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A 647 OULU 2014 UNIV ER S IT Y OF OULU P. O. BR[ 00 FI-90014 UNIVERSITY OF OULU FINLAND U N I V E R S I TAT I S S E R I E S SCIENTIAE RERUM NATURALIUM Professor Esa Hohtola HUMANIORA University Lecturer Santeri Palviainen TECHNICA Postdoctoral research fellow Sanna Taskila MEDICA Professor Olli Vuolteenaho 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 PUBLICATIONS EDITOR Publications Editor Kirsti Nurkkala 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. 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Geophysics 52: 267–278. 73 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 ACTA UNIVERSITATIS OULUENSIS SERIES A SCIENTIAE RERUM NATURALIUM 630. Stibe, Agnis (2014) Socially influencing systems : persuading people to engage with publicly displayed Twitter-based systems 631. Sutor, Stephan R. (2014) Large-scale high-performance video surveillance 632. Niskanen, Alina (2014) Selection and genetic diversity in the major histocompatibility complex genes of wolves and dogs 633. Tuomikoski, Sari (2014) Utilisation of gasification carbon residues : activation, characterisation and use as an adsorbent 634. Hyysalo, Jarkko (2014) Supporting collaborative development : cognitive challenges and solutions of developing embedded systems 635. Immonen, Ninna (2014) Glaciations and climate in the Cenozoic Arctic : evidence from microtextures of ice-rafted quartz grains 636. Kekkonen, Päivi (2014) Characterization of thermally modified wood by NMR spectroscopy : microstructure and moisture components 637. Pietilä, Heidi (2014) Development of analytical methods for ultra-trace determination of total mercury and methyl mercury in natural water and peat soil samples for environmental monitoring 638. Kortelainen, Tuomas (2014) On iteration-based security flaws in modern hash functions 639. Holma-Suutari, Anniina (2014) Harmful agents (PCDD/Fs, PCBs, and PBDEs) in Finnish reindeer (Rangifer tarandus tarandus) and moose (Alces alces) 640. Lankila, Tiina (2014) Residential area and health : a study of the Northern Finland Birth Cohort 1966 641. Zhou, Yongfeng (2014) Demographic history and climatic adaptation in ecological divergence between two closely related parapatric pine species 642. Kraus, Klemens (2014) Security management process in distributed, large scale high performance systems 643. Toivainen, Tuomas (2014) Genetic consequences of directional selection in Arabidopsis lyrata 644. Sutela, Suvi (2014) Genetically modified silver birch and hybrid aspen – target and non-target effects of introduced traits 645. Väisänen, Maria (2014) Ecosystem-level consequences of climate warming in tundra under differing grazing pressures by reindeer Book orders: Granum: Virtual book store http://granum.uta.fi/granum/ A 647 OULU 2014 UNIV ER S IT Y OF OULU P. O. BR[ 00 FI-90014 UNIVERSITY OF OULU FINLAND U N I V E R S I TAT I S S E R I E S SCIENTIAE RERUM NATURALIUM Professor Esa Hohtola HUMANIORA University Lecturer Santeri Palviainen TECHNICA Postdoctoral research fellow Sanna Taskila MEDICA Professor Olli Vuolteenaho 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 PUBLICATIONS EDITOR Publications Editor Kirsti Nurkkala 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