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SCIENTIAE RERUM NATURALIUM
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SCIENTIAE RERUM SOCIALIUM
ACTA
ELECTRICAL CONDUCTIVITY
STRUCTURE OF
THE LITHOSPHERE IN
WESTERN FENNOSCANDIA
FROM THREE-DIMENSIONAL
MAGNETOTELLURIC DATA
University Lecturer Veli-Matti Ulvinen
SCRIPTA ACADEMICA
Director Sinikka Eskelinen
OECONOMICA
Professor Jari Juga
EDITOR IN CHIEF
Professor Olli Vuolteenaho
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ISBN 978-952-62-0709-4 (Paperback)
ISBN 978-952-62-0710-0 (PDF)
ISSN 0355-3191 (Print)
ISSN 1796-220X (Online)
UN
NIIVVEERRSSIITTAT
ATIISS O
OU
ULLU
UEEN
NSSIISS
U
Maria Cherevatova
E D I T O R S
Maria Cherevatova
A
B
C
D
E
F
G
O U L U E N S I S
ACTA
A C TA
A 647
UNIVERSITY OF OULU GRADUATE SCHOOL;
UNIVERSITY OF OULU,
OULU MINING SCHOOL
A
SCIENTIAE RERUM
RERUM
SCIENTIAE
NATURALIUM
NATURALIUM
ACTA UNIVERSITATIS OULUENSIS
A Scientiae Rerum Naturalium 647
MARIA CHEREVATOVA
ELECTRICAL CONDUCTIVITY
STRUCTURE OF THE LITHOSPHERE
IN WESTERN FENNOSCANDIA
FROM THREE-DIMENSIONAL
MAGNETOTELLURIC DATA
Academic dissertation to be presented with the assent of
the Doctoral Training Committee of Technology and
Natural Sciences of the University of Oulu for public
defence in Kuusamonsali (YB210), Linnanmaa, on 12
December 2014, at 12 noon
U N I VE R S I T Y O F O U L U , O U L U 2 0 1 4
Copyright © 2014
Acta Univ. Oul. A 647, 2014
Supervised by
Doctor Maxim Smirnov
Doctor Toivo Korja
Professor Pertti Kaikkonen
Reviewed by
Doctor Heinrich Brasse
Doctor Josef Pek
Opponent
Professor Thorkild Rasmussen
ISBN 978-952-62-0709-4 (Paperback)
ISBN 978-952-62-0710-0 (PDF)
ISSN 0355-3191 (Printed)
ISSN 1796-220X (Online)
Cover Design
Raimo Ahonen
JUVENES PRINT
TAMPERE 2014
Cherevatova, Maria, Electrical conductivity structure of the lithosphere in western
Fennoscandia from three-dimensional magnetotelluric data.
University of Oulu Graduate School; University of Oulu, Oulu Mining School
Acta Univ. Oul. A 647, 2014
University of Oulu, P.O. Box 8000, FI-90014 University of Oulu, Finland
Abstract
The lithospheric conductivity in the westernmost Fennoscandia has been studied using
magnetotelluric (MT) data. The western margin of Fennoscandia was significantly affected in
Paleozoic by the Caledonian orogeny and later by the rifting of Laurentia and the opening of the
Atlantic Ocean c. 80 Ma ago. Magnetotelluric studies have been carried out in two target areas in
southern Norway and in western Fennoscandia. The first study resulted in 2-D geoelectric models
of two profiles stretching from Oslo to the Norwegian coast. The interpretation suggests that the
basement is in general very resistive with a few conductive upper crustal layers, representing the
alum shales, and middle crustal conductors possibly imaging the remnants of the closed ocean
basins. A more extensive MT study was performed within the project "Magnetotellurics in the
Scandes". Measurements were carried in summers of 2011 to 2013, resulting in an array of 279
MT sites. The data allowed us to derive 2-D geoelectric models for the crust and upper mantle as
well as 3-D models for the crust. The inversions revealed a resistive upper crust and a conductive
lower crust, two upper crustal conductors in the Skellefteå and Kittilä districts, highly conducting
alum shales in the Caledonides and a conductive upper crust beneath the Lofoten peninsula. The
thickness of the lithosphere is around 200 km in the north and 300 km in the south-west. The
Palaeoproterozoic lithosphere is the thickest, not the Archaean, on contrary to a generally accepted
hypothesis. A better image of the lithosphere will help to evaluate the proposed mechanisms of the
exhumation of the Scandinavian Mountains.
The theoretical part of this study is the development of a new multi-resolution approach to 3D electromagnetic (EM) modelling. Three-dimensional modelling of MT data requires enormous
computational resources because of the huge number of data and model parameters. The
development of the multi-resolution forward solver is based on the fact that a finer grid resolution
is often required near the surface. On the other hand, the EM fields propagate in a diffusive manner
and can be sufficiently well described on a grid that becomes coarser with depth. Tests showed
that the total run time can be reduced by five times and the memory requirements by three times
compared with the standard staggered grid forward solver.
Keywords: Caledonides, electomagnetic forward modelling and inversion, electrical
conductivity, Fennoscandia, lithosphere-asthenosphere boundary, magnetotellurics
Cherevatova, Maria, Maankamaran sähkönjohtavuus Fennoskandian länsiosassa
magnetotelluuristen tutkimusten valossa.
Oulun yliopiston tutkijakoulu; Oulun yliopisto, Kaivannaisalan tiedekunta
Acta Univ. Oul. A 647, 2014
Oulun yliopisto, PL 8000, 90014 Oulun yliopisto
Tiivistelmä
Olemme tutkineet litosfäärin sähkönjohtavuutta Fennoskandian länsiosassa magnetotelluurisen
(MT) menetelmän avulla. Fennoskandian länsireuna muokkautui merkittävästi paleotsooisena
aikana Kaledonidien vuorijonopoimutuksessa sekä myöhemmin mesotsooisena aikana Laurentia-mantereen repeytyessä ja Atlantin valtameren syntyessä noin 80 miljoonaa vuotta sitten. MTtutkimukset tehtiin Etelä-Norjassa ja Fennoskandian luoteisosassa. Ensimmäisessä tutkimuksessa kallioperän sähkönjohtavuutta kuvattiin kaksiulotteisilla (2-D) johtavuusmalleilla, jotka ulottuvat Oslosta Norjan rannikolle. Mallien tulkinta viittaa siihen, että maan kuori on pääosin hyvin
eristävä lukuun ottamatta muutamaa kuoren ylä- ja keskiosassa olevaa johdekerrosta. Yläkuoren
johteet edustavat alunaliuskeita ja keskikuoren johteet todennäköisesti suljetuissa merialtaissa
syntyneitä hiilipitoisia sedimenttikerrostumia. Laajempi MT-tutkimus tehtiin ”Magnetotellurics
in the Scandes” -hankkeessa. Mittauksia tehtiin 279 mittauspisteessä kesinä 2011–2013. Saadun
aineiston avulla voitiin laatia 2-D inversiomallit kuoresta ja ylävaipasta sekä 3-D inversiomalli
kuoresta. Tulosten mukaan täällä kuoren yläosa on eristävä kun taas kuoren alaosa on sähköä
hyvin johtava. Edellisen lisäksi malleissa näkyy yläkuoren johtavat muodostumat Skellefteån ja
Kittilän alueilla, korkean johtavuuden alunaliuskeet Kaledonidien alueella sekä johde Lofoottien alla. Litosfäärin paksuus on noin 200 km mittausverkon pohjoisosassa ja noin 300 km lounaassa. Tämän mukaan litosfääri on paksuin varhaisproterotsooisen litosfäärin alueella, ei
arkeeisen litosfäärin alueella vastoin yleistä hypoteesia.
Tutkimuksen teoreettisessa osassa kehitettiin sähkömagneettiseen mallinnukseen uusi monitasoiseen diskretisointiin perustuva menetelmä. MT-aineiston 3-D käänteisongelman ratkaisu ja
siihen liittyvä suora mallintaminen vaativat suuren laskennallisen kapasiteetin, koska havaintojen ja mallin kuvaamiseen tarvittavien parametrien määrä on erittäin suuri. Moniresoluutio-algoritmi perustuu siihen, että mallin hienojakoisempaa diskretisointia tarvitaan yleensä lähellä maan
pintaa kun taas syvemmälle edettäessä, sähkömagneettisen aallon diffuusin etenemisen vuoksi,
malli voi olla karkeampi. Tietokonesimulaatioiden mukaan suoritusaika on viidennes ja muistitarve kolmannes verrattuna tavanomaiseen suoran laskennan ”staggered grid” -diskretisointiin.
Asiasanat: Fennoskandia, Kaledonidit, litosfääri-astenosfääri -raja, magnetotelluriikka,
sähkömagneettinen mallinnus ja inversio, sähkönjohtavuus
To my parents, Alvina and Igor
8
Acknowledgements
I would like to express my greatest gratitude to the people who have helped and supported me throughout my dissertation. I am grateful to my main supervisor Dr. Maxim
Smirnov for his extraordinary support in this thesis process. He has always been patient
in countless discussions, which helped me to better understand the magnetotelluric
method and moved me up to a new level. He shared with me his huge experience in
managing and carrying out magnetotelluric field campaigns providing me an opportunity
to enjoy the field works in Norway and Sweden. He guided and supported me in my
theoretical exercises and codes development, teaching me to be details oriented. I would
like to express my deep regards to my supervisor Dr. Toivo Korja for guiding me in
geological interpretation of my models. He has always been open minded providing
many interesting discussions about various social aspects. This PhD research would
have been impossible without my first supervisor from Saint-Petersburg, Dr. Stanislav
Vagin. I would like to thank him for attracting my interest to geophysics. I would like to
thank Prof. Pertti Kaikkonen for his support with various administrative issues. All
other colleagues and PhD students in our department for thorough discussions during
seminars and in person. I would like to thank the Academy of Finland and the University
of Oulu Graduate School for opportunity to study at University of Oulu and participate
in national and international conferences where I met many famous scientists. Owing to
Dr. Smirnov and financial supports from the Academy of Finland I was able to visit Prof.
Gary Egbert at Oregon State University (Corvallis, Oregon, USA) for several times. I
greatly acknowledge his lessons on 3-D modelling and inversion, which sometimes were
smoothly moved to “Block 15”. I would like to thank him for amazing excursion in
San Francisco. I would also like to thank Dr. Anna Kelbert for many useful discussions
during my stay in Corvallis. Finally, I would like to thank my family who encouraged
me to study and provided a lot of support during these years. Special thanks to my
daughter for being a locomotive for all things I do.
Oulu, November 2014
Maria Cherevatova
9
10
Abbreviations
MaSca
ToSca
2-D
3-D
MT
EM
VTM
PDE
FD
SVD
DLS
CG
NLCG
RMSD
LAB
BBMT
LMT
SSD
NSVB
SSVB
LJZ
TIB
WGR
“Magnetotellurics in the Scandes” project
TopoScandiaDeep project
Two-dimensional
Three-dimensional
Magnetotellurics
Electromagnetic
Vertical Magnetic Transfer Function
Partial Differential Equations
Finite Difference
Singular Value Decomposition
Damped Least Squares
Conjugate Gradients
Non-linear Conjugate Gradients
Root-mean Square Deviation
Lithosphere-asthenosphere boundary
Broad-band magnetotelluric
Long period magnetotelluric
Southwest Scandinavian Domain
Northern Svecofennian Volcanic Belt
Southern Svecofennian Volcanic Belt
Luleå-Jokkmokk Zone
Trans-Scandinavian Igneous belt
Western Gneiss Region
11
12
List of original publications
This thesis is based on the following articles, which are referred to in the text by their
Roman numerals (I–IV):
I
II
III
IV
M. Cherevatova, M. Smirnov, T. Korja, P. Kaikkonen, L. B. Pedersen, J. Hübert, J. Kamm,
T. Kalscheuer, Crustal structure beneath southern Norway imaged by magnetotellurics,
Tectonophysics, 628, 55-70, 2014.
M. Cherevatova, M. Yu. Smirnov, A. G. Jones, L. B. Pedersen and MaSca Working Group,
Magnetotelluric array data analysis from north-west Fennoscandia
(submitted manuscript).
M. Cherevatova, M. Yu. Smirnov, T. Korja, L. B. Pedersen, J. Ebbing, S. Gradmann,
M. Becken and MaSca Working Group, Electrical conductivity structure of north-west
Fennoscandia from three-dimensional inversion of magnetotelluric data
(submitted manuscript).
M. Cherevatova, G. Egbert and M. Smirnov, A multi-resolution approach to electromagnetic
modelling (manuscript).
Reprints were made with permission from the publishers.
An additional extended abstract published in the proceedings of 3DEM5:
– M. Cherevatova, G. Egbert, M. Yu. Smirnov & A. Kelbert, 3-D electromagnetic
modelling using multi-resolution approach, 2013, 5th International Symposium on
Three-Dimensional Electromagnetics (3DEM5), Extended Abstract, Sapporo, Japan.
13
14
Personal Contributions
The papers included in this thesis are the result of a combination of efforts from the
various authors. The individual contributions of the author of this thesis are listed below.
Paper I Performed the data processing, strike and dimensionality analysis, 2-D
inversions and sensitivity tests. Participated in the discussions and interpretation. Wrote
most of the manuscript.
Paper II Co-organized the fieldwork, participated in data acquisition. Performed the
data processing, analysis, 2-D inversions and sensitivity tests. Participated in multiple
strategical discussions. Contributed substantially to the interpretations and wrote most
of the manuscript.
Paper III Performed 3-D inversions, compared 2-D and 3-D models and contributed
substantially to the interpretations. Wrote most of the manuscript.
Paper IV Co-author of the 3-D modelling code on Matlab. Developed and implemented a new ideas within a new multi-resolution approach, tested results. Wrote most
of the manuscript.
15
16
Contents
Abstract
Tiivistelmä
Acknowledgements
9
Abbreviations
11
List of original publications
13
Contents
17
1 Introduction
19
2 Electromagnetic theory
21
2.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Data acquisition and processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Error floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Dimensionality and distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Forward modelling of MT data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Discretization of the forward problem. Finite Difference
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Inverse theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.2 Occam’s inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.3 Data space Occam - Dasocc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.4 Non-linear inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Electrical properties of rocks
37
4 Geological background of the study area
41
4.1 Precambrian continental crust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Caledonides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Thickness of the crust and lithosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Summary of papers
47
5.1 Crustal structure beneath southern Norway imaged by magnetotellurics . . . 47
5.2 Magnetotelluric array data analysis from north-west Fennoscandia. . . . . . . . 51
5.3 Electrical conductivity structure of north-west Fennoscandia from
three-dimensional inversion of magnetotelluric data . . . . . . . . . . . . . . . . . . . . . 57
17
5.4 A multi-resolution approach to electromagnetic modelling. . . . . . . . . . . . . . . . 58
6 Discussion and conclusions
63
7 Future developments
67
References
69
Appendices
73
Original publications
79
18
1
Introduction
The magnetotelluric (MT) method provides delineation of the Earth’s structures ranging
from the depths of a few kilometres to the depths of a few hundreds kilometres scanning,
consequently, the crust, mantle lithosphere and lithosphere-asthenosphere boundary.
Magnetotellurics is a passive non-invasive geophysical electromagnetic (EM) technique
based on measuring Earth’s natural EM field at the surface. Measuring EM field in array
at different frequencies makes it possible to get information about three-dimensional
(3-D) distribution of electrical resistivity at different depth (Simpson & Bahr 2005).
Magnetotellurics is used to solve problems such as the structure of the upper mantle,
which are of fundamental interest, or problems like geothermal and mining exploration,
which are in the field of applied geophysics. Mining exploration is highly demanded in
Finland.
The current thesis is a part of the “Magnetotelluric in the Scandes” (MaSca)
project (the full title of the project “Three-dimensional structure and properties of
the Fennoscandian lithosphere from electromagnetic magnetotelluric measurements.
Multivariate analysis and three-dimensional inversion of synchronous electromagnetic
array data”). The project targets at development and application of the MT method to
study Earth’s structure in the Fennoscandian Shield and on its margins.
The Fennoscandian Shield is defined as the elevated Precambrian north-west
segment of the East European Craton. The absence of the sedimentary cover makes the
Fennoscandian Shield very attractive for MT, as the method is extremely sensitive to the
presence of conductive material within the resistive rocks. The sedimentary rocks are
generally conductive and screen deeper studies. Magnetotelluric data can image the
electrical resistivity of the entire lithospheric column and are therefore one of the most
important data sources for understanding the structure, composition and evolution of the
lithosphere as well as its dynamics.
The research is a natural continuation of the work done in other parts of Fennoscandia.
The long term aim is to construct a 3-D conductivity model for the entire cratonic
Fennoscandia and regions next to it, and use these data to model structures, properties and
dynamics of the Fennoscandian lithosphere in particular and the continental lithosphere
and upper mantle in general. Results from this research bring important information
on the nature of the passive continental margin of the western Fennoscandia. New
19
data also bring important information from an area, which has experienced a recent
(< 30 Ma) intra-plate uplift far from any convergent boundaries of lithospheric plates
(Chalmers et al. 2010). Development of a new approach to carry out the 3-D modelling
and inversion of MT data later will be routinely applied to electromagnetic array data.
The main objectives of the current study are:
– Obtain high-quality MT data in the westernmost, north-west Fennoscandia and
southern Norway;
– Obtain 2-D and 3-D resistivity models of the lithosphere in the transition zone from
the stable Precambrian cratonic interior towards the extended and thinned continental
margin under the Caledonides and the Scandinavian Mountains in the west and
towards oceanic lithosphere further to the west;
– Update the previous knowledge about the thickness of the lithosphere and outline the
lithosphere-asthenosphere boundary;
– Discover the electrical conductivity structure of the crust;
– Improve existing 2-D inversion code and develop a new multi-resolution approach for
3-D EM modelling.
In the following chapters I summarize a few aspects of electromagnetic theory,
discuss on the electrical properties of rocks, give a brief outline of the geology and
tectonic settings of the study area and summarize the major findings in the research. In
Paper I, 2-D inversion results from two MT profiles in southern Norway are presented
and discussed. A damped least squares (DLS) algorithm was improved by author
and used for the inversions. The details of the implementation are given in Paper I
and in Section 2.5.1 of the present thesis. The Papers II and III are dedicated to
MaSca measurements. In Paper II, measured data as well as strike and dimensionality
analyses are presented. Two techniques are compared: the Q-function (Zhang et al.
1987) and the phase tensor (Caldwell et al. 2004). The analyses revealed generally
the 3-D behaviour of the data, yet an average strike was identified. The 2-D inversion
of the determinant of the impedance tensor is presented for the crust and upper
mantle. In Paper III, the 3-D inversion models of the MaSca data are discussed. The
interpretation of both models is given in Papers II and III. The theoretical Paper IV is
dedicated to a novel multi-resolution approach. The first results on the accuracy and
speed-up of the new solver are described in the paper.
20
2
Electromagnetic theory
2.1
Basic concepts
The basic principles of the magnetotelluric method were developed in the 1950s
by Tikhonov (1950) and Cagniard (1953). The complex impedance tensor, defined
initially by Berdichevsky (1960), describes the relationship between orthogonal
components of the electric and magnetic field measured at the surface of the Earth
simultaneously:
"
# "
#"
#
Ex
Zxx Zxy
Hx
=
(1)
Ey
Zyx Zyy
Hy
The skin depth is used as a criterion for the penetration of the electromagnetic wave
and describes the depth where signal decays to 1/e of its original strengths. The skin
depth in media is given as
p
(2)
δ ≈ 500 T ρ,
where T is the period in seconds and ρ is the apparent resistivity of the medium
penetrated, in Ωm (Cagniard 1953). Natural EM field induced in the Earth have
periods ranging from 10−3 to 105 s, assuming an average resistivity of the Earth’s
crust and mantle of 100 Ωm, the depth ranging from a few tens of metres to several
hundreds kilometres.
Apparent resistivity is one of the most frequently used parameters for displaying
MT data and defined as
1
ρ(ω) =
|Z(ω)|2 ,
(3)
ω µ0
where ω is the angular frequency and µ0 is the magnetic constant. Due to the
impedance tensor is complex we can also define the impedance phase
φ (ω) = arg(Z(ω)).
(4)
The vertical magnetic transfer function (VTM or tipper) relates the vertical
magnetic field component Hz and horizontal magnetic field components Hx , Hy
according to relationship (Parkinson 1959)
"
#
h
i H (ω)
x
Hz (ω) = Wx (ω) Wy (ω)
.
(5)
Hy (ω)
21
Fig 1. Schematic figure of an MT site installation after Smirnov et al. (2008).
Induction arrows are vector representation of the complex tipper and used to infer
the presence, or absence of lateral variations in conductivity.
2.2
Data acquisition and processing
From the definition of penetration depth (2), a period range associated with a particular
depth range of interest can be estimated, assuming average bulk conductivities known
a priori for the study area. Different names of the MT method are used depending
on the period (frequency) range of investigation. We measure broad-band (BBMT)
magnetotellurics in the period range from 0.003 s to 1000 s and the long period
(LMT), where the period of EM-field variations is from 10 up to 105 s. To tackle
the problem of the industrial noise, at least two sites are recording simultaneously
(Smirnov et al. 2008), so called remote reference sites. These sites are separated by a
sufficiently large distance that the industrial noise is uncorrelated between the sites.
The distances between sites in the array are defined by the depth range of interest
and horizontal adjustment length, which is the lateral distance to which an MT transfer
function of a given period is sensitive. Thus, BBMT which covers shorter periods
should be installed closer (we use the average distance of 5 km) and LMT with longer
distances (around 25 km). A sketch of an MT station installation is shown in Fig. 1.
22
The system is normally oriented in geomagnetic coordinates. One instrument
consists of an electronic box with the EarthData Ltd. data recorder. For BBMT
site, magnetic field components are measured using three broad-band induction coil
magnetometers Metronix, Germany. The common duration of the measurement
was about 12 h usually during night hours. While the continuous sampling rate is
20 Hz, a night time burst mode was recorded with 1000 Hz during two hours after
midnight when the influence of cultural noise is expected to be lower. For LMT, the
magnetic field is measured using three-component flux-gate magnetometers LEMI
from Lviv, Ukraine. Long-period data were acquired over more than one month, data
are sampled at 1 Hz. Electric field is measured in both cases with two electric dipoles
of normally 100 m long and with Pb/PbCl2 electrodes. Sensors are usually buried at
the depth of a few tens of centimetres to reduce temperature variations, and in the
case of electrodes, to ensure wet environment and low contact resistance of electrodes.
These dipoles are typically configured orthogonal to each other, with one dipole
oriented in the magnetic north-south (N-S) direction, and the other in the magnetic
east-west (E-W) direction. Measurements require non-polarisable electrodes in which
electrochemical effects (which modify the potential difference that is registered)
are avoided as far as possible. A GPS receiver is a standard part of any modern
geophysical equipment. It provides accurate timing and is needed for synchronization
between simultaneous sites as well as for the determination of the exact site location.
The systems are also equipped with an option to use solar panels as a power source
instead of standard 12 V batteries. To configure the system in the field, a computer
with network connection is used.
All transfer functions were estimated using multi-remote reference technique
(Smirnov 2003) as well as novel multivariate analysis technique (Smirnov & Egbert
2012), thus providing stable transfer function in the period range from 0.003–105 s at
BBMT+LMT sites and 0.003–1024 s at BBMT only sites.
2.2.1
Error floors
Error floors are used to avoid an underestimation of the calculated real errors.
We estimate only random standard errors, but the bias error cannot be estimated.
Therefore, error floors are used to take into account bias error. In the 2-D inversion,
23
the error floor for the apparent resistivity and impedance phase are defined as
σρ
ρ
= 2 · ∆|Z|
|Z|
σφ =
∆|Z|
|Z|
,
(6)
e.g., 1% relative error on the impedance corresponds to 2% on apparent resistivity
and 0.57◦ on phase data. With 3-D inversion the impedance components are inverted,
thus it is impossible to assign different error floors to apparent resistivity and
phase, as we did for the 2-D inversion. Accordingly, we set the relative errors as
εZxy = εZxx = σ · |Zxy | and εZyx = εZyy = σ · |Zyx |, where σ is a standard deviation.
Error floor for tipper is defined relatively to unit value. For example, 5% error floor
would correspond to absolute error of 0.05.
2.3
Dimensionality and distortion
For a 2-D Earth, conductivity is constant along one strike direction: σ (y, z). For the
ideal 2-D case, electric and magnetic field are mutually orthogonal: an electric field
parallel to strike (x direction) induces magnetic field only perpendicular to strike
and in the vertical plane (E-polarization), whilst a magnetic field parallel to strike
(y direction) induces electric field only perpendicular to strike and in the vertical
plane (H-polarization). In the measured coordinate system x axis directed to the north
and y to the east (Simpson & Bahr 2005). For a 2-D Earth with the x and y direction
aligned along strike, the diagonal components of the impedance tensor are zero and
impedance tensor becomes:
"
# "
#"
#
Ex
0 Zxy
Hx
=
.
(7)
Ey
Zyx 0
Hy
However, with real data it is not possible to find a direction in which the diagonal
components are zero, due to galvanic effects from small scale local inhomogeneities or
to 3-D induction. There are several approaches on how to determine the dimensionality
of the structure and find correct strike direction (if 2-D). Swift (1967) rotated the
impedance tensor
Z = Rθ Zr R|θ ,
(8)
searching for the EM strike as the angle θ , that maximizes the off-diagonal elements
and minimizes the diagonal elements of the impedance tensor. In eq.8, R is the
rotation matrix, | denotes the transpose of R and Zr is the rotated impedance in
24
the coordinates system aligned with the EM strike. Swift (1967) also provided a
rotationally invariant misfit parameter referred to as Swift skew:
µ=
|Zxx + Zyy |
,
|Zxy − Zyx |
(9)
which indicates the appropriateness of the Swift model to measured data.
Electromagnetic data containing galvanic effects can often be described by
a decomposition model in which the data are decomposed into a non-inductive
response owing to local multi-dimensional heterogeneities and a response owing to
an underlying 1-D or 2-D regional structure. In such cases, determining the strike
involves decomposing the measured impedance tensor into matrices representing the
inductive and non-inductive parts:
"
#"
#
d11 d12
0 Zxy
r
Z = DZ =
,
(10)
d21 d22
Zyx 0
where D is the real frequency independent distortion tensor and Zr is the undistorted
regional impedance tensor. Bahr (1988) suggested that in the coordinate system of the
regional strike (the strike of the regional structure), the tensor elements in the columns
of the tensor should have the same phase. Therefore, the EM strike can be found
by rotating impedance to the coordinate system where the same phase condition
is fulfilled either in the left or in the right column of the impedance tensor. Bahr
(1988) proposed a rotationally invariant parameter termed phase-sensitive skew (or
3D/2D skew) as an adhoc measure of the extent to which an impedance tensor can be
described by the decomposition model (local 3-D heterogeneities and 2-D regional
structure):
p
Im((Zxy + Zyx )(Zxx − Zyy )∗ ) − Im((Zxy − Zyx )(Zxx + Zyy )∗ )
η=
,
(11)
|Zxy − Zyx |
where Im(xy∗ ) = Re(x)Im(y) − Re(y)Im(x). Conclusions about the underlying geometry are made by applying certain thresholds to these parameters. Values of Swift’s
skew above 0.1 indicate more complex dimensionality, whereas for Bahr’s 3D/2D
skew a threshold of 0.3 for the 2-D approximation is recommended (Simpson & Bahr
2005).
According to approach of Zhang et al. (1987), the distorted impedance tensor Z
can be expressed in terms of undistorted impedance tensor Zr as
Z ≈ (I + Ph )Zr ,
(12)
25
where I is the identity matrix and Ph is the distortion matrix. Note, that magnetic field
is relatively free of galvanic distortion caused by local structure, whereas the electric
field will be distorted by real distortion tensor Ph (Zhang et al. 1987). In the case
of the decomposition model, 2-D regional structure and 3-D local structure, in the
coordinate system aligned with the strike
"
#
r
r
Pxy Zyx
(1 + Pxx )Zxy
Z=
.
(13)
r
r
(1 + Pyy )Zyx
Pyx Zxy
Thus the column elements are related by real constants ζ and γ
"
#
r
r
ζ Zyx
Zxy
Z=
,
r
r
Zyx
γZxy
(14)
where
ζ=
Pxy
(1+Pyy ) ,
γ=
Pyx
(1+Pxx )
(15)
and the diagonal components of the impedance tensor are related to the off-diagonal
impedance elements in the same columns as:
Zxx = ζ Zyx
.
Zyy = γZxy
(16)
Estimates ζ , γ and the regional strike angle θ can be determined by minimizing
the target functional with respect to the strike angle θ :
!
1
1
1
2
2
|Zxxi j − ζi Zyxi j | + 2 |Zyyi j − γi Zxyi j | , (17)
Q=
× ∑
2
4N − 3N p − 1 ∑
σyx
σxyi j
i j
ij
where the indexes i and j refer to stations and periods over which the averaging
is done, N = N p Ns with N p denoting the number of periods and Ns – number of
stations. Therefore, the strike direction is defined by searching for the minimum of
√
the Q-functional. For a fixed strike, Q is used as a measure of whether the data
support the assumption of a 2-D regional structure underlying a local 3-D distorter.
The impedance elements are also weighted by errors (σxy or σyx in eq.17), we used
the fixed error floor of 5% of the corresponding off-diagonal impedance elements
instead of real errors because the estimated errors are generally too small (Smirnov &
√
Pedersen 2009). For these conditions, the expectation value of Q will be unity. The
√
stable strike direction for all sites and periods and small Q-function estimates are
necessary and sufficient conditions of data to be described by 2-D regional model.
26
The magnetotelluric phase tensor introduced by Caldwell et al. (2004) is widely
used nowadays to justify 2-D interpretation. The phase tensor geometry provides
information about directionality and dimensionality of the regional structure. The MT
phase tensor defined by the relation
!
Φ
Φ
xx
xy
Φ = X−1 Y =
,
(18)
Φyx Φyy
where X and Y are real and imaginary parts of the complex impedance tensor Z
and Φ is real. The phase tensor has the following properties: (i) the phase tensor is
unaffected by the electric effects of galvanic distortion, (ii) its skew is zero when
the regional structure is 2-D, (iii) it gives the information about the direction of the
regional structure. The phase tensor can be depicted graphically as an ellipse, with
the major and minor axes of the ellipse representing the principle axes of the tensor
(Fig. 2). The orientation of the major axis is specified by the skew angle β and angle
α that expresses the tensor’s dependence on the coordinate system:
"
#
Φmax
0
T
Φ = R (α − β )
R(α + β ),
(19)
0
Φmin
where Φmax and Φmin are the lengths of ellipse semi-axes and the skew angle
Φxy − Φyx
1
β = arctan
2
Φxx + Φyy
and
Φxy + Φyx
1
.
α = arctan
2
Φxx − Φyy
(20)
(21)
The angle θ = α − β is the strike or its perpendicular.
The strike angles found with the described mathematical strategies have a 90◦
ambiguity and have to be supported by tipper data (Section 2.1). In the Parkinson
convention the vectors point towards anomalous internal concentrations of current
(Parkinson 1959). In the Wiese convention (Wiese 1962) the vectors point away from
internal current concentrations. We follow Wiese convention. When the structure is
2-D all these vectors are parallel and perpendicular to the strike direction.
The traditional 2-D inversion requires decomposition of the measured impedance
tensor into E- and H-polarizations, which is not always straightforward. In reality, the
strike often changes with period and from site to site. An alternative approach is the
use of the determinant of the impedance tensor (Ranganayaki 1984):
p
Zdet = Zxx Zyy − Zxy Zyx .
(22)
27
y
Φmax
β
Φmin
α
x
Fig 2. Graphical representation of the phase tensor. The direction of the major axis of the
ellipse, given by the angle θ = α − β , defines the relationship of the tensor to the measured
coordinate system (x, y). Re-drawn from Caldwell et al. (2004).
The 2-D inversion of the determinant in a 3-D environment was carefully studied
and promoted by Pedersen & Engels (2005). The authors concluded that in realistic
3-D situations the determinant has certain advantages over traditional 2-D inversion.
The determinant is an invariant under rotation, i.e. independent of the strike direction.
Nevertheless, a good estimate of a regional strike direction is still needed to correctly
scale the model, that is, sites must be projected onto a profile that is perpendicular to
the strike direction. The second advantage is that the phase of the determinant is free of
electric galvanic distortion, which follows from the fact that det(DZ) = det(D) det(Z).
At the same time, the phases of the main impedance components will remain unaffected by galvanic distortion only in case of an ideal regional 2-D structure. A
regional structure can be considered as ideal in a 2-D sense when there is no change
in strike with depth and 3-D effects are negligible.
2.4
Forward modelling of MT data
2.4.1
Maxwell’s equations
For the MT technique the natural EM source field utilized, being generated by
large-scale ionospheric current systems that are relatively far away from the Earth’s
28
surface. These current systems induces near-vertical plane waves on the surface of
the Earth and EM induction in the conductive Earth is independent of the source
morphology. This assumption can be violated in the areas near the auroral and
equatorial electrojets where the source effect may hamper MT data (Viljanen et al.
1999, Egbert 2002, Lezaeta et al. 2007).
Forward electromagnetic modelling requires the solution of the quasi-static
Maxwell equations in conducting media. At the frequencies used in magnetotellurics,
displacement currents are negligible and diffusion dominates. Thus, essential for
MT method Faraday’s and Ampere’s laws in frequency domain (assuming a time
dependence of eiωt )
∇ × E = −iω µH,
(23)
∇ × H = σ E,
where E is the electric field, H is the magnetic field, σ is the conductivity, µ is the
√
magnetic permeability, ω is the angular frequency, i = −1 (Simpson & Bahr 2005).
The magnetic field can be eliminated from eq. 23, yielding the second-order elliptic
system of partial differential equations (PDEs) in terms of the electric field alone
∇ × ∇ × E = −iω µσ E,
(24)
with the tangential components of E specified on all boundaries. The secondary field
H are then computed directly from the first-order Maxwell’s equations (23):
H = −(iω µ)−1 ∇ × E.
2.4.2
(25)
Discretization of the forward problem. Finite Difference
method.
Forward modelling of MT data involves simulation of the electromagnetic induction
process with a computer program and requires the solution of the Maxwell’s equations
for a given resistivity distribution. Currently, there are three main approaches to
solve the system of Maxwell’s equations: with finite differences, finite elements and
with the integral equation method (Avdeev 2005). These approaches differ in the
discretization of the model space and the numerical solution strategy. We discuss
most explicitly a finite difference modelling approach. Numerical schemes for solving
Maxwell’s equations are often elegantly formulated in terms of a pair of vector field
defined on conjugate grids (Egbert & Kelbert 2012). In the staggered grid convention
29
of Yee (1966) the discretized electric field vector components are defined on cell
edges (Fig. 3). The magnetic field are naturally defined on the discrete grid on cell
faces (Smith 1996, Siripunvaraporn et al. 2002). Thus, the discrete representation
of second order PDEs derived from Maxwell’s equations 24 can be expressed as a
discrete system on the interior nodes as
[C† C + diag(iω µσ (m))]e = 0.
(26)
Here diag(v) denotes a diagonal matrix with the components of the vector v on the
diagonal, C is the discrete approximation of the curl operator (mapping interior cell
edge vectors to interior cell faces), C† is the discrete adjoint of the curl (mapping
interior cell face vectors to interior cell edges). Although e is the full solution vector
(including boundary components). Finally, σ (m) represents the mapping of the model
parameters m (conductivity or resistivity (or its logarithm)) on cell edges, where the
electric field components are defined. The discrete magnetic and electric field are
related via
h = (−iω µ)−1 Ce.
(27)
The EM induction forward problem can also be formulated in terms of magnetic field
according to eq. 25. With this formulation H would be the primary field defined on
edges, and the electric field E = ρ∇ × H would be the dual field (Mackie et al. 1994).
The numerical discretization of the frequency domain EM partial differential
equation is written generically as
Ae = b,
(28)
where the vector b gives appropriate boundary and forcing terms for the particular
EM problem, e is the Ne (number of edges) dimensional vector representing the
discretized electric and/or magnetic field and A = C† C + diag(iω µσ (m)) is an
Ne × Ne coefficient matrix which depends on the M dimensional model parameter m
(Egbert & Kelbert 2012). The system (28) can be solved according to the nature of A.
For 3-D modelling, A can be very large, and the solution of this system infeasible with
direct methods. Therefore, nowadays often iterative methods and pre-conditioners are
used (Avdeev 2005).
The resulting solution of eq. 28 is used to compute predicted data, for example, an
electric or magnetic field component, impedance tensor or apparent resistivity at a set
of site locations. The inversion problem requires computation of the Jacobian matrix,
therefore the forward problem is an essential part of the inversion.
30
Ey(i+1,j,k)
Ex(i,j,k)
Hz(i,j,k)
Ey(i,j,k)
Ez(i+1,j+1,k)
Ez(i+1,j,k)
Hy(i,j,k)
Ez(i,j,k)
Ex(i,j,k+1)
Ex(i,j+1,k)
Hx(i+1,j,k)
Hy(i,j+1,k)
Hx(i,j,k)
Ez(i,j+1,k)
Ey(i+1,j,k+1)
Hz(i,j,k+1)
Ey(i,j,k+1)
Ex(i,j+1,k+1)
Fig 3. Staggered finite difference grid for the 3-D MT forward problem. Electric field components defined on cell edges the magnetic field components can be defined naturally on the
cell faces (Egbert & Kelbert 2012).
2.5
Inverse theory
The goal of the geophysical inversion is to find Earth model that explain the geophysical observations by minimizing a misfit function. The function characterizes
the difference (or similarities) between observed and predicted data calculated by
using an assumed Earth model. In the electromagnetic inversion, particularly magnetotellurics, we have measured data in form of horizontal and/or vertical transfer
function (Section 2.1) for which we try to find the model parameters (resistivity or
conductivity) that can explain our measured data.
In inverse problems, a vector of N measurement readings d = [d1 , d2 , ..., dN ]|
with its corresponding standard errors σ = [σ1 , σ2 , ..., σN ]| is related to a vector of
M model parameters m = [m1 , m2 , ..., mM ]| through a generally non-linear forward
operator F (N × M):
F(m) = d + e,
(29)
31
where e is the prediction error. Equation 29 can be solved directly using the non-linear
forward operator F(m), or this non-linear function has to be linearised first, before the
inverse problem is solved.
The least squares method is designed to solve linear inverse problems Fm = d + e.
The basic strategy is to find the estimated model parameters mest that minimize a
particular measure of the length of the prediction error, e = dobs − d pre , namely its
Euclidean length E = e| e, where d pre = Fmest is the predicted data and dobs is the
observed data. The problem is the calculus problem of the locating the minimum of
the function E and is solved by setting the derivatives of E to zero
∂E
= 0,
∂mj
(30)
where m j is one of the model parameters, j = 1, ..., M. In matrix notation solution will
be
mest = [F| F]−1 F| d.
(31)
For large problems, the computational cost of computing F| F can be prohibitive.
In this case, an iterative solver, such as bi-conjugate gradient algorithm, is preferred.
In almost all geophysical methods, the number of the model parameters exceeds
significantly the number of the data (N M), therefore there is infinite number of
solutions with zero prediction error. This means that many columns of matrix F are
zero or almost zero, thus F| F is singular. Hence, an additional a priori information is
required to single out precisely one solution. One of the simplest kind of a priori
assumption is the expectation that the solution to the inverse problem is simple, where
the notion of simplicity is quantified by some measure of the length of the solution
L = m| m. In this case, we can write a regularization functional Φ(m) (Tikhonov
1950), which considers an original minimization problem (30) and the minimum
length constraint:
Φ(m) = E + λ L = e| e + λ m| m,
(32)
where λ is the regularization or trade-off parameter that determines the relative
importance given to the prediction error E and the solution length L. Consequently,
both E and L have to be minimized in Φ(m). If λ is large enough, the data misfit is
de-emphasized, leading to a smoother model. If λ is set to zero, the prediction error
will be minimized, but no a priori information will be provided to single out the
model parameters. It is possible, however, to find some compromise value of that by
trial and error.
32
By minimizing Φ(m), we obtain
mest = [F| F + λ I]−1 F| d.
(33)
This estimate of the model parameters is called the Damped Least Squares (DLS)
solution (Menke 1989) as λ dampens the singular values in F| F.
The obvious generalization of L is
L = (m − ma )| C−1
m (m − ma ),
(34)
|
where ma is the a priori model parameters and C−1
m = Wm Wm is the inverse covariance matrix of the model parameters, Wm is the weighting matrix of the model
parameters (Appendix 1.2). By suitably choosing the a priori model vector and the
covariance matrix Cm , a wide variety of measures of simplicity can be quantified.
Weighted measures of the prediction error can also be useful
E = e| C−1
d e,
(35)
|
where the matrix C−1
d = Wd Wd is the data covariance matrix (Appendix 1.1) that
defines relative contribution of each individual error to the total prediction error.
Equation 33 with weighting matrices and a priori model is then
−1
est
−1
[F| C−1
= F| C−1
d F + λ Cm ]m
d d + λ Cm ma .
(36)
The linear methods do not work when linearity is invalid, but the problem can
be linearised by expanding F(m) in a Taylor series around the trial solution mn and
keeping the first two terms
∂ Fi (m)
F(m) ≈ F(mn ) +
(m − mn ) = F(mn ) + J∆m.
(37)
∂mj
m=mn
Hence, linearised system of equations:
∆d = J∆m,
(38)
where ∆m is the parameter change vector, r = d − F(mn ) is the prediction error e or
data residual, J is the Jacobian or sensitivity matrix, i = 1, ..., N, j = 1, ..., M. The
Jacobian (N × M) matrix describes the perturbations ensuing for N forward responses
F(m) due to perturbations of M model parameters
Ji j (m) =
∂ Fi (m)
.
∂mj
(39)
33
Then, the linearized Tikhonov’s functional (32) for mn+1 = mn + ∆m is expressed as
| −1
| −1
Φlin
λ (mn+1 , r) = (r−J∆m) Cd (r−J∆m)+λ (mn+1 −ma ) Cm (mn+1 −ma ) = min.
(40)
After minimizing equation 40, the solution ∆m of thus formulated minimization
problem is given through the normal equations
| −1
| −1
−1
J Cd J + λ C−1
m ∆m = J Cd r − λ Cm (mn − ma )
(41)
to yield a new trial solution mn+1 . Equation (41) represents the Gauss-Newton
(GN) least square solution. Since the Taylor-expansion to linearise the forward
modelling operator in eq. 37 is only an approximation, an iterative procedure is sought
to improve mn . The usual procedure used in the Gauss-Newton algorithm can be
addressed as follows:
– solving the forward modelling problem and computing root-mean square deviation
(RMSD):
s
RMSDmn =
1 N (di − Fi (mn ))2
;
∑
N i=1
σi2
(42)
– if RMSD is greater than a desired value, then compute the sensitivity matrix and
solve the normal equations 41;
– steps 1 and 2 are repeated until convergence is reached.
2.5.1
Singular Value Decomposition
There are many variants to solve the basic system of equations 41 numerically. In
the standard damped least squares, ma = mn and the model covariance matrix is an
identity matrix Cm = I. Therefore, eq. 41 reduces to
(J|W JW + λ I)∆m = J|W rW ,
(43)
where JW = Wd J is the normalized Jacobian matrix and rW = Wd r. This system can
be solved using singular value decomposition (SVD) of JW (Golub & van Loan 1996)
JW = UΓV| ,
(44)
where U is an N × N matrix with the data eigenvectors, V is an M × M matrix which
comprises the model parameter eigenvectors and Γ is an N × M diagonal matrix
34
whose diagonal elements γ are the non-negative singular values ordered from the
highest value to the lowest one (γ1 ≥ γ2 ≥ ... ≥ γi ... > 0). In every iteration, the
estimated model corrections can therefore be expressed as:
∆m = Vdiag
γi
U| rW .
γi2 + λ
(45)
In equation (45), λ dampens the very small eigenvalues and defines the trade-off
between model regularisation and data misfit. The optimal selection of λ produces
a minimum model norm and minimizes the data misfit at the same time. At every
iteration, the search for the best parameter λ is performed using the L-curve criterion
(Hansen 1998).
In the current thesis the regularization terms and a priori model ma are considered
(eq. 41) and SVD is applied to the normalised weighted Jacobian
|
Wd JW−1
m = UΓV .
(46)
Hence, the estimated model
∆m
= W−1
m Vdiag
2.5.2
λ
γi
|
−1
U rW − Wm Vdiag 2
V| Wm (mn − ma ). (47)
γi2 + λ
γi + λ
Occam’s inversion
In the Occam approach (Constable et al. 1987), eq. 41 is solved for mn+1 and
re-written as
−1 −1 | −1
mn+1 = (J| C−1
d J + λ Cm ) J Cd d̂n + ma ,
(48)
where d̂n = d − F(mn ) + J(mn − ma ). The unique feature of the Occam approach
is that the parameter λ , is used in each iteration both as a step length control and a
smoothing parameter. That is, equation 48 is solved for a series of trial values of
λ , and the prediction errors E = e| e for each λ is evaluated by solving the forward
problem (Section 2.4). An advantage of this approach is that λ is determined as part
of the search process, and at convergence one is assured that the solution attains at
least a local minimum of the model norm L = m| m, subject to the data fit attained
(Parker 1994).
35
2.5.3
Data space Occam - Dasocc
The Occam scheme can also be implemented in the data space (Siripunvaraporn &
Egbert 2000, Siripunvaraporn et al. 2005). The solution to eq. 48 can be written as
mn+1 = J|n βn ,
(49)
βn = (λ Cd + Jn Cm J|n )−1 d̂n ,
(50)
where βn is an unknown expansion coefficient vector of the basis functions (Cm J|n ).
The inverse problem thus becomes a search for the N real expansion coefficients βn+1 ,
instead of the M-dimensional model mn+1 .
2.5.4
Non-linear inversion
Computing the full Jacobian J is a very demanding computational task for multidimensional EM problems, since the equivalent of one forward solution (Section 2.4)
is required for each row (or column) of J (eq. 39). An alternative is to solve the normal
eqs. 48 or 49 with a memory efficient iterative solver such as conjugate gradients
−1
(CG). This requires computation of matrix-vector products such as (J| C−1
d J + λ Cm )
(e.g. Mackie & Madden (1993)). Alternatively, the penalty functional can be directly
minimized using a gradient-based optimization algorithm such as non-linear conjugate
gradients (NLCG) (e.g. Avdeev (2005), Egbert & Kelbert (2012)). Due to NLCG
avoids explicit use of J and consequently, the cross product J| J, it seems to be
a realistic option to solve the inverse problem in the 3-D MT case. However, in
comparison with Gauss-Newton, the NLCG requires more CG steps for convergence
(Meqbel 2009). With this NLCG approach, one must evaluate the gradient of the
non-linear penalty functional
| −1
Φλ (mn , d) = (d − F(m))| C−1
d (d − F(m)) + λ (mn − ma ) Cm (mn − ma )
with respect to variations in the model parameters m
∂ Φ = −2J| r + 2λ mn .
∂ m mn
(51)
(52)
The gradient is then used to calculate a new “conjugate” search direction in the model
space. After minimizing the penalty functional along this direction using a line search
which requires at most a few evaluations of the forward operator, the gradient is
recomputed.
36
3
Electrical properties of rocks
Electrical resistivity, or its inverse, electrical conductivity, is a material specific
parameter with a large variation between different types of rocks, that span over
ten magnitudes, from one million Ωm for a competent unfractured batholith to one
millionth of an Ωm for the most conducting sulphides and graphite (Jones 1999, Bahr
1988). Sea water has a resistivity of 0.3 Ωm, and highly saline brines can be as low as
0.005 Ωm (Nesbitt 1993). Typical ranges of values for certain rock types and for
some conducting phases (saline fluid and graphite films) are given in Fig. 4 (Simpson
& Bahr 2005). Magnetotelluric method is extremely sensitive to presence of the
conductive material within the resistive rocks, that makes the estimation of electrical
resistivity an excellent tool to image the subsurface.
Resistivity ( Ω m)
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107
Dry igneous rocks
based on laboratory measurements
Old sediments
Young sediments
Sea water
0.5-2 M saline fluids at 350oC
1 vol.% saline fluids at 350oC (25 Sm-1 in 105 Ωm rock matrix)
Connected
5 vol.% interconnected (0.5-2M) saline fluids at 350oC
Isolated
Graphite
1 vol.% graphite film (2x105 Sm-1 in 105 Ωm rock matrix)
Connected
Isolated
Dry olivine at 1000-1200o
Dry pyroxene at 1000-1200o
Wet olivine at 1200o
Partial melt at 1200o
IImenite+garnet
MT
Mid-lower continental crust
Upper continental mantle
Upper oceanic mantle
Fig 4. Electrical resistivities of rocks and other common Earth materials (after Simpson &
Bahr (2005))
37
There are two dominant types of conduction currents in the crust and upper
mantle: electronic conduction and ionic conduction. Electronic conduction (electrons
and polarons as charge carriers) is the dominant conduction mechanism in most solid
materials, and is a thermally-activated process governed by the appropriate activation
energy for the material, Boltzmann’s constant, and the absolute temperature. Ionic
conduction (ions as charge carriers) is the dominant conduction mechanism in fluids,
but is also important for olivine at high temperature (1100–1200◦ C) (Jones 1999).
In order for conductive materials (e.g., graphite, saline fluid or partial melt) to
enhance conductivities, the conductive component must form an interconnected
network within the resistive host medium. Archie (1942), considering the electrical
resistivity of an electrolyte-bearing rock, proposed the empirical formula:
σ = aφ m Sn σ f ,
(53)
where σ (S/m) is the bulk conductivity, a is a proportionality factor, φ is the fractional
porosity of the interconnected pore space, m is a cementation exponent, S is the
fraction of the pores containing water, n is a saturation constant and σ f is the
conductivity of the electrolyte (e.g., brine) (Hashin & Shtrikman 1962).
Rocks are made of various minerals with different orientations including some
fluid phases and grain-boundaries. Grain-boundaries are present in any polycrystalline
aggregates and may enhance or may reduce electrical conduction. When electronic
conduction dominates, grain-boundaries act as barriers for the electric current. When
ionic conduction plays an important role, grain-boundaries may enhance conduction if
grain-boundary diffusion is enhanced. Fluids in general have high ionic conductivity
and therefore if fluids are present they enhance the electrical conductivity of a rock.
Both aqueous fluids (in the crust) and silicate or carbonatite melts are often invoked
to explain high electrical conductivity. Similar to the influence of a fluid phase,
the influence of graphite is sometimes proposed. Graphite has a very high (and
anisotropic) conductivity and therefore even a small amount of carbon can enhance
conductivity if carbon assumes a connected phase. Melts in general have higher
electrical conductivity than minerals. However, the existing of partial melting does not
necessary mean that electrical conductivity anomalies are caused by partial melting.
In order for partial melting to explain high electrical conductivity, a certain amount
of melt is required. One needs to explain if such an amount of melt can exist in a
realistic environment (Karato & Wang 2013, Karato 2014).
38
Structure and composition of Earth’s middle and lower crust are expected to be
laterally heterogeneous (Rudnick & Fountain 1995). The resistivity of the continental
lower crust is normally within the 102 -103 Ωm, with the top of the conductive zone
in the mid-crust (Jones 1992, 2013). The most important variables that may affect
electrical conductivity are: (i) the major element composition (and mineralogy), (ii)
temperature, (iii) the water content, and (iv) the degree of partial melting. The rocks in
the continental middle and lower crust have mafic composition, the dominant minerals
being orthopyroxene, clinopyroxene, and plagioclase (+ some hydrous minerals).
Electrical conductivity in the lower crust mineral depends strongly on minerals
(plagioclase has significantly lower conductivity than clino- and orthopyroxene)
(Karato & Wang 2013).
In contrast to the lower crust, the distribution of temperature and major element
chemistry in the mantle is rather uniform except for the lithosphere. The results of the
laboratory experiments and its comparison with the MT inversion results allow to
make the following conclusions about the distribution of electrical conductivity in
the Earth’s mantle: (i) electrical conductivity in the asthenosphere is ∼ 10−2 S/m
on average, but locally near the top of the asthenosphere conductivity reaches ∼
10−1 S/m. The electrical conductivity of the asthenosphere can be explained by
a modest amount of water (∼ 10−2 wt%) that is consistent with the geochemical
inference. There is no need for partial melting to explain this commonly observed
conductivity of the asthenosphere. Most of large lateral variation in conductivity is
likely due to the lateral variation in the water content. (ii) Electrical conductivity
in the lithosphere-asthenosphere transition zone is generally higher than that in the
asthenosphere. On average, it is ∼ 10−1 S/m, but in the south-central Europe, the
conductivity is low ∼ 10−2 -10−1 S/m, whereas in the eastern Asia, conductivity is
high (∼1 S/m) (Karato & Wang 2013).
Overall, the electrical resistivities of rocks and other common Earth materials span
14 orders of magnitude (Fig. 4). Dry crystalline rocks can have resistivities exceeding
106 Ωm, whilst graphite-bearing shear zones might have resistivities of less than
0.01 Ωm. Such a wide variance provides a potential for producing well-constrained
models of the Earth’s electrical conductivity structure.
39
40
4
Geological background of the study area
The Fennoscandian (Baltic) Shield is located in Fennoscandia (Norway, Sweden
and Finland), north-west Russia, under the Gulf of Bothnia and the Baltic Sea. The
Fennoscandian Shield is defined as the elevated Precambrian north-west segment
of the East European Craton. It is composed mostly of Archaean and Proterozoic
rocks and contains the oldest rocks of the European continent. The Fennoscandian
Shield consists of three quite distinct units viz. the Archaean Domain (3.1-2.9 Ga
Saamian orogeny, Lopian orogeny 2.9-2.6 Ga) in the north-east, the Svecofennian
Domain (2.0-1.75 Ga) in the east, and Caledonian orogen (0.6-0.4 Ga) in the west
(Gaál & Gorbatschev 1987). The western margin of the Fennoscandian Shield was
reworked by Sveconorwegian and Caledonian orogenies 1.25-0.9 and 0.6-0.4 Ga ago,
respectively. (Gaál & Gorbatschev 1987).
4.1
Precambrian continental crust
The Archaean Domain is a typical Neoarchaean granitoid-greenstone province
consisting of granitoid gneiss complexes and supracrustal rocks ranging in age
between 3.1-2.6 Ga. The Svecofennian Domain documents well the formation of
continental crust during early Proterozoic time. In the north-east, along the edge of the
Archaean Domain, there is a continental-margin volcanic belt (Northern Svecofennian
volcanic belt – NSVB). The belt consists of Palaeoproterozoic supracrustal rocks
of partly recycled Archaean rocks with an inferred Archaean basement (Gaál &
Gorbatschev 1987). Another volcanic belt follows the southern boundary of the Shield
across the Baltic Sea – Southern Svecofennian volcanic belt (SSVB) (Gorbatschev &
Bogdanova 1993). After formation the Precambrian continental crust underwent
reworking by subsequent orogenic events. The final major tectonic event in the
area was the intrusion of 1.8 Ga Trans-Scandinavian Igneous Belt (TIB) Revsund
granitoids. There is not a distinct margin between the Archaean and Proterozoic
Domains, but rather a combination of Archaean and Proterozoic material in a belt
some 100 km wide called Luleå – Jokkmokk Zone (Fig. 5).
The Precambrian rocks along the coast of northern Norway are probably a westerly
extension of Archaean and Palaeoproterozoic basement complexes. On the mainland,
the basement rocks are concealed beneath a thick pile of the Caledonian nappes.
41
10˚
20˚
30˚
40˚
68˚
68˚
Archaean
Domain
NSVB
LJZ
64˚
Ca
le
do
nid
es
64˚
60˚
TIB
Southwest
Scandinavian
Domain
56˚
500 km
10˚
Svecofennian
Domain
60˚
SSVB
East European
Platform
56˚
40˚
20˚
30˚
Legend
Archaean Domain
Palaeoproterozoic Svecofennian Domain;
Northern Svecofennian Volcanic Belt (NSVB),
Southern Svecofennian Volcanic Belt (SSVB)
Luleå-Jokkmokk Zone (LJZ)
Palaeoproterozoic Trans-Scandinavian Igneous Belt (TIB)
Mesoproterozoic Southwestern Scandinavian Domain
Caledonides
Sedimentary cover (Neoproterozoic - Phanerozoic)
East European Platform
Fig 5. Main tectonic domains in the Fennoscandian Shield (modified from Koistinen et al.
(2001))
42
The oldest dated rocks consist of tonalite and tonalitic gneiss occurring on islands
north of Tromsø. Many of the well-known mountainous areas in Nordland consist of
Precambrian rocks which form tectonic windows in the Caledonian nappes. The
bedrock in these windows was uplifted and locally deformed during crustal extension
after the Caledonian orogeny. The basement in the 160 km-long Lofoten Wall reaches
far above sea level because, in the period following the Caledonian orogeny, it was
uplifted along a series of faults mostly trending north-east-south-west. The rocks of
Lofoten are approximately as old as those in western Tromsø (Ramberg et al. 2008).
The Meso- to Neoproterozoic Southwest Scandinavian Domain (SSD) resulted
from the collision of Baltica with another major plate, possibly Amazonia (Gaál &
Gorbatschev 1987, Bingen et al. 2008a,b). The Southwest Scandinavian Domain is
divided into the Eastern Segment and four western terranes. The Eastern Segment is a
parautochthonous unit lithologically related to TIB whereas the western terranes,
viz. Idefjorden, Kongsberg, Bamble and Telemarkia, were transported significantly
towards the east (Fig. 6).
The average corrected heat-flow of the Archaean Domain in north-eastern
Fennoscandia is 35±6 mW m−2 , slightly lower than the global average heat-flow
of the Archaean provinces of 41 ±11 mW m−2 (Nyblade & Pollack 1993). The
40 mW m−2 isoline runs roughly parallel to the Luleå–Jokkmokk Zone, generally
through Svecofennian crust. Within the Svecofennian Domain, the heat-flow increases
gradually south-westwards, from 40 mW m−2 in the north-east to nearly 60 mW m−2
in the south-west. The average heat-flow of the Svecofennian Domain is 47±13
mW m−2 , similar to the average heat-flow of Palaeoproterozoic crust (Nyblade &
Pollack 1993). The heat-flow map shows a rather sharp increase from ca. 40 to 50
mW m−2 going from Archaean to Svecofennian crust, whereas the heat-flow is rather
stable around 50 mW m−2 in the rest of the Svecofennian Domain. The Caledonides
yield an average heat-flow of 58±9 mW m−2 (Slagstad et al. 2009).
Later, in Neoproterozoic to Early Palaeozoic time, large parts of Baltica were
covered by fluvial and shallow-sea sediments including organic carbonaceous black
shales having a carbon content up to 10%. The latter, so-called alum shales, comprises
one of the most widespread lithostratigraphic units in Scandinavia (Andersson et al.
1985, Bergström & Gee 1985, Gee 2005). The organic-rich alum shale beds provided a
well-lubricated surface acting as a detachment plane for the nappes in the Caledonian
orogeny along the length of the entire mountain belt (Gee 1980, 2005). Black shales
thus formed a parautochthon in the Caledonian orogeny above the basement rocks of
43
the SSD that remained largely unaffected during the Caledonian orogeny, i.e. formed
an autochthonous basement for the Caledonian nappes.
4.2
Caledonides
The Scandinavian Caledonides (Fig. 5) were formed as the result of a closure of
the Iapetus Ocean and a continental collision of the Baltica and Laurentia in the
Late Silurian (Ramberg et al. 2008). In the Caledonian orogeny (540-400 Ma), the
Precambrian rocks in the western margin of Baltica were thrust beneath Laurentia to
ultra-high pressure depths. The underthrust rocks of Baltica were heated, metamorphosed and deformed (Andersen 1998) whereas the rocks of the Neoproterozoic to
Early Palaeozoic accretionary wedge were transported to the east/north-east over
Baltica as the Caledonian nappes. The Caledonian nappes are generally divided into
lower, middle, upper and uppermost allochthons (nappe series) (Gee & Sturt 1985,
Andersen 1998, Ramberg et al. 2008). The major nappe thrusting was completed
between 400 and 405 million years ago. Today the remnants of the Caledonian
orogen are preserved as a relatively thin cover above the Precambrian basement which
stretches north-east from Ireland and Scotland in the south to Svalbard in the north
(Ramberg et al. 2008).
In the Devonian, the Caledonian mountains were eroded. The extensional collapse
of the Caledonides resulted in rapid tectonic denudation of the orogen, exhumation of
high- to ultra-high-pressure metamorphic rocks and provided a structural template for
the formation of Devonian supra-detachment sedimentary basin (Andersen 1998).
The extensional shear zones and fabrics indicate W- to NW- directed translations
(Gee et al. 2008). Extension-related structures in the north are not as well studied as
in central and southern Norway.
During the Cenozoic, Scandinavia was uplifted twice and mountains were formed
once more (the Scandinavian Mounatins – the Scandes). One uplift event took place
mostly in the Palaeogene and is related to the emplacement of magma from the
Iceland plume and marginal uplift at the start of sea-floor spreading. The second
uplift event occurred during the late Neogene and Quaternary and, besides the uplift
of basin margin areas, also accelerated the subsidence of some basins. In the present,
Fennoscandia and adjacent areas are largely affected by high uplift rates. The rates of
present vertical uplift in Fennoscandia range from close to zero along the Norwegian
coast to more than 8 mm/yr in central parts of the Gulf of Bothnia. Over some areas
44
of the Scandes a significant vertical component of 1-4 mm/yr is observed (Ebbing
et al. 2012).
4.3
Thickness of the crust and lithosphere.
The Archaean Domain and most of the Svecofennian Domain have crustal thicknesses
of 40-50 km. The region of the anomalously thick crust in the Fennoscandian Shield
is located at the Archaean-Proterozoic boundary (Korja et al. 1993), Luleå-Jokkmokk
Zone in Fig. 5. In the Caledonides, a crustal thickness is 32 km at the Norwegian
coast and increases to 43 km beneath the Central Scandinavian Mountains (Ebbing
et al. 2012).
Today, several types of lithosphere are defined, viz. rheological, seismic, electrical,
thermal and elastic lithosphere. Seismic lithosphere is defined as the outer shell with
higher seismic velocities than in the underlying upper-mantle shell (low velocity
zone), the asthenosphere whereas electrical lithosphere is defined as the resistive
outer shell overlying a highly conducting shell (high conductivity zone) in the upper
mantle, also called the electrical asthenosphere (Korja 2007). The boundary between
the lithosphere and asthenosphere, the LAB, is a fundamental boundary in the plate
tectonics separating the rigid material of the plates from ductile convecting material
below on which plates ride. As the best resolved parameter in MT is the top of the
conducting layer the LAB can be detected with high-quality MT data regardless of
the primary cause of enhanced conductivity in asthenosphere. As a consequence, MT
method is well suited to map the depth to LAB (Jones 1999) for basement areas. There
is certain disagreement between electric and seismic LAB estimates for Precambrian
Europe. The depth to the electric LAB is 237±66 km and to seismic LAB is 169
±35 km (Jones et al. 2010). The thickest seismic lithosphere is defined beneath the
Archaean Domain. Thick lithosphere keel extends to 250 km. The region of the
anomalously thick crust in the Fennoscandian Shield is located at the suture between
Archaean and early Proterozoic blocks. This region, where both the crust and the
lithosphere have anomalous thickness, suggest that the crustal and lithospheric roots
could have formed during the same tectonic event that they may represent a unique
preserved remnant of an ancient continent-continent or continent-ocean collision
zone (Artemieva & Thybo 2008). In contrast to seismic studies (Jones et al. 2010),
the thickest electrical lithosphere is in the Palaeoproterozoic Svecofennian Domain
45
(300 km) not in Archaean; the lithosphere is thinning towards the Atlantic and Arctic
Oceans as well as to the east to 100 km (Korja 2007).
46
5
Summary of papers
5.1
Crustal structure beneath southern Norway imaged by
magnetotellurics
In Paper I, 2-D resistivity models are presented for two MT profiles in southern
Norway, ToSca’09 and ToSca’10 (Fig. 6). These measurements were established
within the framework of the TopoScandiaDeep (ToSca) project (Stratford et al.
2008). The major aim of the project is to fill in the gap of MT measurements in
the western margin of the Fennoscandian Shield and investigate the relationship
between the Caledonian cover and the Precambrian basement as well as their internal
conductivity structure. Accordingly, the profiles cross the major tectonic structures of
the Caledonian orogen and western margin of the Precambrian Baltica: the Southwest
Scandinavian Domain, the allochthonous Western Gneiss Region, the Caledonides,
and the Oslo Graben (Fig. 6).
Two profiles were measured in 2009 and 2010 along 280 km and 350 km long
transects, respectively. Altogether 39 broad-band MT sites were acquired along the
ToSca’09 profile, yielding an average site spacing of 7 km. The ToSca’10 profile
consists of 64 BBMT sites with an average site spacing of around 5.5 km. Broad-band
MT data were collected in the frequency range from 0.001 to 300 Hz with the
MTU2000 MT system developed at Uppsala University, Sweden (Smirnov et al.
2008). All data were processed using the robust remote reference code by Smirnov
(2003) to obtain impedance tensors at every site. However, there was no permanent
remote reference site installed and we used only sites measured along the profile for
mutual remote referencing.
The strike and dimensionality analysis of Zhang et al. (1987) was used to assess
to what extent the data can be interpreted in a 2-D sense on a regional scale and to
define proper profile direction (see Section 2.3). Dimensionality and strike analysis
indicates generally 3-D behaviour of the data. However, the majority of the used
data distinguishes a preferable strike direction, which is supported by the geology of
the region. Hence, we choose to invert the determinant of the impedance tensor to
mitigate 3-D effects in the data on our 2-D models (Section 2.3).
The damped least squares (DLS) solver was used within the 2-D inversion code
EMILIA by Kalscheuer et al. (2010). However, the existing classical DLS approach
47
64˚
4˚
6˚
8˚
10˚
To
Trondheim
a
Sc
Caledonian orogen
Östersund
Upper Allochthon
0
’1
63˚
14˚
12˚
Middle Allochthon
Ålesund
Lower Allochthon
FC
MT
62˚
Parautochthon
WGR
Allochthonous basement
TIB
61˚
Valdres
1
a
Bergen
Transscandinavian Igneous Belt (TIB)
Terrain boundary in SSD
1 Vardefjell Shear Zone
k
r
a
m
FG
e
l
e
MUSZ
T
B
8˚
Oslo
Post−Caledonian
Main extensional shear zone
Caledonian Thrust Front
ef
6˚
n
er ent
st
gm
Ea
Se
en
rd
jo
K
Id
59˚
4˚
Autochthon (SSD)
Mjøsa
i
ToSc
a’09
60˚
58˚
Precambrian basement
10˚
12˚
Magnetotelluric site
14˚
200 km
Fig 6. Main geological units of the Caledonian orogen and the Precambrian basement in south-western Fennoscandia, the location of the ToSca magnetotelluric sites
(black/magenta dots and 2-D model profiles (black lines)). Caledonian orogen: Upper allochthon – greenstones, gabbros, mica schists from the Iapetus Ocean (Neoproterozoic –
Silurian); Middle allochthon – basement nappes of Baltica origin (Mesoproterozoic – Cambrian); Lower allochthon – sandstones, phyllites, mica schists, shales from the periphery of
the Baltica basement (Mesoproterozoic – Palaeozoic); Parautochthon – shales, alum shales,
sandstones and carbonate rocks, transformed during thrusting (Cambrian – Silurian); Allochthonous basement (WGR) – Precambrian basement variably affected by the Caledonian
orogeny; Autochthon – basement unaffected by the Caledonian orogeny (Neoproterozoic –
Cambrian). Post-Caledonian extension: FG – Faltungsgraben, MTFC – Møre-Trøndelag Fault
Complex (Devonian). Southwest Scandinavian Domain: MUSZ – Mandal-Ustaoset Fault and
Shear Zone, B – Bamble, K – Kongsberg Terrains (Meso- to Neoproterozoic). The map is
compiled and simplified after Andersson et al. (1985), Gorbatschev & Bogdanova (1993),
Ramberg et al. (2008), Bingen et al. (2008b) and Ebbing et al. (2012).
48
Conductances: red = 0−20 km; blue = 20−40 km; black circles = 0−40 km
Oslo
AS
0
a11
a08
a81
a82
a10
AS+Ph
Graben
lg ρ (Ωm)
4
SSD
10
Depth (km)
a07
MUSZ
a52
a53
a02
a01
b04
a12
Scandinavian Domain
a61
LA
FG
Southwestern
a06
MA
a50
UA
a
a3013
a41
a43
Caledonian nappes
a92
lg S
4
3
2
1
0
SSD
20
PC
PC
PC
3
SSD?
30
Moho
2
40
U2
50
1
60
West
East
0
70
0
50
100
150
200
250
Distance (km)
Fig 7. Interpretation of the crustal conductivity along the ToSca’09 profile. Integrated crustal
conductance (top panel), surface geology from Fig. 6 (middle) and final interpreted model
(bottom). Abbreviations: UA – Upper allochthon, MA – Middle allochthon, LA – Lower
allochthon, FG – Faltungsgraben (extensional shear zone, red pattern), MUSZ – MandalUstaoset Fault and Shear Zone, PC – Precambrian conductor, U2 – upper mantle conductor,
and SSD – the Precambrian Southwest Scandinavian Domain. Moho depth (solid line) after
Stratford & Thybo (2011), RMSD = 1.3.
was improved by author, by adding the smoothness constraints to the model covariance
matrix (see Section 2.5.1 and/or the publication for the details).
The improved DLS scheme was tested on a synthetic data set COPROD2S1
(Varentsov et al. 2002) for the determinant of the impedance tensor. The obtained
results were compared with the models derived from classical DLS, Occam’s and
Rebocc inversion algorithms. The comparative analysis shows that improved DLS is
feasible and allow simple implementation of the different smoothness constraints
(first derivative or second derivative covariance matrices of the model parameters
(Appendix 1.2)). As the synthetic tests are successful, the new DLS was used for the
real data inversion of the ToSca data. The final inversion models for ToSca’09 and
ToSca’10 profiles are presented in Fig. 7 and 8, respectively.
The main findings of the interpreted models are the following:
– the Precambrian basement (autochthonous Southwest Scandinavian Domain and
allochthonous Western Gneiss Region, with respect to the Caledonian orogen) is
generally resistive;
49
WGR
Depth (km)
AS+Ph
d03
d07
d11
d09
d12
d17
d20
d24
AS
PC
FG
WGR−C
20
d22
d32
d30
d36
d42
d41
d46
d44
d51
d48
d55
0
10
30
n a p p e s
LA
FG
d61
d60
d64
C a l e d o n i a n
UA
MA
Western Gneiss Region
MTFC
lg S
Conductances: red = 0−20 km; blue = 20−40 km; black circles = 0−40 km
5
4
3
2
1
0
−1
3
C1
Moho
SSD?
2
40
50
60 West
70
0
lg ρ (Ωm)
4
U1
1
East
0
50
100
150
200
250
300
Distance (km)
Fig 8. Interpretation of the crustal conductivity along the ToSca’10 profile. Integrated crustal
conductances for three different depth ranges (top panel), surface geology from Fig. 6 (middle) and the final interpreted model (bottom). Abbreviations: UA – Upper allochthon, MA
– Middle allochthon, LA – Lower allochthon, FG – Faltungsgraben, MTFC – Møre-Trøndelag
Fault Complex (extensional shear zones; red pattern), AS+Ph – Alum Shale and Phyllite, SSD
– the Precambrian Southwest Scandinavian Domain, WGR-C – region of the Precambrian protoliths of the Western Gneiss Region (WGR), eclogites and non-rotational high strain zones.
C1 – crustal conductor, U1 – upper mantle conductor, PC – Precambrian conductor, question mark – low resolution region. Moho depth (solid line) after Stratford & Thybo (2011),
RMSD of the interpreted model = 1.25.
– the western part of Western Gneiss Region is more conductive (WGR-C) possibly
due to a large scale shearing in the post-Caledonian extension;
– the Southwest Scandinavian Domain contains several sub-horizontal layers, the
high conductivity of which implies an electronic conduction mechanism (graphite,
sulphides);
– the mid-crustal conductors (PC) in the Southwest Scandinavian Domain represent
the remnants of closed ocean basins (e.g., peripheral foreland basins, retroarc
basins, back-arc basins) formed during the accretions and collisions of various
Sveconorwegian terranes;
– the shallower conductors (AS) most likely, according to the data and models
from the Jämtland region, represent highly conducting Late Precambrian to Early
Palaeozoic shallow-sea sediments, so called alum shales, that mark the boundary
between the Precambrian basement and the overlying Caledonian nappes and
represent a decollement along which the Caledonian nappes were overthrust.
50
14˚
70˚
16˚
18˚
20˚
26˚
24˚
22˚
Legend
Tromsø
Archaean
Plutonic rocks and gneisses (3500-2500 Ma)
n
Palaeoproterozoic
fo
te
69˚
Lo
Sedimentary and volcanic rocks (2500-1960 Ma)
Sedimentary and volcanic rocks (1950-1800 Ma)
Plutonic rocks (1960-1840 Ma)
Plutonic rocks (1850-1660 Ma)
Volcanic and intrusive rock belonging to TIB
and Revsund-Sorsele suite (1650-1820 Ma)
68˚
Kiruna Kiruna
Neoproterozoic and Mesoproterozoic
67˚
Sandstone and shale
Caledonides
Mo i Rana
66˚
Bothnian
Bay
65˚
64˚
14˚
16˚
18˚
20˚
22˚
24˚
26˚
Middle Allochthon
Upper Allochthon
Uppermost Allochthon
Lower Allochthon
Basement variably affected by the orogeny
Sedimentary cover (Devonian)
Caledonian Thrust Front
Devonian extensional shear zones
Normal faults
MaSca array MT sites
200 km
Fig 9. Main geological units of the Caledonian orogen and the Precambrian basement
in north-west Fennoscandia. Simplified geology map compiled after Gorbatschev & Bogdanova (1993), Ramberg et al. (2008) and Ebbing et al. (2012).
5.2
Magnetotelluric array data analysis from north-west
Fennoscandia.
The following Papers II and III are dedicated to the project “Magnetotellurics in the
Scandes” (MaSca) (Section 1). The project focuses on the investigation of the crustal
and upper mantle lithospheric structure in the transition zone from stable Precambrian
cratonic interior to passive continental margin beneath the Caledonian orogen and the
Scandinavian Mountains in western Fennoscandia (Fig. 9). Prior to this project, there
were no MT data acquired in the westernmost Fennoscandia, except ToSca profiles
(Section 5.1) and across the central Scandinavian Mountains (Korja et al. 2008).
The MaSca array covers 350 km by 480 km area and stretches from Tromsø and
Bodø (Norway) in the west to Kiruna and Skellefteå (Sweden) in the east (Fig. 5).
51
The project commenced in the summer of 2011 and data collection continued until
summer 2013, resulting all together in 70 synchronous long period (LMT) and 236
broad-band (BBMT) sites. The average separation between LMT sites is 30 km, and
10 km between BBMT; however there are gaps of 50-80 km in places because of
inaccessibility due to the high topography in the region of the Scandes. We also used
other available data sets from the MaSca area. These are six MT sites from BEAR
array (Varentsov et al. 2002, Lahti et al. 2005), four vertical component measurements
from the IMAGE observatories (http://space.fmi.fi/image/), and two small arrays of
Finnish Geological Survey (GTK) (Lahti et al. 2012). All transfer functions were
estimated using multi-remote reference technique (Smirnov 2003) as well as novel
multivariate analysis technique (Smirnov & Egbert 2012), thus providing stable
transfer function in the period range from 0.003-105 s at BBMT+LMT sites and
0.003-1024 s at BBMT only sites. The array is located at high geomagnetic latitudes
therefore application of multivariate technique was vital to validate source field free
transfer functions (see Section 2.4).
One of the major topic of the current paper is the strike and dimensionality
analysis of the MaSca data. We used two techniques for analysis: the Q-function
analysis following the approach of Zhang et al. (1987) and the phase tensor analysis
(Caldwell et al. 2004) (see Section 2.3). A comparison of the two techniques shows a
great consistency between the results. Maps of the Q-function rose diagrams show: (i)
unstable strike in the Caledonides, more consistent NS strike in the Precambrian part
for shorter periods and N45◦ E regional structure. It also agrees with the structural
data, having a dominant geological strike in a NNE–SSW direction in the west, but
contradict to strike N45◦ W for geological mapping in the east. The tipper data are
doubtful with regard to strike direction, as they are affected by the coast effect from
the Atlantic Ocean and Skellefteå anomaly. Therefore, the induction vectors can
not be used as supporting information for strike determination. Since there is no
preferable strike direction for all periods and sites, we choose to invert the determinant
of the impedance tensor, which is invariant under rotation. The selected azimuth of
the profiles is N135◦ E, however we also present the model for azimuth N45◦ E in the
east. We separated inversions into crustal- and lithospheric-scale. To resolve the
crustal features, we selected 7 profiles Crust 1-7 and 4 profiles Lithos 1-4 to resolve
the thickness of the lithosphere (Fig. 10).
The final 2-D inversion crustal models after sensitivity tests are shown in Fig. 11.
The average distance between the profiles is 50 km. A homogeneous half-space initial
52
(a) 70˚ 14˚
69˚
5
6
16˚
18˚
3
4 M30
20˚
14˚
(b)
Crust1
2
16˚
18˚
20˚
22˚
24˚
70˚
Lithos1
2
a12
P08
a07
o21
69˚
68˚
68˚
w10
o12
7
67˚
26˚
24˚
22˚
3
o01
67˚
M10
o52
z08
S08
66˚
65˚
66˚
65˚
4
S02
64˚
64˚
14˚
16˚
18˚
20˚
22˚
24˚
14˚
16˚
18˚
20˚
22˚
24˚
200 km
Fig 10. Location of the magnetotelluric stations plotted on top of the geological map. (a)
Crustal-scale 2-D inversion profiles: Crust 1–7. (b) Lithospheric-scale 2-D inversion profiles:
Lithos 1–4.
model and an ocean a priori model were used for all inversions. The a priori model
was obtained from the map of integrated conductance (S-map) of Fennoscandia (Korja
et al. 2002). Due to the expected 3-D effects and possible static shifts, higher error
floors for apparent resistivity were assigned than for impedance phases (Section 2.2.1).
The phase of the determinant of the impedance tensor is not affected by the electric
effects of galvanic distortion. Thus, more weight is given to the phases. In this study,
we present the final models obtained using the Rebocc algorithm (Siripunvaraporn &
Egbert 2000). The specific inversion parameters are given in the Table 1.
The interpretation of the obtained models shows:
– the Svecofennian resistive rocks extend to the lower crust under the Caledonian
Thrust Front thickening to 50 km westwards;
– the upper crust in Norway is resistive and consists of the rocks of Caledonian
nappes, underlined by Precambrian basement. The basement below the Caledonides
is supposed to belong to the Northern Svecofennian volcanic belt and composed
mostly of Archaean and Palaeoproterozoic rocks and partly of Caledonian nappes;
– the middle to lower crust in the east is more conductive. An enhanced conductivity
in the lower crust may be due to metamorphosed carbon- and sulphide-bearing
53
HC
NW
SE
z13
z08
z07
B16
M41
350
C5−3
Cc
0
10
20
30
40
50
C5−5
HC
NW
SE
C6−4
C6−3
NW
SE
Depth(km)
0
10
20
30
40
50
C7−2
C7−1
S04
q12
400
q16
B15
S05
S06
350
S07
300
S08
g61
q30
250
q37
200
S11
S10
q43
S09
n15
(g)
150
o52
o51
S13
100
S15
S14
50
0
10
20
30
40
50
q06
C6−2
450
q13
q17
q25
q22
q29
RL
C
Moho
400
S03
C6−1
350
q31
300
q52
250
q42
q50
q41
q40
200
n08
o62
n18
o58
n17
150
S17
n14
0
10
20
30
40
50
n20
d43
a15
(f)
100
o65
50
0
10
20
30
40
50
RL
C7−4
C7−3
Moho
NW
SE
150
200
250
300
350
400
0
10
20
30
40
50
450
Distance(km)
Fig 11. Final altered models for the crustal profiles: (a)–(g) Crust 1–7. Abbreviations: RL –
resistive layer, KC – Kittilä conductors, HC – horizontal crustal conductor, C – other crustal
conductors. The bold black line shows the Moho depth for Fennoscandia from (Grad et al.
2009). The red line indicates the Caledonian Thrust Front.
54
Depth(km)
RL
C
300
z05
z04
o62
250
M18
200
M42
150
n21
o65
o73
M19
a16
a20
d13
C5−1
Moho
100
Depth(km)
M15
350
z01
M38
z08
300
M17
250
M18
M42
200
RL
Cc
50
o03
o04
0
q03
C
Moho
150
a08
a07
a11
a18
M14
100
1
0
10
20
30
40
50
C3−5
HC
2
rr01
M15
c16
j28
B23
M39
x05
C3−4
3
400
q14
q15
C3−3
350
SE
50
a19
o05
B31
o04
o06
e04
P01
M37
o14
o13
o11
x02
x01
300
NW
0
10
20
30
40
50
o02
o01
e02
P01
P04
o13
j13
M32
j12
c06a
w02
w01
250
w10
w07
w06
a08
a09
M11
RL
C
(d)
0
10
20
30
40
50
200
lg(ρ, Ωm)
4
0
10
20
30
40
50
SE
150
Moho
0
(e)
400
KC
Moho
NW
a18
M30
a21
0
10
20
30
40
50
350
RL
100
(c)
300
S02
C2
250
B22
Cs
200
o25
o15
M31
P03
SE
P10
o31
a12
y05
P09
M34
P08
a13
0
10
20
30
40
50
0
10
20
30
40
50
KC
Moho
NW
150
(b)
GT3
k13
k11
k10
k09
k08
k07
k06
k05
k03
k02
k01
RL
C1
o21
o18
P06
o17
o27
o26
Depth(km)
0
10
20
30
40
50
k16
k19
Caledonian Thrust Front
(a)
Name
Crust 1
Crust 2
Crust 3
Crust 4
Crust 5
Crust 6
Crust 7
Ns
27
28
23
13
22
22
33
T, s
0.003-8192
0.003-8192
0.003-8192
0.003-46341
0.003-2896
0.003-2896
0.003-8192
Distance, km
305
340
385
370
450
400
380
δ ρa , %;δ φ , ◦
ρinit , Ωm
My × Mz
RMSD
20; 6
30; 6
20; 3
30; 6
20; 3
40; 11
45; 11
104
234× 97
225× 85
225× 85
205× 93
243× 80
221× 80
275× 85
1.2
1.7
1.7
1.3
1.0
1.1
1.1
102
102
103
103
103
102
Table 1. Crustal-scale 2-D inversion parameters. Abbreviations: Ns – number of sites; T –
periods range; Distance – profile length; δ ρa ,%; δ φ , ◦ – error floor for apparent resistivity and
impedance phase in percent and degrees, respectively; ρinit – resistivity of the initial model
(half-space); My × Mz – model mesh size, My and Mz – number of the model parameters in y
and z directions, respectively.
–
–
–
–
sedimentary rocks, transported into deep crustal levels by tectonic processes (Korja
et al. 1996, 2002, 2008);
in the north-east, a region of the enhanced conductivity of a few thousands of
Siemens is observed in the upper and middle crust (the Kittilä conductor). The
conductor in the Kittilä district dips from the near surface (2 km) to the middle
crust (25 km) north-eastward. The highest conductivity in this region is related to
NS elongated graphite- and sulphide-bearing schists (Lahti et al. 2012);
the alum shales are observed along the Caledonian Thrust Front as small conductive
units in the upper crust;
an extensive regional middle to lower crustal conductor is observed in the west of
the Caledonian Thrust Front. The conductor is interpreted as the mica shists of
the uppermost allochthon, which can contain the conductive material. Another
possible explanation is the alum shales transported to a greater depth westward of
the Caledonia Thrust Front;
the upper crust beneath the Lofoten is conductive. One possible explanation of
the enhanced conductivity is the presence of the saline fluids in complicated fault
system within the Lofoten.
The lithospheric-scale inversion is aimed at resolving the thickness of the lithosphere or depth to lithosphere-asthenosphere boundary (LAB). Three parallel profiles
Lithos 1–3 with the azimuth of N135◦ E and one perpendicular profile Lithos 4 with
the azimuth of N45◦ E were selected for 2-D inversion. The average distance between
55
100
200
200
300
300
400
400
350
2
400
(d)
c16
M39
M38
B23
j28
z15
M41
300
M17
250
w05
M42
200
M02
a07
a09
M11
150
lg(ρ, Ωm)
3
Depth(km)
o02
M22
P01
M37
P02
j12
o15
M32
P03
j13
100
M19
M12
a18
M30
M14
a20
o21
o18
P06
o17
M24
P05
k09
k07
k06
k05
k03
SE
0
100
0
1
0
100
200
300
300
400
400
400
B15
S06
350
q31
S07
300
q52
q48
g60
250
S10
(c)
200
S09
q42
q50
q41
q40
150
o51
100
S15
50
S17
0
(d)
S03
?
200
q13
100
q18
(b)
(d)
0
w12
Depth(km)
P10
y03
y04
y05
k18
k17
M34
k15
NW
(a); azimuth 135 deg
0
0
?
100
200
200
300
300
400
400
Depth(km)
300
350
400
450
(a)
o05
M10
o06
j04
c16
B22
z13
q14
(b)
q17
(c)
SW
250
NE
0
0
100
100
?
200
200
300
300
Depth(km)
200
o03
150
(d); azimuth 45 deg
100
400
400
0
50
100
150
Distance(km)
200
250
300
Fig 12. Final 2-D lithospheric-scale inversion models for profiles Lithos 1–3 (a)–(c) with the
azimuth of N135◦ E and profile Lithos 4 (d) with the azimuth of N45◦ E (see Fig. 10 for location).
The polar colour scale is from 10 Ωm to 1000 Ωm. White colour corresponds to 100 Ωm and
indicates lithosphere-asthenosphere boundary.
56
the parallel Lithos 1–3 profiles is 140 km and the azimuth is the same as in case
of crustal-scale inversion (Fig. 10). On the other hand, MaSca array crosses the
Archaean-Proterozoic boundary (Luleå– Jokkmokk Zone in Fig. 5), which is NW-SE
directed. Therefore, we undertook inversion for the perpendicular profile Lithos 4.
The results of the 2-D inversion from four MT profiles in north-western Fennoscandia
suggest that the thickest lithosphere is in the Palaeoproterozoic Svecofennian Domain
(Fig. 12). This is supported by the previous studies of Korja (2007) and Jones (1999),
but contradict to seismic observations (Section 4.3), suggesting thickest lithosphere
in the Archaean Domain. From our results, the thickness of the Palaeoproterozoic
lithosphere reaches 300 km depth and the Archaean lithosphere is 200 km to 250 km.
The thickness of the lithosphere according to Korja (2007) studies is 250 km.
5.3
Electrical conductivity structure of north-west Fennoscandia
from three-dimensional inversion of magnetotelluric data
In Paper III we present 3-D electrical resistivity models of the crust in the north-west
Fennoscandia. The parallel version of the ModEM code was employed (Egbert &
Kelbert 2012, Meqbel 2009) for 3-D inversion (Section 2.5.4). The 3-D inversion is
very expensive in terms of computational time and memory requirements. Thus, some
compromises are required with regards to grid size. To improve resolution properties
of the inversion we subdivided the entire array into smaller sub-arrays with finer grid
resolutions. In the paper we present the inversion results for the entire array and for
one of the selected sub-arrays in the Kiruna district.
For the entire MaSca array four impedance components were inverted (Zxx ,
Zxy , Zyx , Zyy ) together with vertical transfer function (Hz , tippers). A total of 201
stations were carefully chosen from a complete data set of 284, and 13 periods in
the range of 16 – 65536 s for impedance tensor and from 16 to 512 s for tipper. All
together 67 stations had vertical transfer functions among sites used for 3-D inversion.
For 3-D inversion we set error floors 10% of the off-diagonal components for all
impedance elements and 5% for vertical component (Section 2.2.1). We aligned
the 3-D numerical grid with the average geoelectric strike of N45◦ E. The mesh was
discretized to be fairly regular in the centre, with cell widths of approximately 10 km
and comprises a total of 62 cells in the x direction, 74 cells in the y direction, and
80 cells in the z direction. The thickness of the uppermost layer was chosen to be
500 m. Layer thickness was increased with depth by applying a scaling factor of 1.1
57
between each layer. A 1000 Ωm half-space starting model (ocean included) resulted
in geologically consistent models with relatively low RMSD. The final RMSD of this
model is 4.13.
In general, the interpretation of the final 3-D inversion models revealed similar
structures as for 2-D crustal models. A resistive upper crust in the east and in the
west. The central part, beneath the Caledonian Thrust Front, is strongly resistive
down to the upper mantle (Fig. 13b). The lower crust is more conductive, due to the
presence of the graphite-bearing rocks and sulphides. Interestingly, the Skellefteå
conductor (SC) is located in the upper crust, contrary to the 2-D inversion model,
where the strong conductor (C7-4 in Fig. 11g) was placed to the middle crust. The
3-D image of the conductor SC agrees with the earlier studies in this region by Hübert
et al. (2013), where feature SC was interpreted as shallow conductive graphitic shales
within the Vargfors group. Another upper crustal conductor in the Kittilä region
is well seen in 3-D models. This conductor represents the southern margin of the
Kittilä greenstone belt. The enhanced conductivity there related to graphite- and
sulfide-bearing schists, which are visible also in the airborne electromagnetic data of
the study area (Lahti et al. 2012). The highly conductive upper crustal conductors
(CI) within the Caledonides may represent the alum shales or uppermost allochthon
mica schist, which may also contain the conductive material. Similar to 2-D inversion
models, the middle crustal regional conductor (C) is observed beneath Caledonides in
the west. The conductor possibly associated with Caledonian related processes or later
opening of the Atlantic Ocean, which also might have affected the lower crust. The
upper crust below the Lofoten peninsula is conductive, as confirmed by 2-D models.
The enhanced conductivity can be caused by the presence of saline fluids within
cracks and faults in the basement. To conclude, the 2-D and 3-D inversion models
represent nearly the same features and do not contradict to each other. Therefore,
the 2-D inversion of the determinant of the impedance tensor in 3-D conditions is
feasible.
5.4
A multi-resolution approach to electromagnetic modelling.
In Paper VI, we present a multi-resolution approach for 3-D electromagnetic forward
modelling. Development of the technique is motivated by the fact that the finer grid
resolution is often required in the near surface to adequately represent near surface
inhomogeneities and topography. On the other hand, the EM field propagate in
58
x
(a)
N
y
z
Caledonian Thrust Front
LC
Depth (k
m)
CI
CI
0
Kiruna
RL
C
40
RL
80
0
HC
100
SC
400
Skellefte
200
Di
300
sta
n
300
ce
(
)
km
200
km
)
400
100
500
(
ce
tan
s
Di
0
(b)
LC
0
Depth (k
m)
CI
Caledonian Thrust Front
CI
C
RL
Kiruna
C
40
RL
80
0
HC
100
400
200
Di
sta
n
ce
(k
m
)
200
400
100
lg resistivity
0
1
2
3
300
SC
300
4
500
s
Di
)
km
(
ce
tan
0
Fig 13. Perspective view of 3-D inversion model, with NE (a) and NW (b) slices across the
modelled space. Abbreviations: RL – resistive layer, HC – horizontal mid-crustal conductor,
KC – Kittilä conductor, SC – Skellefteå conductor, LC – Lofoten conductor, C – mid-crustal
conductor, CI – upper crustal conductor. Continuous red line – Caledonian Thrust Front.
59
Sub-grid 1
Sub-grid 2
Sub-grid 3
Fig 14. Example of the multi-resolution grid composed of three sub-grids.
a diffusive manner and can be sufficiently well described on a grid that becomes
gradually coarser with depth. With a conventional structured finite-difference grid the
fine discretization required to capture rapid variations near the surface and is continued
to all depths, resulting in high computational costs. We adopt a multi-resolution
finite-difference scheme that allows us to decrease the horizontal grid resolution with
depth, as is done with vertical discretization. A multi-resolution approach, therefore,
provides a means to significantly decrease the number of degrees of freedom and
hence improve on computational efficiency without significantly compromising the
accuracy of the solution.
We represent the multi-resolution grid as a vertical stack of sub-grids. Each
sub-grid represent a standard staggered grid (Fig. 14a). With this approach we are
able to apply operators developed for a standard single resolution finite-difference
scheme within each sub-grid (2.4). The major difficulty in discretizing Maxwell’s
equations on locally refined meshes lies in discretizing the ∇ × ∇× operator around
interfaces between varying cell sizes. To maintain efficiency of iterative solvers, and
to simplify adjoint sensitivity calculations for inversion applications it is desirable
to preserve symmetry of the discretized operators. In particular, the discretized
operator ∇ × ∇ × would ideally be self-adjoint, and the operators ∇ · and ∇ should
be adjoints of each other. A very simple scheme that preserves these symmetries, is
to construct the simpler ∇ × operator on the multi-resolution grid, and then define
∇ × ∇ × in terms of the product of ∇ × and its transpose. However, this scheme is
only first order accurate. We are thus exploring use of more accurate formulations
60
which will necessarily result in an induction operator (A) that is non-symmetric. The
idea here is to develop a more accurate and physically sensible adjoint to the ∇ ×
operator, based on interpolation of field components near interfaces. We will present
results comparing solution accuracy and convergence rate for solutions obtained on a
full high resolution staggered grid and those obtained with symmetric and asymmetric
operator formulations.
61
62
6
Discussion and conclusions
In this thesis the results of the magnetotelluric studies from the westernmost and
north-western Fennoscandia are presented. The high quality broad-band and longperiod magnetotelluric data were measured within the project “Magnetotellurics in
the Scandes” (MaSca). The MaSca array located between E13◦ to E23◦ and N64◦ to
N69◦ , and covers an area of 350 km (NW-SE) by 480 km (NE-SW). All together
70 simultaneous long period and 236 broad-band MT sites were measured and MT
transfer functions in the period range of 0.003-105 s were obtained. Measured time
series of natural variations of electromagnetic field recorded in the MaSca project will
be made available for public use after the project completion.
In Paper II, the processed data, analysis of the strike and dimensionality and 2-D
inversion models are presented. Two techniques for the strike and dimensionality
analysis were used and compared: the Q-function analysis (Zhang et al. 1987) and the
phase tensor (Caldwell et al. 2004). Both methods show consistent results: (i) the data
are generally exhibit 3-D behaviour, (ii) an average strike of N45◦ E of the regional
structure can be identified in the Precambrian part, (iii) more complicated strike in the
Caledonides. Despite of the complicated structure and clear 3-D behaviour of the data
we undertook 2-D inversion using the determinant of the impedance tensor (Section
2.3). A consequent comparison of the 2-D inversion of the determinant and 3-D
inversion, presented in Paper III, proves reliability of our approach to 2-D inversion,
when data exhibit 3-D behaviour.
The lithospheric-scale inversion models, presented in Paper II, reveal a thick
Palaeoproterozioc lithosphere of 300 km thickness in the Svecofennian Domain and
thinner Archaean lithosphere of 200 km – 250 km. These new estimates agree with
the previous compilations by Jones (1983), Korja (2007) and allow us to update
Fennoscandian (and European) LAB map and database and use it to tectono-geological
interpretations of the conductivity data.
The 3-D inversion and interpretation of the MaSca data are discussed in Paper III.
The MaSca array covers large area of around 200 thousands of square kilometres,
thus some compromises are required with regards to grid size. To obtain 3-D models
with finer discretization of the modelled domain, we sub-divided the entire area into
smaller sub-areas. In Paper III, together with the inversion of the entire area we
63
present the inversion for one smaller sub-array near Kiruna. A comparison of the
two 3-D inversion models show that similar features are observed in both models.
However, the Kiruna sub-array inversion provides more contrast model with respect
to resistivity.
The interpreted 2-D and 3-D models reflect similar crustal structures: (i) a resistive
crust in the Precambrian region, which thickens below the Caledonian Thrust Front;
(ii) the lower crust is more conductive; (iii) two upper crustal conductors: in the
Skellefteå district (interpreted as shallow conductive graphitic shales within the
Vargfors group) and the Kittilä conductor related to graphite- and sulfide-bearing
schists; (iv) highly conductive alum shales in the upper crust in the Caledonides
and along the Caledonian Thrust Front; (v) the upper crust beneath the Lofoten is
conductive, an enhanced conductivity there can be caused by he presence of saline
fluids within cracks and faults in the basement; (vi) the middle to lower crustal
conductor under the Caledonian nappes, which can be associated with Caledonian
related processes or later opening of the Atlantic Ocean, which also might have
affected the lower crust.
In Paper I, we present 2-D inversion of the data from southern Norway. Two
profiles ToSca’09 and ToSca’10 were measured in the summers of 2009 and 2010
across the major tectonic structures of the Caledonian orogen as well as the western
margin of the Precambrian Baltica. Two-dimensional models were derived by
inverting the determinant of the impedance tensor using weighted damped least
squares (DLS) (Section 2.5.1). The existing classical DLS scheme was improved by
author by adding a possibility to use the a priori model parameters and the different
model covariance matrices. These changes allowed us to improve the solution
significantly. The tests with the synthetic data set COPROD2S1 (Varentsov et al.
2002) show that the new DLS provides the reliable models, competitive with the
standard Occam’s and Rebocc inversion algorithms. The real ToSca data were inverted
using improved DLS scheme. The major findings are the following: (i) resistive rocks,
extending to the surface, image the autochthonous Southwest Scandinavian Domain
and the allochthonous Western Gneiss Region; (ii) near-surface conductors, which are
located between the resistive Caledonian nappes and Precambrian basement, delineate
highly conductive alum shales; (iii) a deeper, upper to mid-crustal conducting layer in
the Southwest Scandinavian Domain may depict the remnants of closed ocean basins
formed during the accretions and collisions of various Sveconorwegian terranes.
64
The theoretical part of the thesis is dedicated to the development of a new multiresolution approach for 3-D EM modelling. The theoretical issues as well as results
are presented in Paper IV. The multi-resolution approach is motivated by the fact that
the finer grid resolution is often required in the near surface to adequately represent
near surface inhomogeneities and topography. On the other hand, the EM field
propagate in a diffusive manner and can be sufficiently well described on a grid that
becomes gradually coarser with depth. We adopt a multi-resolution finite-difference
scheme that allows us to decrease the horizontal grid resolution with depth, as is done
with vertical discretization. A multi-resolution approach, therefore, provides a means
to significantly decrease the number of degrees of freedom and hence improve on
computational efficiency without significantly compromising the accuracy of the
solution. In Paper IV, we have discussed the discretization of the forward problem on
the multi-resolution grid and defined operators on the interfaces. The accuracy of
the multi-resolution forward solver is within 2% compared to original 3-D solution
and allows computations to be speed up by several times; the scheme also has a
smaller memory foot print. Therefore, inversion based on the multi-resolution grid
forward solver might provide a significant computational advantages, and also allow
greater flexibility in terms of model discretization and better resolution of near-surface
features. Future work will involve testing alternative solvers for the system of linear
equations as we seek to construct more accurate solutions without sacrificing speed.
65
66
7
Future developments
The work presented here focuses on the inversion of the MT data from western margin
of the Fennoscandia. This thesis present the first data within the MaSca project,
measured in the summers of 2011 to 2013. However, the measurement keeps going
and further extension of the array is made in the summer 2014: to the north-east in
Finland near Sodankylä and to the south along the Blue Road seismic profile. These
data are to be inverted in the coming years. The presence of the reflection seismic
profiles in the extended regions makes it possible to compare the seismic and MT
interpretations. In addition, results from the SCANLIPS 2 seismic profile in northern
Norway will be available in the coming years and they will facilitate modelling of
the processes that shaped the topography of the Scandes (England & Ebbing 2012).
Therefore, the inversion models presented in the current thesis can be compared with
the seismic SCANLIPS 2 models.
A multi-resolution approach is still under the intensive development. In the future
we are planning to test different applications of the adjoint operator, different iterative
solvers and pre-conditioners to improve the computational properties of the solver
as well as accuracy. The major aim for development the multi-resolution forward
modelling is to use it in the 3-D inversion. Faster forward computations allow us
faster inversions. Therefore, we aim at integration of the multi-resolution solver into
inversion code. It is still under discussion whether the multi-resolution inversion code
will be developed as a separate code in Matlab, or it will be implemented within the
framework of the existed ModEM 3-D inversion code in Fortran.
67
68
References
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74
Appendix 1 Covariance matrix
1.1
Data covariance matrix
Frequently, some measurements are made with more accuracy than others. Therefore,
it is desirable that more accurate measurement have a great weight in computation of
the prediction error functional, than inaccurate observation. Accordingly, weighted
prediction error functional can be written as
EW = e| W|d Wd e = e| C−1
d e,
where Wd defines the relative contribution of each individual error to the total
|
prediction error and C−1
d = Wd Wd is the data covariance matrix. In the case of
uncorrelated data the data covariance matrix:




Cd = 



σ12
0
0
0
σ22
0
0
0
0
0
σ32
...
0
...
...
...
0
0
0
. . . σN2




,



where σd = (σ1 , σ2 , ..., σN )| represents a vector of the standard deviations and N is
the number of the data.
1.2
Covariance matrix of the model parameters
A similar approach could be used in evaluating the regularization term L in Tikhonov’s
functional (32). The weighted minimum length functional LW , or stabilizing functional, may vary significantly, depending on the form of the model parameter
weighting matrix Wm and expresses an a priori information about the unknown
physical model.
75
For example, one can consider the difference of the laterally adjacent model
parameters as approximation of the first order derivative (Menke 1989):




Wm m = 




m1
−1 1
0 ...
0

m2
0 −1 1 . . .
0 

 m

0
0 −1 1
0  3

..
...

 .
0
0
0
−1 1
mM





.



The weighted minimum length functional then takes form LW = m| W|m Wm m =
|
−1 is the covariance matrix of the model parameters.
m| C−1
m m, where Cm = (Wm Wm )
This matrix can be interpreted as a weighting factor that enters into the calculation of
the length of the vector m.
The weighting matrix can also represent the discrete Laplacian of the model
parameters, which allow stronger smoothing of the inversion models:




Wm m = 




m1
1 −2 1 . . .
0


0 1 −2 1
0   m2

m3
0 0
1 −2 1
0 


..
...

 .
0 0
0 . . . 1 −2 1
mM





.



A difference of the model parameters between vertically adjacent cells as well as
weighting factor on the cell sizes can be added (Constable et al. 1987). In this case,
weighted regularization term has the form
LW = m| W|y Wy m + m| W|z Wz m,
where

−1 0

 0 −1


Wz = 



...
0
..
.
0
...
−1
0
1 0
0 1
0
0
..
.
...

...

... 


,


1 
Wz acts to difference the model parameters between vertically adjacent blocks.
76

wy1
0
wy2





Wy = 




..






,




.
wyMz
0
where wyi – horizontal roughening matrix for layer i, which differences the model
parameters between laterally adjacent blocks in layer i






wyi = 




−∆zi /∆y
∆zi /∆y
−∆zi /∆y
0

∆zi /∆y
..
.
0





.



−∆zi /∆y ∆zi /∆y 
...
0
77
78
Original publications
I
II
III
IV
M. Cherevatova, M. Smirnov, T. Korja, P. Kaikkonen, L. B. Pedersen, J. Hübert, J. Kamm,
T. Kalscheuer, Crustal structure beneath southern Norway imaged by magnetotellurics,
Tectonophysics, 628, 55-70, 2014.
M. Cherevatova, M. Yu. Smirnov, A. G. Jones, L. B. Pedersen and MaSca Working
Group, Magnetotelluric array data analysis from north-west Fennoscandia (submitted
manuscript).
M. Cherevatova, M. Yu. Smirnov, T. Korja, L. B. Pedersen, J. Ebbing, S. Gradmann,
M. Becken and MaSca Working Group, Electrical conductivity structure of north-west
Fennoscandia from three-dimensional inversion of magnetotelluric data
(submitted manuscript).
M. Cherevatova, G. Egbert and M. Smirnov, A multi-resolution approach to electromagnetic modelling (manuscript).
Reprinted with permission from Tectonophysics (I).
An additional extended abstract published in the proceedings of 3DEM5:
– M. Cherevatova, G. Egbert, M. Yu. Smirnov & A. Kelbert, 3-D electromagnetic
modelling using multi-resolution approach, 2013, 5th International Symposium
on Three-Dimensional Electromagnetics (3DEM5), Extended Abstract, Sapporo,
Japan.
Original publications are not included in the electronic version of the dissertation.
79
80
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630.
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