frequencies.

2.5.2 Impedance

The impedance is one of the two tools we got to estimate the resistivity structures of theearth, and was first used byWait 1954. The use of the impedance is a very important sim-plification, in the sense that one can disregard the source-moment and geometry of thesource. The impedance is also independent of the amplitude of the electric and magneticfields. Going back to equation (2.13), and expanding the curl of E gives

∇×E = (∂Ez∂y

− ∂Ey∂z

)x− (∂Ez∂x

− ∂Ex∂z

)y + (∂Ez∂x

− ∂Ex∂y

)z = −iωµ(Hx, Hy, Hz). (2.28)

In a homogeneous half-space there is no variation in conductivity except in the z direc-tion, therefore ∂

∂x =∂∂y=0, and

−∂Ey∂z

x+∂Ex∂z

y = −iωµ(Hx, Hy, Hz). (2.29)

Considering equation (2.29) component-wise gives Hz =0. The vector components havethe following relations to each other,

∂Ex∂z

= −iωµHy (2.30)

∂Ey∂z

= iωµHx. (2.31)

As a conclusion from equations (2.30) and (2.31) we find that the electric and magneticfields in a homogeneous half-space have only horizontal components, E= (Ex, Ey, 0) andH = (Hx, Hy, 0). The 1D solution eq (2.19) is then inserted into equations (2.30) and eq(2.31),

−√

iωµσEx = −iωµHy. (2.32)

The complex impedance Z is now defined (e.g. Bastani, 2001) as

Z =ExHy

=iωµHx√iωµσ

=

√

iωµ

σ= −Ey

Hx. (2.33)

Using eq (2.26) and solving eq (2.33) for the resistivity, gives

ρ =1

ωµ|ExHy

|2 =1

ωµ|Z|2. (2.34)

Resistivity is a constant quantity, however equation (2.34) shows that resistivity is a func-tion of frequency. Hence, the impedance increase as the square root of frequency.

2.6. 2D-CASE 12

2.5.3 Phase

Ward andHohmann (1991) showed by rewriting equations (2.33) and (2.34) the impedancephase for a homogeneous half space can be written as

φ1D = (ωµρ)1/2eiπ/4 = −iωµHy = (ExHy

)ei(ψE−ψH). (2.35)

Equation (2.35) states that the phase of the impedance is 45 corresponding to the phasedifference between Ex and Hy in a homogeneous half space.

2.6 2D-case

That a model is 2D means that the conductivity is a function of two axis (σ(y, z)), imply-ing that the conductivity is constant along one axis which is defined as the strike axis.Typically the coordinate system is defined such that, x goes in the strike direction, z ispositive downwards, and y is the profile direction. When modeling EM data the 2Dmodel approximation is most common. In an electromagnetic wave the vectors of theelectric and magnetic fields, are always perpendicular to each other. Therefore an electricfield parallel to the strike, induces a magnetic field perpendicular to the strike and in thevertical direction. Whilst a magnetic field parallel to the strike, induces an electric fieldperpendicular to the strike in the vertical direction. (e.g. Simpson and Bahr, 2005).

With this in mind, one can decompose Faraday’s and Ampere’s law into two differentmodes. The TE mode (transverse electric) or the E polarization mode, and TM mode(transversemagnetic) or B polarizationmode. Reminding us about that the current vectorJ has the same direction as the electric vector E according to Ohm’s law eq (2.36)

Jk = σEk (2.36)

Expanding the the curl of eq (2.13) and eq (2.14), (neglecting displacement currents in eq(2.14))

∇×E = (∂Ez∂y

− ∂Ey∂z

)x− (∂Ez∂x

− ∂Ex∂z

)y + (∂Ez∂x

− ∂Ex∂y

)z = −iωµ(Hx, Hy, Hz), (2.37)

∇× H = (∂Hz

∂y− ∂Hy

∂z)x− (

∂Hz

∂x− ∂Hx

∂z)y + (

∂Hz

∂x− ∂Hx

∂y)z = σ(Ex, Ey, Ez), (2.38)

makes it easy to overview how to define the different modes.

2.6.1 TE-mode

The definition of the TE-mode is very easy. The TE-mode describes a situation where thecurrent flows parallel to the strike, meaning that electric field also must be parallel to thestrike. An other way of expressing this, is that the TE-mode relates to the Ex-field see

2.6. 2D-CASE 13

figure 2.1. According to the above argument, one can define the TE-mode from eq (2.37)and eq (2.38). Note that in the strike direction x, ∂∂x must be zero.

∂Ex∂y

= iωµHz (2.39)

∂Ex∂z

= −iωµHy (2.40)

∂Hz

∂y− ∂Hy

∂z= σEx (2.41)

For instance, a change in the Ex field in the y-direction gives rise to a magnetic field inthe z-direction. Whilst a change in the Ex field in z-direction give rise to a magnetic fieldin the y direction. In conclusion with the given equations in the TE-mode we find thatE = (Ex, 0, 0) andH = (0, Hy, Hz).

y

Hz

Hy

Ez

Ey

HxEx

TE-mode TM-mode

Vertical contact

Medium 1 Medium 2

x

z

Figure 2.1: Illustration of TE and TM mode for a simple 2D-model.

2.6.2 TM-mode

The definition of the TM-mode is on the other hand describing the situation when themagnetic field is parallel to the strike. With other words the TM-mode is related to theHx-field see figure 2.1.

According to the above argument how the EM fields must relate to each other one candefine the TM-mode from eq (2.37) and eq (2.38).

∂Hx

∂y= −σEz (2.42)

∂Hx

∂z= σEy (2.43)

∂Ez∂y

− ∂Ey∂z

= −iωµHx (2.44)

In conclusion with the given equations in the TM-mode we find that E = (0, Ey, Ez) andH = (Hx, 0, 0) in the TM-mode.

2.6. 2D-CASE 14

2.6.3 Impedance and Tipper vector

The interpretation of Magnetotelluric data is focused on impedance and the tipper vectordata. It was shown by Cantwell (1960) that there exist a linear relationship between theelectric field components (Ex, Ey) and themagnetic field components (Hx, Hy). Moreoverthere also exist a linear relationship between the magnetic field component (Hz) and themagnetic field components (Hx, Hy). This for any given conductivity distribution, andare defined as follow (e.g. Bastani, 2001),

(

Ex

Ey

)

=

(

ZxxHx + ZxyHy

ZyxHx + ZyyHy

)

(2.45)

Hz = T

(

Hx

Hy

)

. (2.46)

The complex impedance tensor Z, is a 2 × 2 matrix and is defined as

Z =

(

Zxx Zxy

Zyx Zyy

)

. (2.47)

The complex tipper vector T, is defined,

T = (A,B). (2.48)

In a Magnetotelluric investigation it is the impedance Z, and the tipperT, one attempt toestimate, these parameters contains the electric and magnetic properties of the Earth.Note that Zxx, Zxy, Zyx, Zyy, A,B, all are complex numbers, and that each Zij have amagnitude and a phase. The impedance phase is an effect of that induction is not aprocess that happens instantaneously but take time (e.g. Simpson and Bahr, 2005). Theimpedance tensor Z, and the tipper vector T, are also known as theMagnetotelluric trans-fer functions.

Now the TE and TM-mode explains how the electric and magnetic fields relate to eachother see figure 2.1. The equations in TE-mode states that Ex, relates only toHy andHz .Look at eq (2.45), note that Ex, only relate to Hx and Hy. Hx is not zero, it has just norelation to Ex hence Zxx must be zero. Similar arguments for Zyy using the equations inTM-mode, the negative sign in eq (2.49) is related to eq (2.40) and eq (2.43). Concludingthat Zyy, Zxx is zero and rewriting the impedance tensor for the 2D-case to

Z =

(

0 Zxy

−Zyx 0

)

. (2.49)

According to eq (2.41) Hz only relate to Hy. Therefore the tipper vector for the 2D-casehas the following form

T = ( 0, B ). (2.50)

2.6. 2D-CASE 15

For a 1D Earth-model there is of course no strike direction, eq (2.33) and eq (2.29) givethe solution for the impedance Z, and the tipper T, for the 1D case

Z =

(

0 Z

−Z 0

)

(2.51)

T = ( 0, 0 ). (2.52)

For a 3D world there are no simplifications of the impedance tensor eq (2.47) and tippervector eq (2.48), and all the components have to be solved.

The general 2D expressions for the apparent resistivity and impedance phase in the the2D-case are given by the following equations (e.g. Simpson and Bahr, 2005),

ρa,ij(ω) =1

µω(Zij(ω))2 (2.53)

φij = arctanIm(Zij)

Re(Zij). (2.54)

2.6.4 Strike direction

If the horizontal field components are measured parallel and vertical to the strike direc-tion the off diagonal elements of the impedance tensor is zero for a 2D structure. Dueto noise and deviations from two-dimensionality thees elements are not really zero inpractise. The most common method to find the strike direction is done by minimizing(Zxx − Zyy)

2 with respect to the angel in the rotated coordinate system (e.g. Bastani,2001). However in this thesis an other method has been used. The above method hasbeen shown to be strongly influenced by local inhomogeneities and local strikes. Whichgive rise to bias results when it’s only the regional strike that is of interest.

Following the Magnetotelluric strike rules article Zhang et al. (1987), inhomogeneitiesand local strikes exist in the subsurface, corresponding to higher frequencies. The re-gional strike is to be found in the deeper structures related to the lower frequencies. Itis shown that the impedance tensor can be decomposed in one distorted and one undis-torted part as

Z ≈ (I + Ph)Z0. (2.55)

I= identity matrixPh = distortion tensor, which depends on the local strike and subsurface inhomogeneities.Z0 = The undisturbed impedance tensor.

In direction of the regional strike the impedance tensor can be expressed as

Z =

(

βZyx Zxy

Zyx γZxy

)

. (2.56)

2.6. 2D-CASE 16

Here Zxx = βZyx and Zyy = γZxy. β and γ are local distortion parameters in the formof real functions which are independent of frequency and dependent on station. Theregional strike is found by rotating Z from 0 to 180o and determining the angle for whichβZyx and γZxy approximate the diagonal elements Zyx and Zxy best. For all tested angles(strike directions), β and γ are found by minimizing the target function

Q = Q(β, γ, φ) = (Z∗xx − βZ∗

yx)2 + (Z∗

yy − γZ∗xy)

2. (2.57)

Where, φ = The try out angel between the profile line and the assumed regional strike.Z∗xx = Z

∗xx = field data

Z∗yx = Z

∗yy = data kernel

β and γ = model parameters

By taking the the least square solution with respect to β and γ will yield an estimate to βand γ.

β =ZT∗xxZ

∗yx

|Z∗yx|

(2.58)

γ =ZT∗yy Z

∗xy

|Z∗xy|

(2.59)

By trying different values of φ in eq (2.57) one chooses the best φ that minimizes Q. Inan ideal case (not real) if φ is correctly estimated, Q would be identically equal to zero.Asterisk denote complex conjugation and T transposition.

2.6.5 Strike rules

When measuring in practise one usually has an idea about the strike and attempts to putthe profile-lines perpendicular to the assumed strike. After an investigating of the strikedirection e.g with the Q-method one might find, that the assumed strike was not verysatisfactory. If that is the case it is not difficult to apply a rotation matrix to adjust themeasured fields to the fields that would be measured if the profile direction was chosencorrectly.

If the electromagnetic components measured in field is expressed as (Hx, Hy, Hz, Ex, Ey),and the electromagnetic components in the rotated coordinate system is (H ′

x, H′y, H

′z, E

′x, E

′y).

A rotation matrix R relates the components in the measured coordinate system to thecomponents in the rotated coordinate system, like this (e.g. Bastani, 2001)

(

Hx

Hy

)

= R

(

H ′x

H ′y

)

(2.60)

(

Ex

Ey

)

= R

(

E′x

E′y

)

(2.61)

2.6. 2D-CASE 17

Assumed strike x

Real strike x’

Profile line y

y’

x’

ϕ

a

a’

Figure 2.2: Illustration of a strike scenario where the real strike has an angel ϕ, to theassumed strike. The station distance is denoted a in the measuring coordinate system(x,y), and a’ in the rotated coordinate system (x’,y’). The dashed lines are perpendicularto y’.

The rotation matrix R is defined as

R =

(

cos(ψ) − sin(ψ)

sin(ψ) cos(ψ)

)

(2.62)

andHz = H ′

z (2.63)

In the measuring coordinate system, we have

(

Ex

Ey

)

= Z

(

Hx

Hy

)

(2.64)

(

Hz

)

= T

(

Hx

Hy

)

(2.65)

Substituting eq (2.60) and eq (2.61) into eq (2.64), and solving for Z’ gives,

R

(

E′x

E′y

)

= ZR

(

H ′x

H ′y

)

(2.66)

The impedance tensor in the rotated coordinate system is now given by (T denotes thetransposition of Z)

Z′ = RTZR. (2.67)

2.6. 2D-CASE 18

Substituting equation (2.60) and (2.63) into eq (2.65) and solving for T’ gives the tipper inthe rotated coordinate system,

T′ = TR. (2.68)

The station distance in the new coordinate system needs to be change as well, (figure 2.2)

a′ = cos(ϕ)a (2.69)

2.6.6 Determinant

Whenmeasuring on top of a 2D structure the strike angel ϕmust be known or calculated.Luckily there are mathematical operators for making the impedance tensor independentof the strike direction. One of them is the determinant of the impedance tensor. This canbe shown by taking the determinant of eq (2.67).

Det(Z′) = Det(RT ) Det(Z) Det(R) = ZxxZyy − ZxyZyx (2.70)

Equation (2.70) shows that the impedance tensor is independent of the angle ϕ, and there-for invariant to rotation. Pedersen and Engels (2005) show how to use the determinant ofthe impedance tensor for interpretation of Magnetotelluric data. Following their deriva-tion one sets

Det(Z′) = Z2DET . (2.71)

To get the same units as eq (2.70). For a 2D case it follows that

ZDET =√

−ZxyZyx, (2.72)

because both Zxy and Zyx are complex numbers they can be rewritten as

Zxy = |Zxy|eiφxy (2.73)

Zyx = |Zyx|eiφyx . (2.74)

Substituting eq (2.73) and eq (2.74) into eq (2.72) gives the solution

ZDET =√

|Zxy||Zyx|e1

2i(φxy+φyx) (2.75)

From eq (2.75) one can directly transform ZDET to apparent resistivity and impedancephase, ρappDET = 1

2(ρappxy + ρappyx ) and φDET = 12(φxy + φyx). According to eq (2.49), Zxy

correspond to the TE-mode and Zyx to TM-mode, whereby the 2D determinant can beconsidered as the arithmetic mean of TE and TM mode data.

Chapter 3

Radio Magnetotellurics (RMT)

3.1 Introduction

In most parts of the world, a large number of radio-transmitters can be used for geo-physical purposes. The RMT technique utilizes radio signals covering the band from 14kHz to 250 kHz.These transmitters provide excellent far-field or plane wave signals, thatalso have very good signal to noise ratios and are used to estimate the Magnetotellurictransfer functions equation (2.47) and (2.48). At a distance from the transmitter the fieldcomponents consist of a ground wave, sky wave and a space wave. The dominant com-ponent is the ground wave which is guided along the air-ground interface, (e.g. Persson,2001). The wavelength of the electromagnetic field ranges from 20 km at 15 kHz to 1 kmat 300 kHz. The skin depths for a homogeneous half space of 1000 Ωm would be 130mand 30m for the corresponding frequencies according to equation (2.27). This means thatthe wavelength can be considered to be very large compared with the penetration depth,which is the plane wave assumption (Pedersen et al., 2006).

3.2 Transmitter direction

The powerful transmitters used in RMT method are coupled to the ground either as ver-tical or horizontal electric dipoles. An EM-wave from a transmitter like that will inducean electric field in the direction of the wave propagation, see figure 3.1. For an ideal 2DEarth-model the linear relationship between the electric field components (Ex, Ey) andthe magnetic field components (Hx, Hy) are reduced to

(

Ex

Ey

)

=

(

ZxyHy

ZyxHx

)

, (3.1)

according to eq (2.45) and (2.49). The TE-mode is related to the Ex field, and the the TM-mode is related to the Hx field. Means that a distant transmitter in the direction of thestrike will give rise to only electric polarization or TE-mode. Whilst a distant transmitterin a direction of the profile line, will give rise to only magnetic polarization or TM-mode(figure 3.2). Of course one utilizes transmitters independent of transmitter direction. Atransmitter that is not in the strike or profile line direction will give rise to electric and

3.3. RMT FIELD LAYOUT 20

magnetic component, of both TM-mode and TE-mode this to equation (3.1).

Strike x

Profile y

Ex

HyHx

Ey

TE-mode

TM-mode

Direction of wave propagation

Transmitter 1.

Transmitter 2.

Figure 3.1: Illustration how 2D induction depends on the direction of the transmitters.Transmitter 1, is in the direction of the strike and give rise to Electric polarization orTE-mode. Transmitter 2, is in the direction of the profile line, and give rise to magneticpolarization or TM-mode,

3.3 RMT field layout

In the RMTmethod the electric andmagnetic field components are measured at the Earthsurface. The horizontal electric components Ex and Ey are measured with two couplesof electrodes, which are put into the ground perpendicular to each other. The horizontalelectric field components are measured in two well defined directions (x, y). Typicallyone set one electrode pair parallel to the strike (x) and the other pair parallel to the pro-file line (y). The magnetic components Bx, By and Bz are measured simultaneously withthree magnetic sensors orthogonal to each other. The horizontal magnetic field compo-nents are measured in the same direction (x, y) as the horizontal electric field compo-nents. Figure 3.2 shows a schematic RMT field layout.

3.4 Estimation of the MT transfer functions

The RMT instrument detect the frequencies of the measured EM components in differentbands with a well defined bandwidth, usually one octave (2:1). The classical way of es-timating the impedance tensor eq (2.47) and the tipper vector eq (2.48) is to assume thatthe noise is concentrated to the electric field components. If we neglect the noise in themagnetic components then the estimated impedance tensor and tipper vector Z, and T,will be bias free (Pedersen, 1982).In order to estimate the transfer functions, one needs at least two transmitter that areclose in frequency, to solve eq (2.45) and eq (2.46) with respect to the unknown transferfunctions. Transmitters having the same direction in the same frequency range will ren-der in identically equations. A fundamental assumption when estimating the transferfunctions Z, and T, is that they are constant in the band, in which they are estimated.

3.4. ESTIMATION OF THE MT TRANSFER FUNCTIONS 21

y

x

Hx

Hy

Hz

+

+

-

+

+

Ey

Ex

Processing Unit

1.2.

3.

4.

5.

7.

6.

AF-BOX

Figure 3.2: Illustration of a RMT field layout. TheEx field is measuredwith the electrodes(1) and (2), and theEy field is measured with the electrodes (3) and (4).The magnetic fieldcomponents Bx By and Bz , are measured with the magnetic sensors (5). The electrodesand the magnetic sensors are connected to the analogue filter box (6). The analog signalsare transferred to the central processing unit (7)

This is called the band averaging method. This is done by making the following func-tions, F (Z) = |E−ZHh|2 and ,G(T) = |Hz−THh|2 (h denotes horizontal) from eq (2.45)and eq (2.46). These functions are minimized, and solved with respect to the unknowntransfer functions. The least square solution follows by,

Zest = (HThHh)

−1HThE (3.2)

Test = (HThHh)

−1HThHz. (3.3)

The impedance for a homogeneous half space given by eq (2.33) shows that the impedancevaries as the square root of frequency. It means that over a bandwidth of one octave theimpedance vary pretty much for the lowest and highest frequency in the band. The ap-parent resistivity will be more or less biased corresponding to that the true frequency isnot correctly measured. This is the reason why the TSVD-method has been developedintroduced in the next section.

3.5. TSVD 22

3.5 TSVD

The TSVDmethod has been used to estimate the magnetotelluric transfer functions (Bas-tani and Pedersen, 2001). The TSVD method has proved to give a smoother estimate ofthe transfer functions than the corresponding band averaging estimate. The impedanceelements Zxy and Zyx are represented as

Zxx(Tn) =M∑

m=1

iaxxmTn + iTm

(3.4)

Zxy(Tn) =M∑

m=1

iaxymTn + iTm

. (3.5)

Where Tn= the known periods for the electromagnetic components n=1,2,....m= the number of coefficients to be estimated for each impedance tensor element.Tm= selected periods usually picked such that there are 10 periods T1, T2... in each fre-quency interval of factor 10.axxm , a

xym = the unknown coefficients to be estimated.

If we now substitute the impedance elementsZxyandZyx into eq (2.45) theEx componentcan be rewritten as

Ex(Tn) = Hx(Tn)M∑

m=1

iaxxmTn + iTm

+Hy(Tn)M∑

m=1

iaxymTn + iTm

. (3.6)

This is an under-determined problem, meaning that there are more unknowns than thereare equations. The given equation (3.6) is then solved with respect to the unknown co-efficients with the SVD. The estimated coefficients are then back substituted into equa-tion (3.4) and (3.5). The frequencies used in the TSVD processing are interpolated overthe selected periods Tm, usually 15 frequencies covering the measured frequencies withthe period Tn. Another application of the TSVD processing is to secure data quality. Ifthe natural inverse of the SVD does not fit the data to a specific value, the station containsdata with a low signal to noise ratio and bad quality, will be discarded.

Chapter 4

Controlled source tensorMagnetotelluric (CSTMT)

4.1 Introduction

A problem with the RMT method is the dependency of receiving a sufficient number oftransmitters. Also that these transmitters doesn’t always have the adequate frequencyrange for the investigation. The lowest frequency used in the RMT method is 14 kHzwhich may not be sufficiently low for getting the desired penetration depths. This prob-lem can be solved by the use of a mobile transmitter, where the operator freely can choosethe desired frequencies for the investigation. The instrument used for the survey in thisthesis is the EnviroMT system, developed at Uppsala University.

4.2 Using a source in practice

The theory for RMT is valid for CSTMT as long as the plane wave condition is fulfilled.However, some additional techniques are used in CSTMT to improve data quality andget better signal to noise ratios. In the RMT method one needs information from at leasttwo transmitters which are close in frequency with a reasonable spread. In CSTMT onemake use of a mobile source consisting of two dipoles, transmitting in two perpendiculardirections for each frequency. The dipoles are transmitting one at the time.

The plane wave assumption is almost always valid in the RMT method. That is notalways the case when using a controlled source. Especially not when executing groundmeasurements in Swedish conditions. It has been shown for a homogeneous half spacethat a source receiver distance greater than four skin depths eq (2.27) are sufficient toachieve plane wave conditions (Goldstain and Strangway, 1975). The terminology for ex-pressing plane wave and non-plane wave in a physical context is far- field and near-field.The skin depth equation (2.27) derived for a homogeneous half space can be used as ruleof thumb when deciding the transmitter receiver distance.

4.3. ENVIROMT 24

4.3 EnviroMT

The EnviroMT system employs a double dipole source. The term controlled source tensorMagnetotelluric (CSTMT) was first proposed by Li and Pedersen (1991). The EnviroMTis a frequency domain instrument made for ground measurements. The measurementscan be done in two different modes. The RMT mode which make use of distant trans-mitters in the frequency band 14-250 kHz. The CSTMT which operates the source in thefrequency band 1-100 kHz and combines the the signals from both the source and thedistant transmitters.

The magnetotelluric data are stored in a data file in the EnviroMT in fixed columns cor-responding to the xy and yx tensor elements. In chapter 3.1 it stated that the impedanceelement Zxy is related to the Ex component and the TE-mode. The impedance elementZyx is related to the Hx component and the TM-mode. This means that all the xy-components measured in the field e.g Zxy, Txy are related to the TE-mode. Whilst allthe yx-components e.g ρyx, φyx are related to the TM-mode.

4.3.1 The source

The source consists of two perpendicular horizontal magnetic dipoles and are remotelycontrolled by the central processing unit. The magnetic dipoles are made of 24 meterlong rubber cables, which are set up in rectangular loops with a length of 9 meter andheight of 3 meter see figure (4.1). The reason for using horizontal magnetic dipoles isthat they are easier to install than electric dipoles, they also have less coupling to nearbyconductive structures than electric dipoles. The magnetic dipoles are also expected toprovide a better plane wave condition compared with the electric dipoles (Bastani, 2001).

Figure 4.1: Illustration of a source set up (left) and a transmitter set up (right)

4.3.2 Transmitter

The transmitter is mainly used in the frequency range 1-14 kHz. But can be used togenerate frequencies up to 100 kHz. The transmitter box consists of a signal generator, apower switch device, a relay box, a GPS clock and a radio modem. Both the transmitter

4.3. ENVIROMT 25

and the central processing unit are equipped with a GPS clock see figure (4.1). The GPSinstruments are used to synchronize the time in the transmitter and receiver in the centralprocessing unit. The radio modems are used for the communication between transmitterand receiver.

Figure 4.2: Illustration of a measurement field lay out (left) and of the central processingunit (right)

4.3.3 Electric field sensors and Analogue filter (AF) box

The electric field components are measured in two perpendicular direction. The potentialdifferences are measured in each direction with pairs of electrodes which are coupled tothe ground at a distance of 5 to 10 meters. One electrode pair is marked N and S and theother pair is marked E and W. The default coordinate system where the N/S electrodesand E/W electrodes are oriented in the x and y axis respectively. The N/S electrode pairshould be set in the strike direction (x-axis) and E/W in the profile direction (y-axis). Theelectrodes are connected to the AF-box which are used for amplifying and filtering theincoming signals (figure 4.2).

4.3.4 Central processing unit

The central processing unit is the heart of the EnviroMT instrument. It contains hardwarecomponents which involve the acquisition, processing, interpretation and storage of data.The central processing unit has an integrated radio modem and is connected to the AF-box and GPS (figure (4.2).

Chapter 5

Forward modeling and inversion

5.1 The forward model

The translation from data to model is executed in the inversion process. In order to doan inversion one needs a forward model in the first place. To calculate the 2-D forwardmodel response we use the finite difference approximation (FDA). (Aprea et al., 1997).Here follows a very brief description of the method. The conductivities and fields aresampled at the nodes of a finite grid. The field between the nodal points are locally ap-proximated using low degree polynomial, determined from the values at a small numberof nodal points usually five. For the TE-mode, Maxwell’s equations in a good conductorplus Ohm’s law reduce to

∇2Ex = iωµσEx. (5.1)

and the equivalent equation for the TM-mode is

∇ · ρ∇Hx = −iωµσHx. (5.2)

U

R

D

L

σ1σ2

σ3 σ4

Α1

Α3 Α4

Α2

y

z

0

∆U

∆D

∆R∆L

A

Figure 5.1:

If we consider a rectangular box with the nodal point ’0’, and its neighboring nodes areR (right), L (left), U(up) and D (down) as illustrated in figure 5.1. We start by integratingeq (5.1) over the rectangular area A in figure 5.1, whose sides intersect the grid halfway

5.1. THE FORWARDMODEL 27

between O and each adjacent nodes with the use of Gauss Theorem eq (5.3),∫

A(∇ · ∇Ex)dA =

∫

(n · Ex)dl. (5.3)

The right hand side of equation (5.3) can be approximated by the use of the centered firstdifferences. For instance the line-integral along the right edge of A in figure 5.1 becomes

∫

∂Ex∂y

dl = (∆U + ∆D

2)ER − E0

∆R(5.4)

This is done for all four sides of A and we add up the contributions. Substituting theright-hand side of eq (5.1) into the right hand-side of equation (5.3), and we are allowedto do the following approximation

iωµ

∫

AEx(x, y)σ(x, y)dA ≈ iωµEx

4∑

i=1

σiAi + error. (5.5)

Here Ex in the right hand side of eq (5.5) expresses the electric field component in thecentral node. The error term in eq (5.5) can be neglected as long as node spacing is smallcompared with the local skin depth. Combining equation (5.4) and (5.5), of course withthe contribution of all four sides in eq (5.4). Both sides are then rescaled with the totalarea of integrationA = (∆L+∆R)(∆U+∆D)/4. Also defining the effective conductivityin the central node as

σ0 =1

A

4∑

i=1

σiAi. (5.6)

y

z

Boundary c.1D-solution

Boundary c.1D-solution

Boundary c.Wave totally damped

Boundary c. Incident waveExij

Ex(N)

E(Uij)

E(Lij)

E(Dij)

E(Rij)

Figure 5.2:

We can now from a model with uniform grid and a node spacing ∆ and a constant con-ductivity σ0, receive the following equation

1

∆2(EL + ER + EU + ER + ED − 4Ex) − iωµσ0Ex = 0. (5.7)

5.2. OCCAM INVERSION 28

Equation (5.7) is implemented on each node in the way shown in figure 5.2, and decom-posed so it can be expressed in the form

Ax = b. (5.8)

The values of the nodes that lie on the boundary are moved to the right hand side ofeq (5.7) in vector b. The electric field component in the central nodes Ex are collectedin the vector x. The remaining terms in eq (5.7) builds up the matrix A. The boundaryconditions in the forward model are illustrated in figure 5.2 and have the following prop-erties. The electric field components of the nodes that lie on the surface are given by aunit vector (1.0) (the incident wave). The electric field Ex on the left and right sides of the2D model are given by some 1D condition distributed to the leftmost and rightmost cellsrespectively. In the lower end of the model the wave are considered to be totally dampedwhich means that the electric field Ex on such nodes is zero.

The complex system of equations eq (5.8) is then solved with respect to the unknownelectromagnetic field components.

To get the magnetic field there are a some additional steps one has to take which areconcisely presented. A second order Taylor expansion upwards is made on the U(up)node in the z-direction, and then a second order Taylor expansion downwards on theD(down) node in the z-direction. This is done for each central node Exij on the Earth’ssurface. The second order term is approximated by eq (5.1) and back substituted intothe right hand side of eq 5.1 separately. From these two equations one calculates EUijand EDij which are solved with respect to the first derivative ∂Ex

∂z . The derivative∂Ex

∂z isfound in eq (2.40) for the TE-mode and is related to the magnetic field component Hy.

After the electric andmagnetic fields are calculated one obtains the impedance , the phaseand apparent resistivity according to eq (3.1), eq (2.54) and eq (2.53).

A similar procedure can be done for the TM-mode by integrating eq (5.2). The TM-modebecomes more complicated than the TE-mode. This is because ρ varies along the integra-tion path and it is therefore necessary to use the appropriate ρi for each path segment. Asa result there are four effective resistivity’s instead of one.

5.2 Occam Inversion

The Occam inversion code (deGroot Hedlin and Constable, 1987) was originally devel-oped for the interpretation of magnetotelluric data. The investigated model is approxi-mated in a large number of discrete cells (2D) of unknown resistivity.

The measured electromagnetic field components are known and one seeks the resistivityof some discritisized model. The division of the cells are made in such a way that eachcell has a constant resistivity, and usually one chooses to extend the cells in the verticalby 10% and increase the horisontal spacing with 50% outside the profile see figure 5.3(Aprea et al., 1997).

Before the inversion process the forward model is further rearranged to

d = G(m) + e. (5.9)

Where d is a vector (1 × M) which contains the measured data, phase and apparent

5.2. OCCAM INVERSION 29

resistivity ( d = ρa1, ρa2, ρaN , φ1, φ2, φN ). The model parameters m (N × 1) contain theunknown conductivities for each cellm = σ1, σ2, σN . G (N ×M) is the data kernel thatrelates the data (d) to the model parameters m. And e is the estimated errors on themeasured data which usually are assumed to be Gaussian distributed. Equation (5.1) is anonlinear function that we want to solve with respect to the unknownmodel parameters.As the number of unknown model parameter exceeds the number of equations in thesystem (N«M) the problem gets under determined, which leads to an unstable solution.

In the Occam code one wants to fit the data to a specific threshold value (Siripunvarapornand Egbert, 2000), which can be written as ‖ dfwd − d ‖2< δ. Where dfwd is the dataresponse from the forward model which corresponds toG(m). Rewriting the expressiongives ‖ G(m) − d ‖2< δ. At the same time one wants to minimize the regularized normof the model parameters which can be expressed as ‖ Lm ‖2. Here L can by any type ofregularization matrix that penalizes the model parameters to be smoother. What is doneis one minimizing ‖ Lm ‖2 with the constraint that ‖ G(m) − d ‖2< δ. This is done bythe use of Lagrange multiplier (κ2) and by minimizing the following target function

oc =‖ G(m) − d ‖2 +κ2 ‖ L(m) ‖2 . (5.10)

The electromagnetic forward model eq (5.9) is weakly nonlinear which means that theobjective function ‖ G(m)−d ‖2 has an unique solution however, local minima’s do exist.A couples of methods are accessible for solving such a problem and one is to linearize it.This is done by guessing an initial solution of the model,and then perturbing that modelto a new better model until some convergence criteria is reached.

This is done by a Taylor expansion of the data kernel G(m) around the model of the jth

iteration step as

G(mj + ∆m) ≈ G(mj) + J(mj)∆m. (5.11)

Where J is the Jacobian matrix (containing all the derivatives ofG(m)).

The target function eq (5.10) can be rewritten as

oc =‖ G(mj) + J(mj)mj+1 − J(mj)mj − d ‖2 +κ2 ‖ L(m + ∆m) ‖2 . (5.12)

In terms of the updated modelmj+1 after each iteration and the currently predicted datad∗ and data kernelG∗ one can write

ρ1 ρ2

ρm

y

z

Figure 5.3: Illustration of a discritisized model

5.3. REBOCC 30

mj+1 = mj + ∆m, (5.13)

d∗ = d − G(mj) + J(mj)mj , (5.14)

G∗ = J(mj)mj+1, (5.15)

and

oc =‖ G∗ − d∗ ‖2 +κ2 ‖ Lmj+1 ‖2 . (5.16)

Now d∗ and G∗ are constants for each iteration, which means that the solutions to eq(5.16) is just a damped least square from the linear inversion theory (Menke, 1989), and

mj+1 = (J(mj)TJ(mj) + κ2LTL)−1(J(mj)Td∗. (5.17)

The only difference from the linear case is that the damping parameter κ is adjusteddynamically so that the solution will not exceed the allowable misfit. At each iterationthe largest κ is picked such that the χ2-value will not exceed the value of δ. If that can notbe achieved κ is chosen to minimize the χ2-value. Here χ2 = (G(m)−d)C−1

d (G(m)−d)and Cd is the data covariance matrix. Now we can see according to eq (5.16), if κ is smallthat fitting the data will have bigger influence on the inversion process, whilst a bigger κwill give more influence on reducing the model norm.

5.3 REBOCC

REBOCC stands for REduced Basis OCCam’s inversion and is an efficient variant ofthe Occam algorithm. The goal of the inversion is to find the minimum structure model,subjected to a desired misfit level. REBOCC is freely available for academic use, andthe program version used in this thesis is called REBOCC95 (Siripunvaraporn and Eg-bert, 1999), modified by T.Kalscheuer. The measured field data stored in the EnviroMThas to be converted to the right format before inversion in REBOCC. This is done witha matlab code (readmtdata.m) where one chose the desired error floor of the phase andresistivity. This is done to allow a certain error on the Impedance before inversion. Theerror floor obey the following rules, Relative(error on the resistivity) = 2(relative er-ror on the Impedance) and 1%(relative error on the impedance) = (0.57o er-ror on the phase). This program also gives the options to remove frequencies with thecorresponding data for a given profile.

Chapter 6

Interpretation of Magnetotelluricdata

6.1 Interpretation of the phase

The phase eq (2.54) plotted as function of frequency have the following properties for themodels illustrated in figure 6.1. In model a), the EM fields for the higher frequencies onlypenetrates the top layer and the phase is equal to 45 degree. The lower frequencies willeventually penetrate through the top layer and the stratum between ρ1 and ρ1 and thephase will be higher than 45 degree. For even lower frequencies the EM components canbe looked upon as they only existed in the bottom layer and the phase will asymptoticallyconverge towards 45 degree. For more complicated structures the phases is always inrelation to what the phase was at the higher frequencies. If we follow the phase from agood conductor through a poor conductor the phase will be less then 45 degree. If wethen in a deeper structure encounter a good conductor again the phase will rise but notnecessary above 45 degrees. A nice property of the phase is that it reacts slow to changesin the ground, which means that the phase are always smooth. So an other application ofthe phase is to use it as a quality gauge of the data. Single point anomalies in the phasecan always be considered to be bad data and should be disregarded or thrown away.

Poor conductor

Good conduktor

Good conductor

Poor conductor

Homogeneoueshalf space

ρ1

ρ2

ρ1

ρ2

ρ

ϕ>45

ϕ<45

ϕ=45

ρ1>ρ2

ρ1<ρ2

a)

b)

c)

Figure 6.1: Interpretation of the phase, illustrating three 1D model cases a,b,c. ϕ is indegrees.

6.2. INTERPRETATION OF APPARENT RESISTIVITY 32

6.2 Interpretation of apparent resistivity

The apparent resistivity given by equation (2.53) can be defined as the average resistivityof a uniform half space. The apparent resistivity is a function of frequency and eachfrequency corresponds to a specific skin depth see equation (2.27) for the 1D case. Theapparent resistivity could be interpreted as the average of the real resistivity to the skindepth of the corresponding frequency.

6.3 Interpretation of the tipper

The tipper vector defined in equation (2.48) is often plotted as real R or imaginary Q

induction arrows expressed asR = Re(Ax+By) (6.1)

Q = Im(Ax+By). (6.2)

Shown in chapter 2.5.3 the tipper vector has the following properties for a 1D, 2D and a3D model.1D) T=(0,0)2D) T=(0,B)3D) T=(A,B)

One can show how to interpret the tipper or the real induction arrows by the vector repre-sentations of the primary and secondary fields illustrated in figure 6.2. The figure showsa TE-mode case where a good conductor extend infinite by the strike direction. The pri-mary magnetic field denoted by Hp is perpendicular to the electric field component Ex.The electric component will induce a current in the good conductor according to Ohmslaw eq (2.7) and this current will induce a secondary fieldHs according to Ampere’s law.At the measuring points P1 and P2 a receiver will respond to the resulting total magneticfieldHres of the primary and the secondary field. The resultant ofHres have the compo-nents Hy and Hz . The 2D tipper (0,B), corresponds to Hz = BHy according to equation(2.46). Let see how the real induction arrowRwould look like in figure 6.2.

When measuring at P1 the Hz component is positive (z-axis positive downwards), andthe Hy component is negative, making B negative. At measuring P2 the Hz componentis negative and the Hy component is also negative making B positive. From this vectorrepresentation (figure 6.2) one sees that the induction arrows R always will point awayfrom the good conductor.

The real induction arrow R also reveals the dimensionality of the measured structure.The tipper component A is zero in an ideal 2D Earth model and both the componentsA and B is zero in a 1D Earth model. This means that the A component in the tippervector indicates if the measured structure is 3D. However a 3D structure which have norestrictions regarding the components A and B in the tipper vector can behave as both a1D or 2D tipper. But in the very most cases the 3D structure have both the components Aand B in the tipper vector.

6.4. NEAR FIELDS EFFECTS 33

Strike x

Profile y

Transmitter

Hp

ExHs

Hs

Hp

HresHres

Hy

HzHs

Hp

HresHres

HzHy

Good conductor

Induced magnetic field

P1 P2

P1 P2

Figure 6.2: Illustration of a vector representation of the primary field Hp and the sec-ondary field Hs in a 2D TE-mode case. Hres is the resultant of Hp and Hs. The measur-ing point 1, (P1) is situated to the left of the conductor and the measuring point 2, (P2) issituated to the right of the conductor.

6.4 Near fields effects

The near field effects occur when the incident electromagnetic wave can not be consid-ered to be a planewave. Thewave-number q, for the electromagnetic field equation (2.15)can be rewritten for the general case as

q2 = iωµσ + κ. (6.3)

whereκ =

2π

λ. (6.4)

The effective wavelenght λ is roughly equal to the distance between transmitter and re-ceiver. The plane wave assumption is fulfilled if κ << iωµσ, thus the kappa (κ) term canbe neglected and the wave can be considered to be plane. Note that high frequencies andhigh conductivities will do the trick. This is just another way of expressing the a planewave criteria. However non of the theory given in this thesis is valid when dealing witha non plane wave.

6.4.1 Interpretation of the near fields effects

Experimental evidence has shown that the tipper vector will recognize the source as aconductor in the near field and the induction arrow vector R eq (6.1) will point in thedirection of the source. The phase has been shown to have similar properties. In thenear-field the phase will dramatically increase corresponding to a) in figure (6.1), but theincrease in phase is just an artificial effect of the near field and has no geological origin.

6.5. INTERPRETATION OF THE 2D-INVERSION MODELS TE,TM,TE+TM AND THEDETERMINANT DET 34

6.5 Interpretation of the 2D-inversionmodels TE,TM,TE+TMand

the determinant DET

The TE and TM modes respond differently to specific features of geology, which one hasto take into account in the interpretation process (Pedersen and Engels, 2005). Generallyspeaking the TE-mode couples better to conductive than resistive structures, whilst theTM-mode couples better to resistive than conductive structures. In the TE-mode, thecurrent flows along the strike, and it is logical that the TE-mode will be more influencedby 3D effects than the TM-mode where the current flows perpendicular to the strike.Therefore one should expect that it will be harder to fit the data in the TE-mode whichmore likely deviate from the 2D approximation.

The determinant has shown to be practical to use in the sense that it’s invariant to ro-tation, which means that the data to be modeled is the same regardless of the assumedstrike direction used for 2D modeling. It has also shown to be less influenced by 3D ef-fects. This together with the fact that it’s invariant to rotation, makes it easy to fit the datain the inversion process. The 2D determinant can be considered as the arithmetic meanof the TE and TMmode. When interpreting the determinant one should expect to discernthe characteristic features from both the TE and TMmode but more diffusely than the TEand TM mode separately.

The best results will be foundwhen combining both TE and TMmode in a joint inversion.However it is likely that one can not achieve a satisfying data fit in this inversion. Thisdepends on the fact that the TE and TMmode respond differently to specific conductivitymodels and also because of the 2D approximation is inconsistent with the real 3D world.

Chapter 7

Description of the area and Field data

7.1 Field measurement in situ

The measurements were executed in a place called Flintalycka situated on the Halland-såsen Horst (figure 7.1). Three parallel profiles where measured marked with green linesin figure 7.1, which are denoted Line1,Line2 and Line3. Each profile line is measured indirection NE-SW. The two red lines marks the tunnel lines, which are situated approx-imately at a depth of 150 meters. The distance between each parallel profile line is 30meters. Line1 has the length of 450m, Line3 in the middle is 400m long and Line2 in theNW end, is 400m long. The controlled source marks the source in figure 7.1 has an ap-proximate closest distance (the normal distance) to Line1 of 315m, to Line3 of 345m andto Line2 of 375m. The geological knowledge of the measuring area is that the subsurfacecontains a sediment layer of 2-10m thickness and is underlain by bedrock.

7.2 Geological review of the Horst Hallandsåsen

The Hallandsåsen Horst is one of several uplifted blocks of the Earth’s upper crust thatare found in Skåne. The Horst is composed of Precambrian rocks which are flankedby younger sedimentary rocks with the strike in NW-SE direction. It is expected thatthe Horst consists of crystalline and finely grained metamorphic rocks of Precambriangneiss. In the tunnel area there is gneiss with amphibolitic parts and several doleritedikes trending NW-SE. The main fracture zone can be followed from the most southernpart of the cape with an approximate NW-SE strike direction. There are fracture zonesin other directions situated on the cape, however the fracture zones in the tunnel projectarea have strikes in the NW-SE direction. The major faults originated about 500 millionyears ago and can be several hundred meters wide

7.2. GEOLOGICAL REVIEW OF THE HORST HALLANDSÅSEN 36

Figure 7.1: A map over the field-area where the measurements took place

Chapter 8

Result

8.1 Introduction

The magnetotelluric transfer functions are TSVD processed which means that the fre-quencies are interpolated over the real frequency interval. The interpolated TSVD fre-quencies are shown in table 8.1. The stations that were removed in the TSVD processingare station 1 to 4 from Line1, station 20 and 21 from Line2 and station 1 from Line3. Thecontrolled source tensor magnetotelluric (SCTMT) data which combines the signals fromboth the controlled source and distant radio transmitters are denoted COMB.

Real Source frequencies (Hz) TSVD RMT frequencies (Hz) TSVD COMB frequencies (Hz)2000 14143500 20004000 28285000 40008000 5656

10000 800014142 1131320000 1600028284 2262740000 3200056568 4525480000 64000113137 90509160000 128000226274 181019

Table 8.1: Source frequencies, TSVD RMT frequencies, TSVD COMB frequencies

8.2 Phase

The phase given by equation (2.54) is one of the many algorithms that is programed inthe EnviroMT, and are available directly field. The frequencies in table 8.1 are statedlogarithmically (log10f ). The interpretation of the phase is explained in chapter 6.1.

8.2. PHASE 38

8.2.1 TM-mode COMB TSVD, Line1

The low phases in the high frequency range show features of conductive sediment overresistive bedrock in the shallow subsurface. Between stations 19 and 28 the phase is al-most 45o which is the phase of a homogeneous half space. One can suspect that the sedi-ment layer is very thin here. The phase increase in the middle frequency range betweenstation 27 and 46 indicates a conductor at intermediate depths. Near-field effects are veryobvious in the controlled source frequencies, because of the extremely high phases whichare seen between stations 1 and 39. The near field effects are not as evident between sta-tion 39 and 46. There are two possibilities for this, either that the source receiver distanceis greater or that the geology is more conductive between these station The later is morelikely in this case. In the near field, the plane wave assumption is no longer valid, andhow to interpret the magnetotelluric data is more difficult. One can always argue, ac-cording to equation (6.3) that the plane wave assumption is depending on frequency andconductivity which is related to source receiver distance and skin depth. In the TM-mode Line1 and Line3 (figure 8.3) show very similar pattern but the near field effects area bit more pronounced in Line3. The source receiver distance is greater for Line3, so onewould expect the opposite. In agreement with Line1 and Line3 the source receiver dis-tance does not seem to be the reason for the lack of near fields effects at the end of Line1and therefore its must be geological. So one can suspect that there is a conductor in thedeeper structures in this part of Line1.

Figure 8.1: TM-mode COMB TSVD, TE-mode COMB TSVD, Line1

TE-mode COMB TSVD, Line1

The conductive sediment layer is not as resolved as in the TE-mode. The high phasebetween station 12 and 19 shows the presence of a conductor. A result that is of greatsurprise is that the near-field effects are not as evident in the TE-mode as in the TM-mode. Between station station 1 and 21 the phases are very smooth and the higher phasein the lower frequency range can not be directly interpreted as near-fields effects.

8.2. PHASE 39

However, a strong discontinuity can be seen between the controlled source and RMTfrequencies (table 8.1) between station 21 and 40which is interpreted as near-fields effectsin combination with geology.

RMT TSVD, Line1

The phases for RMT are in good agreement with the phases in the combined data set forboth TE and TM mode, figure (8.2). Stations 1-4 are removed in the TSVD processingand these stations obviously contain bad data. More over the phase is employed to spotbad data e.g at station 14 and 19 from and station 38 that are allotted big errors in theinversion so that these data point will be heavily down weighted during inversion.

Figure 8.2: RMT TSVD, Line1

8.2.2 COMB TSVD, Line3

The phases in figure (8.3), follow the same patterns as in Line1 for both TE and TM-mode,which is a indication of good 2D conditions.The low phases between stations 1 and 6in the TM-mode tells about resistive geology at the beginning of the profile. Betweenstations 6 and 10 the TM-mode shows a phase of 45 degrees. However, in the same partof the TE mode profile the phase is low which tells about a conductor in this part of theprofile, which is in agreement with Line1. A conductor is also visible at station 20 inthe TE-mode. This phase is however not in agreement with Line1 and it also show adiscontinuous behavior. The near fields effects follow the same patterns as in Line1 butare a bit more evident at the beginning of the profile.

8.2.3 COMB TSVD, Figure Line2

The phases in Line2 figure (8.4) show similar patterns as in Line1 and Line3. The geologyseems to have a general trend to be more resistive to the west. The conductor seen inthe TE-mode between station 12 and 19 on Line1 is not as evident on Line2, that goes forthe possible conductor in the end of the profile as well. The near-field effects in the TM-mode starts at lower frequencies in Line2 in comparison with Line1 and Line3, which is

8.3. APPARENT RESISTIVITY 40

Figure 8.3: TM-mode COMB TSVD, TE-mode COMB TSVD, Line3

expected as Line2 is furthest away from the source. The discontinuity in the TE-modewhich can be seen on all three lines, from station 19 to 46 on Line1 and from station 11to 21 on Line3 and Line2, is difficult to understand. But as it is at the same place in allthree Lines, a qualified guess is that geology is origin to these low phases correspondingto very resistive structures in the deeper parts.

Figure 8.4: TM-mode COMB TSVD, TE-mode COMB TSVD, Line2

8.3 Apparent resistivity

The apparent resistivity given by equation (2.53) is one of the many algorithms that isprogramed in the EnviroMT, and are available directly in the field. The apparent resis-tivity is plotted with the program transform, and presented logarithmically (log10f ) ina color scale. The frequencies in table 8.1 are stated logarithmically (log10f ). How tointerpret the apparent resistivity is explained in chapter 6.2.

8.3. APPARENT RESISTIVITY 41

8.3.1 Apparent resistivity, Line1

The apparent resistivity figure (8.5), is in good agreement with the phase. The two con-ductors that were recognized in the the phase aremore obvious in the apparent resistivity.Seen in the TEmode between station 12 and 20 and the second conductor between station42 and 46. In the TM-mode the near field effects make a distinct line between the con-trolled source and the RMT frequencies. However, one can clearly see a continuity in thetransition between the RMT and the controlled source frequencies in the apparent resis-tivity. It seems like the real apparent resistivity increases in the near field in both TM andTE-mode. In the TM-mode a conductor is seen between stations 20 and 27 that has beeninterpreted as succession of layers that become more resistive in the deeper structuresaccording to the phase figure 8.2.1. Note that the TE-mode doesn’t show any indicationsof near fields effects in the apparent resistivity.

Figure 8.5: Apparent resistivity, Line1

To estimate the penetration depth one can use the 1D skin depth equation (2.27). Themostresistive features of the profile are shown in the beginning of the TM-mode profile, withan estimated average apparent resistivity of 4000Ωm. That would be equivalent to a skindepth for the lowest controlled source frequency (2kHz) of ≈ 700 m and for the highest(8kHz) of ≈ 350 m and correspond to a source receiver distance for a homogeneous halfspace of 2.8 km and 1.2 km respectively. That is to secure plane wave conditions. Themost conductive parts of the profile can e.g be seen at the end of the TE-mode profile,with an average apparent resistivity of 200Ωm. That would be equivalent to a skin depthfor the lowest controlled source frequency (2kHz) of≈ 150 m, and for the highest (8kHz)of ≈ 80 m. To secure plane wave conditions one needs a source receiver distance of 600m and 320 m respectively. The source receiver distance is approximated 300 to 450 mfor the three lines in the investigation, which implies that the controlled source will be inthe near field at almost all frequencies especially in the resistive parts of the profile. Thelowest RMT frequency will have a skin depth of ≈ 250 m in the resistive parts and ≈ 60m in the conductive parts. The target for this survey is to model the conductive fracturezones, to a depth of 150 m. To achieve this one need a controlled source frequency aslow as 2kHz and a source receiver distance of 600 m to be in the far field, this we don’thave. However for the interesting conductive part we should be able to trust the highestcontrolled source frequency 8kHz with a skin depth of≈ 80 m. For the the resistive partsof the profile the lowest RMT frequency gives the adequate skin depth of ≈ 250 m.

8.4. REAL INDUCTION ARROWS 42

8.4 Real induction arrows

The interpretation of the real induction arrows in equation (6.1) is explained in chapter6.3.

8.4.1 Real induction arrows, Line1

The real induction arrows plotted for the RMT figure (8.6), show the features of fourconductors. The largest induction arrows are seen between stations 30 and 39, pointingaway from the good conductor. This conductor can not be clearly visualized in eitherthe phase or the apparent resistivity figure (8.1) and figure (8.5). In the same part of theTM-mode profile for apparent resistivity (figure 8.5), we see an apparent resistivity of ≈500 Ωm surrounded by much higher resistivities of≈ 3000 Ωm. The induction arrowR isnot an absolute, but a relative tool that only reflect differences between conductivities. Itseems like the real induction arrows are better of detecting differences in conductivity ina more resistive media than the phase and the apparent resistivity. A second conductorcan be seen between station 3 and 8, the length of the induction arrows increase withdecreasing frequency which indicates a more conductive geology with depth. This con-ductor can not be seen in the phase or the apparent resistivity in contrary this part of theprofile show resistive features in the the TM-mode profile of apparent resistivity (figure8.5). Third conductor can be seen more less clearly between stations 14 and 19, and isrecognized as a very clear conductor in the phase in the same part of the TE-mode profile(figure 8.5). Fourth conductor can be seen at the end of the profile.

Line1 figure (8.6), show a overall good 2D conditions (horizontal vectors). 3D effects canbe seen at the boundaries of the conductors and also at the end of the profile from station43 to 46 (Station 1-4 are removed in the TSVD-processing).

The real induction arrowsR plotted for the combined frequencies show increased sensi-tivity to near-field effects. The vectors corresponding to the controlled source frequenciesare very large in comparison with the RMT frequencies and point in direction towardsthe source (obvious near field effects).

Figure 8.6: Real induction arrows, Line1

8.4.2 Real induction arrows, Line3

The 3D effects are overall more evident in Line3 especially for the higher frequencies.The four conductors recognized in Line1 can be seen in Line3 except for the conductor

8.4. REAL INDUCTION ARROWS 43

at the end of Line1. The conductor at the beginning of Line1 can be recognized on Line3only at the lower frequencies, and the induction arrows are here very small indicating 1Dconditions. The conductor between stations 14 and 19 in Line1 is more evident in Line3corresponding here to the stations between 8 and 11.

The near field effects for the combined data are equally evident in Line3 as in Line1, inaddition there is a couple of bad data between stations 9 and 11 and at station 20.

Figure 8.7: Real induction arrows, Line3

8.4.3 Real induction arrows, Line2

Line2 is very effected by 3D effects especially between stations 16 and 21. The conductorthat is seen at the corresponding parts of the profiles Line1 and Line3 can only vaguelybe recognized in Line2 for the reason of 3D effects. The conductor between station 6 and11 has also been recognized in Line1 and Line3. The near-fields effects are as evident inLine2 even though Line2 is furthest away from the source.

Figure 8.8: Real induction arrows, Line2

8.5. STRIKE ANALYSIS, LINE1 44

8.5 Strike analysis, Line1

The strike direction based on the geological investigations done on the Hallandsåsenhorst is an NW-SE going strike. To be sure that the geo-electric strike does not devi-ate much from this a strike analysis was carried out. The method used is explained inchapter 2.4.5. The program for the Q-analysis is written by L.Pedersen and modified byT.Kalscheuer.Strike analysis has been made for the RMT data and combined data separately. Each plotin figure 8.9 to 8.12 is a representation of Q, for the try out angle ( φ ) between the profileline and the assumed regional strike. For the interpretation of Q, one argue that a giventry out angle ( φ ) that gives the lowest values of Q, estimates the real regional strike di-rection best.

In figure 8.9 the try out angel ( φ ) is 90 degree which is the strike assumption from thegeological investigation. For RMT, Q values are high between stations 0 and 15 especiallybetween station 9 and 15 which suggest that 90 degree strike assumption is not the beststrike between these stations. However, between stations 15 and 40 Q values are over-all low except for the highest frequencies and are interpreted as these frequencies onlypenetrates the sedimentary overburden. Which likely deviates from the 2D assumption.

0 10 20 30 40 50

1414

2000

2828

4000

5656

8000

11300

16000

22600

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64000

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181000Q 90−degree Line1 TSVD COMB

Station

Fre

quen

cy

2

4

6

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18

0 10 20 30 40 5010000

14140

20000

28300

40000

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113100

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226300Q 90−degree Line1 TSVD RMT

Station

Fre

quen

cy

2

4

6

8

10

12

14

16

18

Figure 8.9: Strike analysis, Line1

For the combined data, Q values are overall high for the low frequencies. This is an resultof the near field effects. For the RMT frequencies (above 14 kHz) the Q values are lowerthan the corresponding frequencies in the COMB model. This can be explained by thefact that the program uses all frequencies for each single Q-value. If the impedance is bi-ased in the low frequency range the Q values for the higher frequencies will be biased too.

The best strike estimation is given when plotting Q with a try out angle ( φ ) of 80 degreefor the RMT data. This plot gives the overall lowest value of Q in comparison when ( φ )equals 90, 70, or 60 degrees see figure 8.11.

8.5. STRIKE ANALYSIS, LINE1 45

0 10 20 30 40 50

1414

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181000Q 60−degree Line1 TSVD COMB

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Fre

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226300Q 80−degree Line1 TSVD RMT

Station

Fre

quen

cy

2

4

6

8

10

12

14

16

18

Figure 8.10: Strike analysis, Line1

The profile lines in the field extend in the NE-SW direction or more precisely 33o NE,based on the geological map of the area. According to the results from the strike analysis,the best profile direction for 2D measurements would be 43o NE. This also means thatthere is a bias in the estimatedmagnetotelluric transfer functions. This problem can easilybe overcome by a rotation of the coordinate system. However, a 10 degree rotation hasvery little effect on the rotated impedance tensor given by equation (2.67), and Z will bevery close to the rotated impedance tensor Z′, and therefore a rotation is not necessary inthis case1.

0 10 20 30 40 50

1414

2000

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181000Q 70−degree Line1 TSVD COMB

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0 10 20 30 40 5010000

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226300Q 70−degree Line1 TSVD RMT

Station

Fre

quen

cy

2

4

6

8

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14

16

18

Figure 8.11: Strike analysis, Line1

1The data used in the Q analysis are TSVD processed, and station 1-4 on Line1 were removed. This meanse.g that station 1 in the Q analysis, corresponds to station 5 for Line1 in the phase,apparent resistivity andreal induction arrow plots.

8.6. INVERSION 46

8.6 Inversion

The following inversions are made by the use of REBOCC95. Software documentationsand user manual used were written by Siripunvaraporn and Egbert (1999). The ideas be-hind the Occam inversion are explained in chapter 5.2. The interpretation of the differentinversions is explained in chapter 6.5. Bad data in a specific frequency or interval weregiven large data errors, which makes them less important in the inversion process. Thestations are marked with triangular at the top of the profiles.

8.6.1 COMB and RMT TSVD TM, Line1

Because of the near field effects in the combined data set, the five first frequencies in ta-ble 8.1, column COMB-frequencies (TSVD) were removed. The lowest RMS (root meansquare error) in the COMB inversion was found after 8 iterations to 1.7. The lowest RMSfor RMT data only was found after 7 iterations as 2.2, see figure (8.12). The most obvious

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Figure 8.12: COMB-RMS: 1.7, RMT-RMS: 2.2

conductor seen at both profiles (COMB and RMT) is located at profile meter 350. Theconductor has not been very well recognized either in the phase or apparent resistivity,but is evident in the real induction arrows (figure 8.6). The conductor seen at meter 200corresponds well with the apparent resistivity and phase data for the TM-mode (figure

8.6. INVERSION 47

8.1 and 8.5). The skin depths based on 1D calculations should be around 60 m for theconductive parts and more then 150 m for the resistive parts. This signifies that the re-sistive parts starting at 80 m depth in between profile meter 0 to 250 should be reliable.However, whether the conductor between profile meters 300 to 450 really progress intodepths below 60 m is hard to tell. The result can equally be an effect of regularization inthe inversion process in order to fit the data. The inversions from the COMB and the RMTare in good agreement in terms of showing the same features. However, one can see thatthe inversion of the RMT data tries to compensate the resistive and the conductive partsand becomes some average of what is seen in the COMB inversion which is the effect ofregularization.

COMB and RMT TSVD TE, Line1

In order to achieve greater penetration depths all frequencies in table 8.1, column COMB-frequencies (TSVD) are accounted for in the TE-mode inversion. The reason is that thephase and apparent resistivity has shown to be much less influenced by near field effectsin comparison with the TM-mode. The lowest RMS in the combined data set was foundafter 6 iterations to 3.8. The lowest RMS for the RMT was found after 5 iterations to 2.2,see figure (8.13).

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Figure 8.13: COMB-RMS: 3.8, RMT-RMS: 2.2

8.6. INVERSION 48

In the COMB inversion between profile meters 0 and 300, we find a very resistive struc-ture starting from a depth of 60 m and continue downwards. A more conductive partis found in the profile between meters 300 and 450 that goes down to a depth of 140 m.This part of the profile has only revealed it’s conductivity in the real induction arrows(e.g figure 8.6). However, this part is not that conductive ≈ 2000 Ωm, but it makes a dis-tinct difference against its surroundings (to the left) that show very high resistivities of ≈10000 Ωm. At these resistivities one can start to suspect some influence of displacementscurrents. The RMT inversion figure (8.13), is in good agreement with the COMB inver-sion with a few exceptions. The combined inversion is a little bit dislocated downwardswhich probably is an effect of the near field effects. The conductor located in profile me-tre 150 progress deeper in the RMT data inversion than in COMB. The RMT frequenciesdo not receive any information of the resistivity distributions below ≈ 60 m and we canclearly see what regularization does to the inversion model. The inversion tries to finda model that fits the data, in order to that the inversion has increased the depth of theconductor at the same time as it has made the resistive parts more conductive.

COMB and RMT TSVD DET, Line1

The five first frequencies in table 8.1, column COMB-frequencies (TSVD) were removed.The lowest RMS in the combined data set was found after 6 iterations as 1.0. The lowestRMS for the RMT was found after 7 iterations as 1.4, see figure (8.14).

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Figure 8.14: COMB-RMS: 1.1, RMT-RMS: 1.4

8.6. INVERSION 49

The determinant show the characteristic features from the TE and TM-mode very well.Nevertheless it does not seem to be able to resolve the deeper structures very well. Below60 m depth the DET model does not agree with either the TE or TM models. The modelsfigure (8.14), from the combined and the RMT data set are very similar as they should be.

COMB and RMT TSVD TE + TM, Line1

The same frequencies that were used in the combined data set in TM and TE mode areused in the joint inversion of TE and TM mode. This means that all controlled sourcefrequencies are used in the TE-mode whilst in the TM the five first frequencies in table8.1, column COMB-frequencies (TSVD)are removed. The lowest RMS in the combineddata set was found after 10 iterations as 4.3. The lowest RMS for the RMT was foundafter 10 iterations as 3.0, see figure (8.15).

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Figure 8.15: COMB-RMS: 4.3, RMT-RMS: 3.0

The joint inversion of TE and TM models show the best results in terms of summarizeddescription of both TM and of TE mode. Four conductors are visualised, the two con-ductors that are seen between meters 110 and 270 in the comb profile go down to a depthof 70 m and correspond well with both TE and TM models. The most interesting con-ductor is seen in profile between meters 300 and 380. This is the only conductor thatseems to progress down to a depth of 150 meter which is the depth to the tunnel underconstruction. The fact that all frequencies (from 2kHz) are used in the TE mode makes

8.6. INVERSION 50

it reasonable that we should get information from depths up to 150 m. The RMT modelverifies the COMB model by showing very similar features.

8.6.2 COMB TSVD TM and TE, Line3

The five first frequencies in table 8.1, column COMB-frequencies (TSVD) are removedin the TM-mode. The lowest RMS was found after 7 iterations as 1.9. The two firstfrequencies in table 8.1, column COMB-frequencies (TSVD) are removed in the TE mode,and lowest RMS was found after 4 iterations to 1.2, see figure (8.16).

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Figure 8.16: TM-RMS: 1.9, TE-RMS: 1.2

The TM mode model is in good agreement with the corresponding TM model in Line1figure (8.12) and show approximately the same features which indicate good 2D condi-tions. In contrary the TE model shows some features which are not at all in agreementwith the TE-model in Line1.

COMB TSVD TE+TM and DET, Line3

In the joint TE and TM mode inversion, the five first frequencies in table 8.1, columnCOMB-frequencies (TSVD) are removed in the TM-mode data set, whilst only the twofirst frequencies in the same column are removed in the TE-mode. In the determinant

8.6. INVERSION 51

mode (DET) inversion the five first frequencies in table 8.1, column COMB-frequencies(TSVD) are removed. The lowest RMS in the joint TE and TM inversion was found after10 iterations to 3.0, and the lowest RMS for the DET inversion was found after 7 iterationsto 1.2, see figure (8.17).

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Figure 8.17: DET-RMS: 1.2, TE+TM-RMS: 3.0

Even though the TE-mode inversion figure (8.16) shows some odd behavior, the joint TEand TM inversion figure (8.17) is in very good agreement with the corresponding joint TEand TM inversion in Line1 figure (8.15). Primarily there are three conductors that mostlikely are linked together in Line1 and Line3. In both inversions one can see conductorsbetween profiles meters 100-160, 170-290, and 300-380. Predominantly it seems like thereare two conductors that can reach the depths of tunnel lines, and there are the conductorsbetween profile meters 100-160 and 300-380. The DET-model shows features very sim-ilar to the joint TE and TM mode inversion, and the three conductors in discussion areevident also in the determinant (DET) inversion. However, in the DET inversion the lowcontrolled source frequencies are removed and the conductor progressing to a depth of140 m in profile meter 200 is probably regularization’s effects from the inversion.

8.6. INVERSION 52

8.6.3 COMB TSVD TE+TM and DET, Line2

In the joint TE and TM mode inversion, the four first frequencies in table 8.1, columnCOMB-frequencies (TSVD) were removed in the TM-mode data set, whilst the three firstfrequencies in the same column is removed in the TE-mode. In the DET inversion thefour first frequencies in table 8.1, column COMB-frequencies (TSVD) were removed. Thelowest RMS in the joint TE and TM mode inversion was found after 6 iterations as 2.4and the lowest RMS for the DET inversion was found after 5 iterations as 1.3 see figure(8.18).

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Figure 8.18: DET-RMS: 1.3, TE+TM-RMS: 2.4

The same features that were observed in Line1 and Line3 are evident in Line2 as well.The conductors seen between profile meters 100-160 and 170-290 in Line3 figure (8.17),are dislocated to the north in Line2. The real induction arrows (figure 8.7) show 3D effectsthat verify this. None of the conductors in Line2 seem to penetrate down to the tunnellines and moreover the southern part of the profile are strongly influenced by 3D effects,figure (8.8).

8.7. DC-RESISTIVITY SURVEY, LINE1 53

8.7 DC-resistivity survey, Line1

ADC resistivity survey has been executed on the same profile lines as the EnviroMT sur-vey. The DC resistivity profiles have a length of 1000 mwhilst the CSTMT profiles are 450m long. The inversion models shown in figure 8.19 are approximately in the same scale.The programs used for inversion and plotting are not the same for the different methods.The resistivity and the CSTMT models are presented with different colour scales. In ad-dition, the resistivity models are given with topography. The station distance is 10 m, inboth the DC-resistivity and CSTMT survey.

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Figure 8.19: Line1, DC-resistivity inversion (1000m) and CSTMT TE+TM inversion(450m)

The two inversion models show many similar features, but one can also see some differ-ences. The conductors seen in the DC-model between meters 60 and 150, are different incomparison with the SCTMT model. In the DC model, one can see a narrow conductorat the shallow subsurface, whilst in the CSTMT profile (100-160) one can see a conductorthat progresses down to a depth of 100 m. The conductors seen in the DC-model be-

8.7. DC-RESISTIVITY SURVEY, LINE1 54

tween meters 170 and 290, are in better agreement with the CSTMT in terms of shape andlocation. However a big difference can be seen in the depth of which the conductor isresolved. In the DC model, the conductor have a depth that goes from 10 to 50 m, whilstin the CSTMT model the conductor goes from the surface to a depth of 90 m. In the samepart of the DC profile, one can see a lot of resistive features in the subsurface which notare resolved in the CSTMT model. The conductor between meters 300-380 in the CSTMTmodel is also clearly visualised in the DC model. In the CSTMT model, this conductorseems to continue down to a depth of 140 m, whilst in the DC model, it goes to a depthof 70 m.

8.7.1 DC-resistivity survey, Line3

The DC resistivity profile and the CSTMT profile have the same length of 400 m. Theinversion models shown in figure 8.20 are approximately in the same scale. The stationdistance is 10 m, in the DC-resistivity survey and 20 m, in the CSTMT survey, see figure(8.20).

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Figure 8.20: Line3, DC-resistivity inversion (400 m) and CSTMT TE+TM inversion (400m)

8.7. DC-RESISTIVITY SURVEY, LINE1 55

The two inversion models are in good agreement in terms of showing the same features,the differences lie in the actual depth at which the conductors are resolved. The mostconductive features can clearly be seen in both models between meters 100 and 150. Theshapes and sizes of the conductor are similar in both models. Nevertheless a big differ-ence can be seen in the depth at which the conductor is resolved. In the DC model, theconductor goes from 10 to 60 m depth, whilst in the CSTMTmodel, we see the conductorfrom the surface to a depth of 100 m. The conductive zone seen between meters 200 and300, in both models, show the most similar features in terms of size, shape and depth.

8.7.2 DC-resistivity survey, Line2

The DC resistivity profile and the CSTMT profile have the same length of 400 m. Theinversion models shown in figure 8.21 are approximately on the same scale. The twoinversion models are in good agreement in terms of showing the same features, the dif-ferences lie in the actual depth at which the conductors are resolved. The station distanceis 10 m in the DC-resistivity survey and 20 m in the CSTMT survey.

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Figure 8.21: Line2, DC-resistivity inversion (400 m) and CSTMT TE+TM inversion (400m)

Chapter 9

Conclusions

When doing a CSTMT investigation under Swedish conditions one has to be aware ofnear fields effects. However, the near field data give some information about the geol-ogy even when they are interpreted with plane wave assumption. When doing mea-surements along a profile line, the source receiver distance will vary much less, than thedistance it takes to secure plane wave conditions corresponding to the geology. For exam-ple, an electromagnetic field with a frequency of 8 kHz would demand a source receiverdistance of 1.2 km to secure plane wave conditions for a homogeneous half space of 4000Ωm. Whilst if the resistivity was 200 Ωm, a source receiver distance of only 320 m wouldbe adequate to secure a plane wave. This implies that if the near field effects are lessevident in some part of the profile, it generally can be interpreted as a consequence of aconductive geology.

A result that came as a surprise is that the TE-mode seem to be much less effected bythe near filed effects then the TM-mode. The evidence of the near field effects are veryprominent in the TM-mode phase, and are shown by the state of the phase get extremelyhigh. In the TE-mode phase, one can not see any such tendencies. However one cansee discontinuity in one part of the profile, on all three lines. This shows that the dataare a bit disturbed by the near field effects but not nearly as much as in the TM-mode.The apparent resistivity seems to be over all less affected by the near field effects. Butin the TM-mode the apparent resistivity shows higher resistivity than the real apparentresistivity in the near field. In the TE-mode one can not discern any near field effects atall.

The reason for this could be that the TE-mode couples better to conductive structuresthan the TM-mode, which suggest, that the skin depths for the TE and TM mode aredifferent from each other. The TE-mode will have lower skin depth than the TM-mode,corresponding to the above arguments. This would also suggest that the TE-mode needsshorter source receiver distance than TM-mode to secure the plane wave assumption.These arguments could explain why the TE-mode is less influenced by near field effectsthan the TM-mode.

The real induction arrows show some features that are not as clearly displayed in theeither the phase or the apparent resistivity. It seems like the real induction arrows arebetter at detecting differences in conductivity in a more resistive media than the phaseand the apparent resistivity.

57

For the sake of receiving more information about the deeper parts in the inversion. Iallowed more controlled source frequencies in the TE-mode than in TM-mode. This isdone because of the arguments that the TE mode is less affected by the near field effects.It has also been shown that the data fit (RMS) has not been deteriorated, which is anotherindication that the TE mode data are not disturbed by the near field effects. In the jointinversion of TE and TMmode I showed there are no problems of having different numberof frequencies in TE and TM mode.

One has to be aware of the regularisation in the inversion process. Even though theelectromagnetic field only penetrates to a certain depth of which one receive information,the inversion code assigns a resistivity to all cells in the discretized model. REBOCC doesthis with the constraint that themodel should fit the data and have aminimum amount ofstructures. In the TE mode inversion (figure 8.13), the COMB model shows a conductorlocated in profile meter 150, that extends to a depth of 60 m. The same conductor isresolved in the RMT model. But in this model the depth of the conductor is 120 m. Theelectromagnetic field components in the RMT have a maximum skin depth of 60 mwhilstin the COMB inversion with the frequencies used one would expect a skin depth of atleast 120 m. For this reason, one can establish that the deeper parts of the conductor inthe RMT model are just artificial and doe to regularization in the inversion. One can alsosee that the cost of the regularisation is influencing the whole model, where the shallowsubsurface should be exactly the same for the COMB and the RMT models we see thatthe resistivity differs between the models.

When comparing the different inversion models from the CSTMT and the DC-resistivitysurvey one can see a big difference in the depth of which the conductors are resolved. Inthe CSTMT inversion models, many of the conductors which also are recognized in DCinversion models are extending much deeper. In the CSTMT inversion models (TE+TM)there are two conductors that possibly can reach the depth to the tunnel under construc-tion, figure (8.19) and (8.20), which are not at all to be seen in the DC resistivity models.Whether these conductors in the CSTMT inversions (TE+TM) truly extend to the depthat which they are modeled or whether they are in the deeper parts artificial effects ofregularisation can not be said for sure. But considering the frequencies utilised in the TEmode one have very strong arguments that this is a true model.

My results indicate that the CSTMTmethod can to a higher extent resolve conductive fea-tures than the traditional DC resistivity method. This is in fact necessary if the CSTMTmethod should be a competitive geophysical method for land based surveys in compari-son with the DC resistivity method. The CSTMT method has a lot of disadvantages, thatone not have in the robust DC method. Near field effects have to taken into account.Under Swedish condition one need very large source receiver distances to secure planewave conditions which is difficult with the EnviroMT instrument. The CSTMT methodis also very sensitive of electromagnetic disturbance which means that one has to be verycareful that measurements are not executed near power lines or urban areas. Findinga proper place to set up the source can be both difficult and time consuming, also theefficiency of executing the field measurements will increase dramatically if one needs toutilize lower frequencies.

Chapter 10

Acknowledgments

I would like to start to thank Berit Ensted Danielsen for an excellent cooperation doingthe field measurements, also for providing me with the DC resistivity data fromHalland-såsen, thank you verymuch Berit. I want to thank Thomas Kalscheuer for introducingmeto Latex and Rebocc also for giving me clarity in many topics in EM theory. I would liketo thank Laust Börsting Pedersen for always being patient and taking time answering allof my never ending questions in EM theory thank you Laust. I want to thank Nazli Ismailfor helping me with the Enviro MT software and also for the many nice conversations.Many thanks to Lars Dynesius and Mehrdad Bastani for helping and educating me onthe EnviroMT system. I also want to thank Mehrdad Bastani for providing me with yourexcellent PhD thesis. I want to say thanks the sunshine’s in my room at the geophysicalinstitution Hana Karousová and María de los Ángeles García Juanatey. I also would liketo thank María for being opponent on my masters thesis presentation. I would like tosay thanks to my classmates Peter Agerberg and Charlotta Carlsson for the time here atUppsala, thank you very much.

My thanks to all of you for your help.

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