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ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2016 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1394 Airborne Gravity Gradient, Magnetic and VLF datasets Case studies of modelling, inversion and interpretation SAYYED MOHAMMAD ABTAHI ISSN 1651-6214 ISBN 978-91-554-9631-9 urn:nbn:se:uu:diva-300126

Airborne Gravity Gradient, Magnetic and VLF datasets950858/FULLTEXT01.pdfInversion of TMI data is the topic of the third paper where a new type of reference model for 3D inversion

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ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2016

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1394

Airborne Gravity Gradient,Magnetic and VLF datasets

Case studies of modelling, inversion andinterpretation

SAYYED MOHAMMAD ABTAHI

ISSN 1651-6214ISBN 978-91-554-9631-9urn:nbn:se:uu:diva-300126

Dissertation presented at Uppsala University to be publicly examined in Hambergsalen,Geocentrum, Villavägen 16, Uppsala, Monday, 19 September 2016 at 10:00 for the degreeof Doctor of Philosophy. The examination will be conducted in English. Faculty examiner:Professor Thorkild Rasmussen (Luleå University of Technology, Department of Civil,Environmental and Natural Resources Engineering).

AbstractAbtahi, S. M. 2016. Airborne Gravity Gradient, Magnetic and VLF datasets. Case studiesof modelling, inversion and interpretation. Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology 1394. 59 pp. Uppsala: ActaUniversitatis Upsaliensis. ISBN 978-91-554-9631-9.

Northern Sweden is one of the largest hosts for mineral resources in Europe and always has beenan interesting area for researchers from various disciplines of Earth sciences. This dissertationis a comprehensive summary of three case study papers on airborne VLF, gravity gradient andmagnetic data in the area.

In the first paper, tensor VLF data is extracted from an old data set which contains onlythe total and the vertical magnetic components. The anomalous part of the horizontal magneticfield components is computed by a Hilbert transform of the vertical magnetic field. The normalpart of the horizontal magnetic field component is computed as a function of total, vertical andanomalous part of horizontal magnetic fields. The electric field is also calculated for TE modeand impedance tensor and apparent resistivity are computed. In addition tippers are calculatedfor two transmitters and inverted by a 3D inversion algorithm. Comparison of the estimatedmodel and geology map of bedrock shows that lower resistivity zones are correlated withmineralizations.

The second paper deals with the internal consistency of airborne gravity gradient data. Thesix components of the data are estimated from a common potential function. It is shown that thedata is adequately consistent but at shorter land clearances the difference between the estimateddata and the original data is larger. The technique is also used for computing the Bougueranomaly from terrain corrected FTG data. Finally the data is inverted in 3D, which shows thatthe estimated density model in shallow depth is dominated by short wave length features.

Inversion of TMI data is the topic of the third paper where a new type of reference model for3D inversion of magnetic data is proposed by vertically extending the estimated magnetizationof a 2D terrain magnetization model. The final estimated 3D result is compared with themagnetization model where no reference model is used. The comparison shows that using thereference model helps the high magnetization zones in the estimated model at shallow depths tobe better correlated with measured high remanent magnetization from rock samples. The highmagnetization zones are also correlated with gabbros and volcanic metasediments.

Keywords: VLF, gravity gradient, magnetic, airborne, inversion, non-uniqueness

Sayyed Mohammad Abtahi, Department of Earth Sciences, Geophysics, Villav. 16, UppsalaUniversity, SE-75236 Uppsala, Sweden.

© Sayyed Mohammad Abtahi 2016

ISSN 1651-6214ISBN 978-91-554-9631-9urn:nbn:se:uu:diva-300126 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-300126)

Dedicated to my parents

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Abtahi, S.M., Pedersen, L.B., Kamm J. and Kalscheuer, T.

(2016) Extracting geoelectrical maps from vintage very-low-frequency airborne data, tipper inversion, and interpretation: A case study from northern Sweden. Geophysics, 81(5), B135-B147.

II Abtahi, S.M., Pedersen, L.B., Kamm J. and Kalscheuer, T. (2016) Consistency investigation, vertical gravity estimation and inversion of airborne gravity gradient data – A case study from northern Sweden. Geophysics, 81(3), B65-B756

III Abtahi, S.M., Pedersen, L.B., Kamm J. and Kalscheuer, T. (2016) A new reference model for 3D inversion of airborne magnetic data in hilly terrain – a case study from northern Swe-den. submitted to Geophysics

Reprints were made with permission from the respective publishers.

Personal contribution

In Paper I, I did all of the computations. I also wrote most of the text (which was checked and revised in detail by the co-authors) and I creat-ed all of the figures.

In Paper II, I developed the code and did most of the computations (ex-cept nonlinear data inversion). I also wrote most of the text (which was revised in detail by the co-authors) and I created all of the figures.

In paper III, all of the computations were done by me and I also almost wrote almost the entire text of the paper that has been modified and cor-rected in detail by the co-authors. All of the figures are made by me.

Contents

Summary in Swedish .................................................................................... 11 

1. Introduction ............................................................................................... 15 

2. VLF method .............................................................................................. 19 2.1 VLF sources ....................................................................................... 19 2.2. VLF data ............................................................................................ 20 2.3. Scalar and tensor VLF measurements ............................................... 21 2.4. Inversion of VLF data ....................................................................... 22 2.5. Major problems with VLF surveying ................................................ 23 

2.5.1. Variation of normal field ........................................................... 23 2.5.2. The effect of topography ........................................................... 24 2.5.3. Anthropogenic effects ................................................................ 25 

3. Potential field methods ............................................................................. 27 3.1. Gravity gradient method .................................................................... 27 

3.1.1. Full tensor gravity gradient data ................................................ 28 3.1.2. Noise in gravity gradient data .................................................... 28 

3.2. Magnetic method ............................................................................... 29 3.2.1. Magnetic field potential ............................................................. 30 3.2.2. Magnetic and gravity gradient relation ...................................... 30 3.2.2. Major problems with accuracy of magnetic data ....................... 31 

3.3. Inversion of potential field data ......................................................... 31 

4. Geophysical inverse theory ....................................................................... 33 4.1. Linear inverse problems .................................................................... 33 4.2. Model structure functions .................................................................. 36 

4.2.1. Flatness function ........................................................................ 36 4.2.1. Depth weighting ......................................................................... 36 

4.3. Non-linear inverse problems ............................................................. 37 

5. Geology of the study area ......................................................................... 39 

6. Summary of papers ................................................................................... 41 6.1. Paper I ............................................................................................... 41 

6.1.1. Summary .................................................................................... 41 6.1.2. Discussions ................................................................................ 44 

6.2. Paper II .............................................................................................. 45 

6.2.1. Summary .................................................................................... 45 6.2.2. Discussion .................................................................................. 48 

6.3. Paper III ............................................................................................. 49 6.3.1. Summary .................................................................................... 49 6.3.2. Discussion .................................................................................. 52 

Acknowledgments......................................................................................... 53 

Bibliography ................................................................................................. 54 

Abbreviations

1D one-dimensional 2D two-dimensional 3D three dimensional ASL above sea level BSL below sea level EM electromagnetic SGU Geological survey of Sweden VLF very low frequency TE transverse electric TM transverse magnetic Ga billion years (gigayears) LKAB Luossavaara-Kiirunavaara Aktiebolag FTG full tensor gravity gradient TMI total magnetic intensity

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Summary in Swedish

Idén att använda flygburna system for insamling av geofysiska data blev föreslagen efter andra världskriget. Genuppbyggnaden av världens länder krävde stora mängder nya naturresurser som metaller, kol, olja, etc. Industrin kände till att det fanns stora icke-undersökte områden i världen som möjlig-en var rika på mineralresurser, men traditionella prospekteringsmetoder ba-serad på mätningar på marken skulle bli mycket dyra att använda. Dessutom var många av dessa områden väldigt svåra att arbeta i rent logistiskt, såsom Canada eller Sibirien, på grund av deras oåtkomlighet och kalla klimat. Un-der dessa omständigheter blev flygburna geofysiska metoder utvecklad för snabb och effektiv skanning av jorden (Palacky and West, 1973)

I Sverige påbörjade Sveriges Geologiska Undersökning (SGU) ett nation-ellt program för systematiska flygburna geofysiska mätningar i 1965 med magnetiska mätningar med nominell flyghöjd 30 m som senare ändrades till 60 m över landytan (SGU, 2014a). Senast genomförde bolaget Bell-Geospace i 2013 för första gången i Sverige flygburna mätningar av gradient tensorn av tyngdfältet (engelsk förkortning AIR-FTG) i ett område i norra Sverige inom Barent projektet (Bell Geospace Ltd, 2013). Flygburna VLF mätningar påbörjades i 1973 av SGU med användning av enbart en sändare. Senare användes signalen från två sändare för att framställa kartor av för exempel svaghetszoner med godtycklig orientering. Ett exempel är från 1985 när LKAB (Luossavaara-Kiirunavaara Aktiebolag) mätte alla tre komponen-ter av magnetfältet men lagrade bara vertikalkomponentens reella och ima-ginära del tillsammans med storleken av det totala magnetiska fältet. På 90-talet implementerade SGU tensor VLF metoden varvid de framtagna kar-torna visar alla möjliga riktningar av elektriskt ledande strukturer i det sedi-mentära täcket eller i den underliggande kristallina berggrunden. (SGU, 2014b).

Denna avhandling är en utförlig sammanfattning av författarens arbete med flygburna VLF, tyngd gradient och magnetiska data från norra Sverige. Den består av fem kapitel om grundläggande geofysiska aspekter och geolo-gin i området samt om två publicerade artiklar och ett inskickat manuskript. Jag introducerar VLF metoden i det andra kapitlet med en kort beskrivning av den teoretiska bakgrunden. Det tredje kapitlet beskriver grundläggande potentialteori för geofysiska ändamål inklusive tyngd gradient och magne-tiska metoder. Geofysisk invers teori beskrivs kort i det fjärde kapitlet. I det

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femte kapitlet ges en kort beskrivning av geologin i området som studerats. Slutligen sammanfattas i det sjätte kapitlet de tre ingående avhandlingarna.

Avhandling I beskriver problemet att beräkna VLF tensor komponenter ifrån äldre VLF data som bara innehåller real- och imaginärdelen av det ver-tikala magnetfältet och storleken av totalfältet. I avhandlingen förklaras hur man kan beräkna det anomala horisontalfältet ifrån det uppmätta vertikalfäl-tet med hjälp av relationer uttryckt i vågtalsdomänet. Det normala magne-tiska horisontalfältet beräknas med hjälp av storleken på totalfältet och upp-mätta och beräknade vertikala och horisontella anomala komponenterna, respektive. Därefter beräknas tipper vektorerna från de beräknade magne-tiska komponenterna med användning av signalerna från två sändare. Tre-dimensionell inversion av de beräknade tipper vektorerna genomföras med användning av algoritmen utvecklad av Kamm and Pedersen (2014). Jag beräknade det anomala elektriska fältet baserad på relationer givet av Beck-en and Pedersen (2003). Normala elektriska fältkomponenter beräknades utgående från en rimlig skattning av genomsnittliga resistiviteten i måtområ-det. Med de beräknade elektriska och magnetiska fältkomponenter kunde impedanstensorn beräknas och apparent resistivitet av determinantimpedan-sen bestämmas. Karta av apparent resistivitet jämfördes med karta av topo-grafi och svaghetszoner. Zoner med låg och hög apparent resistivitet utvisar betydande korrelation med dessa zoner och med hög topografi, respektive.

I avhandling II undersöks tyngd gradient komponenternas interna konsi-stens inom ett mindre område av det större området som studerades i av-handling I. Genom att uttrycka Laplace’s ekvation i vågtalsdomänet beräk-nas de 6 komponenterna av tyngd gradienttensorn utgående från en och samma potentialfunktion. Jag viste att de uppmätta komponenterna är bias-fria. Jag beräknade tyngdfältet på terrängen från terrängkorrigerade tyngd gradient data och jämför med uppmätta tyngddata. Som ett extra resultat utvecklades en teknik för kontinuering av potentialfält data från en mätyta till en godtycklig yta som ligger över terrängen till exempel en plan yta som inte korsar terrängen. Slutligen estimeras en 3D densitetsmodell med hjälp av algoritmen som beskrivs i Kamm et al (2015) där man använder data uppmätta på en plan yta.

I avhandling III beskrivs en ny metod for estimering av en referensmodell som ska ingå i en 3D modell av magnetfältdata. Först bestäms en 2D magne-tiseringsmodell genom inversion av viktade magnetfältsdata genom en itera-tiv procedur så att en ”bästa” modell med bra korrelation med topografin väljs manuellt. I denna procedur tillåts bara laterala variationer av magneti-seringen och modelldomänen är begränsad till att ligga mellan högsta och lägsta punkt av terrängen. Viktningen av magnetfältsdata definieras från den beräknade normen av tyngdgradienten från terrängen under ett antagande av konstant densitet. Den valda ”bästa” 2D magnetiseringsmodell förlängs neråt så att den vertikala tyngdpunkten blir den samma för alla kolumner i mo-dellen. Härtill används ett genomsnittligt djup som beräknat från avkling-

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ningen av powerspektret som funktion av vågtalet. En bästa 3D modell esti-meras genom att minimera dataanpassningen till en given rms tolerans så att normen av differensen mellan modellen och referensmodellen blir minst möjligt. Magnetiseringsrikningen tillåts att variera fritt. En 3D bild av zo-nerna med störst magnetisering visar en bemärkelsevärdig korrelation med gabbror och metasedimentära bergarrter. Jag fann att de estimerade magneti-seringsriktningarna är helt orealistiska, men detta beteende är oundvikligt när magnetiseringen tillåts variera fritt. Storleken av den estimerade magne-tiseringsfördelning visar emellertid en jämn variation och dess norm er be-tydligt mindre än motsvarande norm om magnetiseringen fastfrysas till en bestämd riktning som i standard inversion a magnetfältsdata.

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1. Introduction

The idea of using airborne systems for geophysical data acquisition was proposed after World War II. Reconstruction of the world demanded a large amount of new raw materials like metals, coal, hydrocarbons, etc. Industry noticed that there were vast unexplored areas in the world that possibly were rich in mineral resources, but traditional prospecting methods in the field would be quite expensive. In addition, many of those areas had difficult con-ditions for field work, like Canada or Siberia because of their harsh and cold climate. In these circumstances, airborne geophysical surveying methods were developed to scan the Earth fast and effectively. Airborne magnetome-ters were the first instruments invented for aerial geophysical reconnaissance surveys based on experiences of detecting of U-boats by flux-gate devices during the war and airborne proton magnetometers were developed later (Parasnis, 1962; Palacky and West, 1973).

Airborne magnetic surveying systems were effective for prospecting iron ores, but were not useful for many other nonmagnetic metals and, instead, EM methods were a better choice for those minerals. The first EM prospect-ing equipment was a device with two mounted coaxial receiver and transmit-ter coils with axis parallel to the flight line direction, installed on a helicopter in 1946. The coils were mounted 10 to 20 m apart each other and the motion of the coils was minimized carefully so that the least possible spurious noise in the in-phase component appeared. Using fixed wing aircrafts was more interesting because of their longer range, higher speed and greater pay-load, but unlike helicopters, it was hard to fix the distance between the coils and, hence, the measured in-phase component was not reliable (Parasnis, 1962). Then, in 1953, dual frequency EM instruments were designed and installed on airplanes. These systems measure only the quadrature (which is approxi-mately purely due to anomalous fields) for two selected low and high fre-quencies. The ratio between low and high quadrature values provides a qual-itative knowledge about the conductivity of underground structures. Eco-nomic success of dual frequency EM methods led to develop VLF (very low frequency) EM instruments which became popular in the 1970s. The main advantage of VLF method was that VLF transmitters have been providing trustable normal fields which can be easily considered as plane waves at large distances and, hence, the computation of the model response becomes much simpler than using transmitters on board the airplane (Parasnis, 1962; Palacky and West, 1973).

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The gravity method is also a capable apparatus in mineral and hydrocar-bon prospecting. For instance, mapping of a salt dome can be effectively done using the gravity method, because the method is sensitive to the high density contrast between the salt and the surrounding rocks (Eckhardt, 1940; Dransfield, 2007). However, unlike magnetic and electromagnetic methods, the use of airborne gravity methods did not attract much interest for several decades because the gravimeters are too sensitive to the aircraft accelerations and, hence, the measured gravity data would have been very noisy. Theoreti-cally it is possible to compute the inertial acceleration of aircraft and subtract it from the measured gravity if the exact location of each measurement point is known. However, due to uncertainty of GPS systems, in practice it is not easy to separate pure Earth gravity from aircraft inertial accelerations (Bruton, 2002; Lane, 2004). The solution to the problem is to measure the gravity gradient components instead. Also, more precise mapping and more accurate shape and depth information can be extracted from gravity gradient data for exploration purposes (Dransfield and Christensen, 2013). By using several acceleration gradient sensors mounted on a rotating disk, the effect of inertial aircraft acceleration will be the same for all of the sensors and, hence, can be substantially compensated (Lee, 2001; Lane, 2004). The first airborne gravity gradient data were successfully measured by the Air Force Geophysics Laboratory for mapping regional gravity in the USA (Jekeli, 2006) and, later, gravity gradients became an effective method for mineral exploration purposes(Bell et al., 1997; Pawlowski, 1998; van Leeuwen, 2000; Dransfield, 2007; Beiki and Pedersen, 2010).

In Sweden, systematic airborne geophysical measurements began in 1965 by the Geological survey of Sweden (SGU) with airborne magnetic meas-urements done nationwide at a nominal flight altitude of 30 m or later 60 m (SGU, 2014a). In 2013, the company Bell Geospace, carried out for the first time in Sweden airborne full tensor gravity gradient (Air-FTG) within Bar-ents project for mapping of northern Sweden (Bell Geospace Ltd, 2013). Airborne VLF surveying was done in 1973 by SGU with only one transmit-ter and later two transmitters were used to construct maps of for example fracture zones with arbitrary orientations. An example is from 1985, when the company LKAB (Luossavaara-Kiirunavaara Aktiebolag) performed a survey in northern Sweden. LKAB measured all components of the magnetic field, but only stored the in-phase and quadrature components of the vertical magnetic field and the magnitude of the total magnetic field. In the 1990s, SGU implemented the airborne tensor VLF method whereby maps can be produced that reflect all possible orientations of conducting structures in the sedimentary cover or in the underlying crystalline crust (SGU, 2014b).

This thesis is a comprehensive summary of the author’s works on air-borne VLF, gravity gradient and magnetic data sets surveyed in northern Sweden. It consists of five chapters about the basic theoretical aspects and geology and two published or accepted papers and one submitted manu-

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script. I introduce the VLF method in the second chapter with a brief back-ground of theory. The third chapter deals with potential field theory in geo-physics including gravity gradient and magnetic methods. Geophysical in-verse theory is described shortly in the fourth chapter. In the fifth chapter, the background geology of the study region is described. Finally, the sixth chapter summarizes the results of the three papers.

Paper I deals with the problem of extracting VLF tensor components from a vintage data set which includes only in-phase and quadrature parts of the vertical magnetic field and the magnitude of the total magnetic field. The paper explains how to extract horizontal anomalous magnetic field data from the vertical magnetic data using relationships in wavenumber domain. Then the horizontal normal magnetic field components are estimated based on the magnitude of the total magnetic field and the horizontal and vertical anoma-lous magnetic field components. Subsequently, I compute the tipper vectors based on the calculated magnetic vector field components using signals from two VLF transmitters. Next, three dimensional inversion of the calculated tipper vectors is done using the approach of Kamm and Pedersen (2014). A plan section of higher and lower resistivity zones of the estimated model is superimposed on a geological map of the area together with the surface wa-ters. It shows a remarkable correlation of lower resistivity zones with surface waters and possible mineralized zones. In addition to tipper inversion, I computed the anomalous electric field components based on a relationship given by Becken and Pedersen (2003). Normal electric field components were computed using a reasonable approximation of the average resistivity of the rocks in the study area. Having total electric and magnetic field com-ponents, the electromagnetic impedance tensor and apparent resistivity of the determinant impedance was calculated. The apparent resistivity map of the study area was superimposed on the topography contour map and fracture zones. The low and high apparent resistivity zones show significant correla-tion with faults and hills, respectively.

In Paper II, the consistency of the airborne gravity gradient components in a small part of the study area of Paper I is investigated. Using the Laplace equation in wavenumber domain, the six components of gravity gradient data are recomputed from the same potential function. Thus, I showed that the data components are not biased. I also computed the gravity acceleration from terrain corrected gravity gradient data on the topography surface and compare the results with the measured gravity data on the ground. As an extra result, a technique for upward continuation of potential field data from a draped surface to an arbitrary surface above the terrain, for example flat surface is derived. Finally, a 3D density model based on the Kamm et al. (2015) approach is estimated using upward continued gravity gradient data to a constant elevation level.

In Paper III, a new method of estimating a reference model for 3D inver-sion of airborne magnetic data is described. First, a 2D model of magnetiza-

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tion is determined by inverting weighted magnetic data in an iterative proce-dure to select some “best” model that correlates well with the topography. In this procedure, only lateral variation is allowed and the model is confined to that part of the terrain lying between the highest and the lowest point of the topography. The weight of the magnetic data is defined based on the com-puted norm of a gravity gradient response of the topography model with an assumed constant density. Then, the inverted 2D magnetization model is extended downward to the desired depth for the 3D model using an exponen-tial function with an average depth calculated from the decay of the azi-muthally averaged power spectrum of the magnetic data. Next, in a 3D in-version, this 3D reference model is used to minimize the model norm of the objective function while at the same time fitting the magnetic data to a given rms tolerance. The magnetization direction is allowed to vary freely. A 3D view of the zones with strongest magnetization is shown and a plan section of these zones is superimposed on the geological map of bedrock. The high magnetization zones show remarkable correlation with gabbros and metased-imentary rocks. I found that the direction of magnetization is unrealistic but this is an inevitable result when the magnetization direction is allowed to vary freely. The magnitude of the estimated magnetization however shows a smooth behaviour a with smaller model norm compared with the standard case of a fixed magnetization direction.

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2. VLF method

The term VLF generally refers to very low frequency band electromagnetic signals with the frequency range of 3 kHz to 30 kHz, but in geophysical prospecting the lowest frequency of the signals is about 14 kHz. This range of frequency allows us to use the quasi-static approximation (Crossley,1981). The method was first described by Wait (1962) and re-viewed in detail by McNeill and Labson (1991). In Sweden the usage of the VLF method began in the 1960s (Paal,1965). Later the SGU started an air-borne VLF surveying project for the entire country. The project is still not complete (SGU,2014b).

2.1 VLF sources VLF transmitters are fixed antennas used by the military for long range radio transmission to communicate with submarines. The antennas consist of ver-tical electric dipoles which generate poloidal electric and toroidal magnetic fields. This so-called transverse magnetic or TM mode wave propagates as a direct or multiply reflected wave within the space between the lower edge of ionosphere and the Earth surface. The wave can be approximated by a plane wave in the far-field zone, e.g. at distances more than 100 km from the transmitter, where the magnetic field component is horizontal and perpen-dicular to the transmitter direction (Bozzo et al.,1994, Pedersen and Oskooi,2004). This so-called “normal field” interacts with the Earth and is refracted vertically according to Snell’s law because of the large contrast in electric conductivity between air and Earth (Wait et al.,1964). In the Earth, the transmitted normal electromagnetic wave generates eddy currents in conductive bodies which produce anomalous electromagnetic fields (see Figure 2.1).

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Figure 2.1. A schematic view of normal VLF wave propagation and its interaction with a conductive body. The anomalous field produced by eddy currents is not show for simplicity. The figure is a redrawn from Gürer et al. (2009).

The anomalous field itself can be decomposed into TM and transverse elec-tric (TE) modes. It is worthwhile to mention that in this dissertation, TM mode refers to polodial electric and torodidal magnetic fields and TE mode indicates poloidal magnetic and torodial electric fields (Becken and Pedersen,2003). TE mode anomalous field is transmitted from the Earth into the air but magnetic fields due to TM mode anomalous fields is zero in the air (Becken and Pedersen,2003, Becken et al.,2008).

Anomalous magnetic fields are not only generated by induction effects (eddy currents) but near surface conductors may generate galvanic currents which in turn produce some anomalous magnetic fields (Zhang et al.,1993, Becken and Pedersen,2003).

2.2. VLF data Based on the type of measurements parameters, VLF methods can be catego-rized in two branches of VLF-R and VLF-Z. In the VLF-R method, one hor-izontal electric field component and one orthogonal horizontal magnetic component are measured and both E- and H-polarizations can be defined similar to magnetotellurics (Kaikkonen,1997, Oskooi,2004). However, the electric fields are dominated by the vertical component of the normal field and a small change in airplane attitude, for example, will cause the vertical field component to be projected onto the measured ( and nominally horizon-tal) component and, hence, using VLF-R for airborne surveying is not appli-

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cable. In the VLF-Z (or VLF-EM) methods only the magnetic field compo-nents are measured. In this thesis, the term VLF method refers to VLF-Z. In this method, the vertical magnetic field component is related to the horizon-tal magnetic field components by

, (2.1)

where and are complex magnetic transfer functions (or tippers) de-fined for a single frequency. Tippers are independent of transmitter direction and dependent only on the Earth’s structure (Pedersen,1991). It is worth-while to mention that in the VLF method is assumed to be purely anoma-lous, because the normal magnetic field is toroidal. Tippers are complex numbers because the anomalous field components have phase delays due to induction effects.

Gharibi and Pedersen (1999) showed that the anomalous part of the hori-zontal magnetic field is the Hilbert transform of the vertical magnetic field. In addition, the anomalous part of the horizontal electric field in the TE mode is computable if the anomalous horizontal magnetic field is known (Becken and Pedersen,2003). If the average resistivity of the study are is known, the normal part of the horizontal electric field can be obtained too. Then, the TE impedance tensor and its corresponding apparent resistivity and phase can be found (Becken and Pedersen,2003, Pedersen et al.,2009).

2.3. Scalar and tensor VLF measurements In practice, signals from one or more transmitters can be used for VLF measurements. The case of using a single transmitter is called “scalar” measurement, because only two or three magnetic components are measured. But it is possible to combine the data from two transmitters in order to derive the tensor of tippers as in magnetotellurics (Pedersen et al.,1994).

In scalar VLF measurements, if the geological strike is assumed to be along the –axis and the VLF transmitter is located in the same direction,

0 and from equation 2.1, the scalar tipper is defined by

(2.2)

and 0. In a 2D structure, is maximal over a vertical resistivity contrast. It varies considerably and can be used as a qualitative tool for re-connaissance. However, if the geologic strike is not along the transmitter direction, equation 2.2 cannot be used, because the normal part of the hori-zontal magnetic field is perpendicular to the direction of the transmitter

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while its anomalous part is perpendicular to the 2D geologic strike direction. Therefore, it is not possible to define a rotation factor for . This problem is usual in airborne VLF measurements where the flying direction is chosen along east-west or north-south for simplicity. In this case if the –axis is assumed to be northward and has an angle with the strike of the 2D struc-ture, the scalar tipper can be defined if all three components of the magnetic field are measured (Pedersen and Oskooi,2004)

. (2.3)

When the Earth is 3D, a definition of scalar tippers is no longer useful and

using a single transmitter will not be informative to find structures not along the transmitter direction. To solve this problem, the Geological Survey of Sweden (SGU) used two transmitters with almost perpendicular directions and compares the field maps so they could see more features lineated along each transmitter direction. However, it was still difficult to combine the maps for more quantitative purposes. Pedersen et al. (1994) introduced the tensor VLF (TVLF) method which integrates two maps into one and pro-vides a theoretically unbiased distributions of conductivity. Using equation 2.1 they solve a system with two equations and two unknowns for two transmitters although they have different frequencies, which can cause bi-asedness. However, Pedersen and Oskooi (2004) studied this difference and showed that this effect is relatively small for the limited frequency range of the VLF method.

2.4. Inversion of VLF data Data inversion is necessary for quantitative interpretation of VLF data. The pioneering algorithm for VLF data inversion was proposed by Beamish (2000) who regularized the 2D inversion for single transmitters data. For two transmitters data, Persson (2001) and Oskooi and Pedersen (2006) used 2D inversion algorithms originally developed for the magnetotelluric method. A 3D inversion of VLF data for a single transmitter data was developed by Kaikkonen et al. (2012). The code I used in paper I was proposed by Kamm and Pedersen (2014) who developed a relatively fast algorithm for inversion of tippers computed from large TVLF data from two transmitters.

However, since a single frequency or a pair of tightly spaced frequencies used, depth resolution is limited and the lateral resolution of conductivity, especially of near surface structures is the main advantage of the VLF meth-od.

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2.5. Major problems with VLF surveying The VLF method provides a relatively cheap apparatus for geological pro-specting. With airborne systems, it can be used for reconnaissance surveys of large areas. However, it has its own disadvantages and problems in survey-ing. The accuracy of data is influenced by the ionosphere, topography varia-tions or accidental noise like that one caused by the presence of powerlines in the area of study. Here, some of the main problems in VLF data surveying are briefly described.

2.5.1. Variation of normal field In the case of using TVLF data, if all three components of the magnetic field are measured, the variation of field strength is only important for the signal-to-noise ratio, but the tippers are independent of variations in signal strength. However, in some applications like the one shown here, where only the ver-tical and total magnetic field are measured (or stored), the time variation of the normal VLF field can influence the estimation of VLF transfer functions. Smooth variations of the normal field can be compensated by assuming that along each profile the average total field (which represents the normal field) is normalized to 1000 ‰. However, large variations in power of transmitter cannot be compensated with that normalization (see Figure 2.2).

Figure 2.2. Variations of total magnetic field along one flight line from Paper I. The two main peaks to the right are due to variations in transmitter strength, because they are not observed on neighboring profiles. Notice that missing samples to the right are replaced by linearly interpolated ones.

To have only small time variations of the transmitter field, it is important to select transmitters with reliable and fixed frequencies. But to determine the fluctuation of the field strength and to correct the fields recorded from an

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aircraft for this fluctuation, a base station needs to be installed at a fixed location (Bozzo et al.,1994). In our case, no base station is available for such a correction, but the extreme example shown in Figure 2 is unique for our data set.

The effects due to wave propagation within the Earth’s waveguide i.e. the space between Earth surface and ionosphere which depends on the shape and variations of the ionosphere, is not so simple to deal with. Vallee et al. (1992) assumed that the ionosphere is like a stratified media with fixed boundaries for their wave propagation model and suggested that the problem can be controlled by removing the corresponding long wavelength variations (Bozzo et al.,1994).

2.5.2. The effect of topography The measured VLF field can be distorted by an uneven topography, because the refraction of the normal field into the slopes is not vertical. This might cause significant error (Bozzo et al.,1994) especially if the hills consist of conductive material, and hence, inductive and galvanic effects within the topography generate anomalous magnetic fields. In airborne VLF measure-ments, another effect of topography appears, because the aircraft pitch changes along profiles crossing the topography. Therefore, the coordinate system of the instrument rotates, and hence, some of the horizontal field appears in the vertical field whereby the in-phase part of the measured verti-cal field will be affected by an amount proportional to the slope (pitch) of the airplane (Pedersen,1988). Of course this problem can be solved if the attitude of airplane in all measurement points is known, but in practice this information is not usually available.

Kamm and Pedersen (2014) studied the effect of variable topographic re-lief on flightlines which are parallel and perpendicular to the topography extension by simulating an airplane flying over a valley and comparing the computed tipper to forward responses for the case of a flat surface. They showed that when the airplane flies parallel to the topography strike (and, hence, the airplane has no pitch and the topography effect is pure) the change of the in-phase part of the tipper is significantly larger than that of the quadrature part. In addition, they showed that the more conductive the topography, the larger the changes in the in-phase component. For the case where the airplane flies perpendicular to the topography, they showed that the topography induction effect is to some extend cancelled by the airplane pitch (see Figure 2.3). They finally concluded that it would be best if the forward modelling were included topography. However, they avoided the correction because it would have needed finer mesh size, and hence, more expensive computations.

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2.5.3. Anthropogenic effects Some variations in measured VLF signals are anthropogenic such as the effects of aerial or buried powerlines, telephone cables, railroads, pipelines or metallic fences. These manmade conductive structures generates anoma-lous fields with a large quadrature component, but they usually can be easily identified because of their regular shape and symmetry (Bozzo et al.,1994). However, removing this kind of distortion from the signals related to the Earth’s structures is not simple. One way that I have used is to downweight the corresponding data or to completely remove them prior to inverting the data.

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Figure 2.3. Topography effect on tipper 𝑇𝑇𝑦𝑦 (𝑦𝑦 is easting) when airplane flies across a valley with 60 m depth and basement rock resistivity of 5000 Ω𝑚𝑚. The cases with the airplane flying across and along the topography are displayed in the top and middle panels, respectively. The resistivity model is shown at the bottom together with flight altitude indicated by a black line. b) Similar to a) but with basement re-sistivity of 1000 Ω𝑚𝑚. The figure is redrawn from Kamm and Pedersen (2014).

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3. Potential field methods

The vector field is a potential field if the scalar potential is har-monic, i.e. 0 in any space without source for the potential . Gravity and magnetic fields are the most common potential functions in geophysics and their gradients are potential functions, too (Blakely,1996). This chapter briefly describes gravity gradient and magnetic methods.

3.1. Gravity gradient method The first gradiometer was invented in 1886 by Lorand Eötvös who estab-lished the gravity gradient method (Dransfield,2007, Mikhailov et al.,2007). Later, the method was utilized successfully for mapping structures like salt domes that have significantly lower density than their surrounding rocks (Eckhardt,1940, Bell and Hansen,1998, Pawlowski,1998). However, during the following several decades, the method was not widely used in practice, because the instruments were expensive and the faster and cheaper gravime-ters were sufficient for geological prospecting (Dransfield,2007, Beiki and Pedersen,2010). The gravity gradient method drew again the attention of researchers after a successful regional gravity mapping program in the USA (Jekeli,2006). In comparison with gravity data, the gravity gradient method is more sensitive to short wavelength anomalies (Barnes and Lumley,2011, Barnes et al.,2011) and gives more precise maps, more accurate depth and more accurate shape information (Dransfield and Christensen,2013). Anoth-er advantage of the gravity gradient method is the fact that it is theoretically insensitive to instrument accelerations and, hence, it can be used properly in airborne measurements (Barnes and Lumley,2011). Meanwhile, as it will be shown later, the gravity gradient tensor has five independent components. Thus, it has better data resolution than gravity data, especially between flight lines (FitzGerald and Holstein,2006, While et al.,2009).

Since the 1980s, the airborne gravity gradient method has become a suc-cessful and a popular tool for mineral and hydrocarbon prospecting (Bell et al.,1997, van Leeuwen,2000, Zhdanov et al.,2004, Dransfield,2007). How-ever, it is worthwhile to mention that hardware and data export is still re-stricted due to the military heritage of the gravity gradient surveying sys-tems, which require an export licence from the US state department (DiFrancesco et al.,2009).

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3.1.1. Full tensor gravity gradient data The gravitational potential of a mass distribution is the superimposition of all small particles with mass of , where is the density at the coordinate and is the volume of the particles. Therefore, the scalar

potential functions | |

and the total potential at the

observation coordinate can be calculated as

| |, (3.1)

where is the gravitational constant and is the total volume of the body. The gravity field is , , and , and are the three orthogonal components of the gravity vector (gravitational acceler-ation). The full tensor of gravity gradient represented by

| |

, (3.2)

and

⁄ , (3.3)

where and are either , and . The gravity gradient tensor is a symmet-ric matrix with a trace of zero, considering that the Laplace equation is satis-fied for noise free data. Therefore, there are five independent components in the tensor. However, the gravity gradient data used in this thesis contain six channels.

3.1.2. Noise in gravity gradient data The existence of noise in gravity gradient data causes the trace of gravity gradient tensor to deviate from zero and data to be inconsistent. This noise might be either due to intrinsic noise of the instruments or dynamically in-duced errors like the air turbulence effect in an airborne gravity gradient measurement. There is also noise due to drift. i.e. the environmental changes like variations of temperature and pressure, but this drift is not a linear func-tion of time (Dransfield and Christensen,2013).

Many researchers attempted to reduce the effect of noise in gravity gradi-ent data and increase the consistency of the tensor components. Some re-searchers tried to apply a low pass filter on each of the tensor components, but it is difficult to find a suitable cutoff wavelength that can separate noise from the signal (Barnes et al.,2011). Pilkington (2012) computed the eigen-

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values of the gravity gradient tensor and tried to find the best combination of the components. An effective tool was using equivalent-source inversion assuming that all of the components have the same source. Then the gravity gradient tensor for the predicted source can be estimated (Dampney,1969, Barnes and Lumley,2011, Barnes,2014). Henderson and Cordell (1971) and Sanchez et al. (2005) used a Fourier transform approach which to some ex-tent is similar to mine.

3.2. Magnetic method The magnetic field is another well-known potential field and magnetic sur-veying is the oldest method for geophysical prospecting. Airborne magnetic measurements were introduced during World War II in order to detect sub-marines (Parasnis,1962). After the war, the method was used for mineral and hydrocarbon explorations (Palacky and West,1973).

Airborne magnetic data are represented by the quantity of the “total field anomaly”, which is actually a confusing term since it is not the real anoma-lous magnetic vector field. The total field anomaly is defined as , (3.4) where and are the magnitudes of the total magnetic field at a local measurement site and a reference site, respectively. The reference value is taken as representative value of the regional field. is not necessarily the magnitude of the anomalous magnetic field, but as shown by Blakely (1996) if the anomalous magnetic field is much smaller than the regional magnetic field, will approximately be the projection of the anomalous magnetic field onto the regional magnetic field direction (see Figure 3.1).

Figure 3.1. Representation of the total magnetic field vector at a local measurement site , the regional magnetic field vector and the anomalous field vector ∆ together with the total field anomaly . is the unit vector pointing along the regional magnetic field.

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It should be noted that in the extreme case when ∆ is orthogonal to the regional magnetic field direction, its projection is zero and would be the dominated by noise. This is particularly important to remember that in the case where remanent magnetization is orthogonal to the inducing field in which case the corresponding maximum of the anomalous field could be orthogonal to the regional magnetic field. Only measurements of all compo-nents of the magnetic field would solve this problem (Stolz et al.,2006).

3.2.1. Magnetic field potential Similar to the gravity potential in equation 3.1, the magnetic potential

is defined as (Blakely,1996)

| |

. 0 , (3.5)

where 4⁄ 10 henry/m is a constant and analogous to in equation

3.1, is the magnetic permeability in vacuum and is the magnetiza-tion vector defined as , (3.6)

where is magnetic susceptibility, is the magnetic field intensity and is the remanent magnetization. Taking the gradient of equation 3.5 gives the magnetic field (or flux)

0

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0. 0 , (3.7)

It is also possible to think about higher order derivatives of the magnetic field and the full tensor magnetic field, which is outside the scope of this dissertation.

3.2.2. Magnetic and gravity gradient relation The gravity potential and magnetic potential are related by Poisson’s relation which combines the equations 3.1 and 3.5 as (Blakely,1996)

∙ .                                                (3.8) 

Pedersen and Bastani (2016) showed that if the ration / of magneti-zation to density is constant, the magnetic field along any arbitrary unit vec-tor is related to the gravity gradient tensor by

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04

1 , (3.9)

where superscript indicates vector transpose and is the gravity gradient tensor defined by equation 3.2.

3.2.2. Major problems with accuracy of magnetic data Modern magnetometers, especially those that have been developed in the 1990s and later, are highly sensitive and have noise levels of 0.1 nT. There-fore the drift error is almost negligible for exploration purposes. However, there are other sources of noise for airborne measurements (Luyendyk,1997).

The aircraft itself is a source of noise due to its remanent and induced magnetizations. However, the remanent magnetization is only a heading error which causes only a base-level shift in the data. Errors with higher frequencies are due to the airplane flying in the Earth’s magnetic field. This so-called manoeuvre noise is being removed by a compensation procedure. These procedures are defined based on parameters like airplane orientation, pitch, roll and yaw at each measurement point. The airplane flies around squares over land with low magnetic relief and the effect of each parameter to reduce the aircraft influence on data is estimated. Using these procedures, the heading error is reduced to a level of 0.25 nT and manoeuvre noise to 0.15 nT (Eppelbaum,1991, Luyendyk,1997).

Short period variations of the Earth’s magnetic field, like diurnal varia-tions, are another source of error. This effect can be corrected using a con-trolling station or using tie point data, i.e. the points which have been meas-ured twice in different flyovers (Eppelbaum,1991, Luyendyk,1997).

Variations of land clearance and topography can be also regarded as noise, but we may add the topography and aircraft altitude as input parame-ters in the modeling.

3.3. Inversion of potential field data Both gravity gradient and magnetic data can be used directly for qualitative exploration purposes, but in order to obtain quantities like the density or magnetization of underground structures or their depth and volume, data inversion is necessary. However, non-uniqueness of estimated density or magnetization is an intrinsic problem in potential field data inversion. Infi-nitely many source distributions can be estimated which produce the same data (Pedersen,1991, Blakely,1996).

Many researchers tried to deal with the non-uniqueness problem in poten-tial field data inversion, but all of the attempts can be classified in three cat-egories. The first trivial solution is to have extra knowledge about geology.

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Some of the researchers preferred to obtain this information from rock sam-ples. They assume that the model has a fixed geometrical shape with a known average magnetization or density (Braile et al.,1974, Safon et al.,1977, Bhattacharyya,1980, Fisher and Howard,1980, Mareschal,1985, Zeyen and Pous,1991, Bear et al.,1995, Henkel and Reimold,1998, Marello et al.,2013). Second group of the researchers attempted to extract infor-mation from the measured potential field data. For instance, one of the most famous techniques is the estimation of average depth using the power spec-trum of the data (Spector and Grant,1970, Pedersen,1978, Maus and Dimri,1995, 1996, Lovejoy et al.,2001, Bansal et al.,2011). The third ap-proach to deal with non-uniqueness is to impose constraints by defining an objective function containing a reference model and/or a depth weighing operator and/or a minimum volume operator (Green,1975, Last and Kubik,1983, Li and Oldenburg,1996, Pilkington,1997, Li and Oldenburg,1998, Portniaguine and Zhdanov,2002, Zhdanov et al.,2004).

The presence of remanent magnetization is another challenge in magnetic data inversion. Most researchers prefer to avoid this challenge assuming that the magnetization is totally induced by the geomagnetic field (Li and Oldenburg,1996, Pilkington,1997, Kamm et al.,2015). This idea simplifies the computations and reduces the cost of inversion but gives unrealistic re-sults in the areas with dominant occurrence of ferromagnetic (and thus high-ly remanent) material. Lelièvre and Oldenburg (2009) suggested to let the magnetization direction vary freely. However, they still need some extra knowledge like values of the Königsberger ratios measured on rock samples to separate the remanence from induced magnetization.

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4. Geophysical inverse theory

Direct sampling is a trivial and very expensive way of extracting knowledge about the rock properties of underground structures. Instead, geophysical inverse modeling is an effective tool which allows us to estimate the Earth’s interior properties with much cheaper mathematical techniques from a num-ber of indirect measurements. Geophysical inverse theory considers the Earth as a mathematical space, within which all of parameters of the under-ground material properties vary. This so-called model space is related to the measured data by a mathematical function called the “forward model”. The main goal of geophysical inverse theory is to retrieve the model parameters from the recorded data, i.e. the process called inverse modeling. However, in practice it is not possible to obtain the true model. One simple reason is that the real Earth has infinitely many degrees of freedom and always some of the model parameters do not affect the data. Therefore, the inverse problem is generally non-unique. In addition, noise in the data, which is usually as-sumed to be Gaussian, is always a source of non-uniqueness (Parker, 1977; Scales and Snieder, 2000). This chapter describes some of the most general aspects of geophysical inverse theory.

4.1. Linear inverse problems Consider the observed data vector , , … , with denoting the number of measurements and the model vector , , … , with

being the number of model parameters (superscript indicate matrix transpose). For a linear model we can write

(4.1)

where the data kernel is a matrix which is independent of the model .

If and is not singular, the estimated model can be computed simply by . But in practice and is almost always singular.

If , which means that the number of unknowns is less than number of equations, the problem can be “purely over-determined” meaning that the number of unknowns is less than number of equations and all unknowns are

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determined by the measurements. Hence it may not be possible to find a particular which fits the data perfectly. The best approximate solution can be found via minimizing data misfit function

, (4.2) where is data weighting matrix. If all of the measurements have equal importance, . However, In general is the symmetric posi-tive definite data covariance matrix. In practice we usually assume that the measurements are independent of each other and hence

0 0 0 000

⋱ 00

0 00 0

00

00

00

⋱0

0, (4.3)

where is data variance and 1, . . , . Then

1 1 (4.4) which is called the “least square solution” provided is non-singular (Lines and Treitel, 1984; Menke, 2012).

If , the problem can be “purely under-determined” which means that the estimated model fits the data perfectly, but infinitely many models can fit the data. In this case, one way to decrease the non-uniqueness is by adding extra information (Parker, 1977), for example in form of a “refer-ence model” and minimizing the perturbed model length function (Menke, 2012)

, (4.5) where is the reference model and is a matrix containing model structure function and which will be described later. Then, the esti-mated model is computed by (Menke, 2012)

1 1 . (4.6)

In geophysics, usually inverse problems are neither over-determined nor under-determined. These cases are called “mixed-determined” where the model parameters can be separated into over-determined and under-determined parts. Here can be calculated by optimizing a combination of data misfit and model length functions

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, (4.7)

where is regularization term and is called a Lagrange multiplier. The model can be computed either based on the model space solution (Menke, 2012)

(4.8) or by data space solution

1 1 . (4.9)

Equations 4.8 and 4.9 are equivalent but for equation 4.9 impose lower computational load in the case when , because is of dimen-sion while is of dimension .

Variations in the value of will change the balance between and . If is too large the model is biased to the reference model with poor data fit and if is too small, data misfit is improved but the model becomes too complex. An empirical good choice of can be obtained by the trade-off curve (Hansen, 1992; Menke, 2012). The value of corresponding to the knee point of the curve is the often considered the best (see Figure 4.1).

Figure 4.1. Selection of the best Lagrange multiplier for a mixed-determined inverse problem. indicates the optimum value for .

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4.2. Model structure functions If the model parameters are independent of each other, then the model struc-tures function can be selected as in equation 4.5. However, in prac-tice the model parameters are related and if the inverse of the model covari-ance was known it could be used as but this information is usually not available. Nonetheless, we may assume that the model parameters obey cer-tain rules which can help to constrain the problem and decrease the non-uniqueness. Here I briefly describe two of the most general model rules.

4.2.1. Flatness function The norm of the first order spatial derivative of the model parameters can quantify the flatness of an analytic function in three dimensional space (Menke, 2012). The flatness function can be expressed as first-order differ-ences between abutting model parameters. For instance, the flatness function for a model space solution of the 1D inverse model is where

1 11 1

1 1⋱ ⋱

⋱ ⋱1 1

. (4.10)

4.2.1. Depth weighting Especially for potential field data inversion, the estimated sources tend to a be accumulated close to the Earth’s surface, since there are infinitely many shallow sources that can explain the data with the same data misfit (Pedersen, 1991; Kamm et al., 2015). Li and Oldenburg (1996) suggested to apply a weighting which compensates the decay of the data kernel at differ-ent depths. They showed that a reasonable choice for depth weighting for gravity gradient and magnetic data inversion is the square root of the kernel function (i.e. ) which can be simulated by

1

03/2 (4.11)

where is depth of the cell and must be obtained by matching the kernel function at the observation point with the function . Kamm et al. (2015) suggested to use forth root of the diagonal of instead.

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4.3. Non-linear inverse problems When the data kernel in equation 4.1 is a function of , the right hand side of the equation will be changed to , where can now be a general non-linear function of . Therefore, none of the equations mentioned so far can be used. However, using a Taylor series expansion in the vicinity of an initial model allows us to linearize the forward problem as Δ Δ (4.12) where

(4.13)

 and  is called the Jacobian which is assumed to be linear (Lines and Treitel, 1984; Menke, 2012). Then, all other equations will be changed ac-cordingly, e.g. is replaced by Δ and with . is model in the next iteration. For instance, equation 4.2 in nonlinear form is  

, (4.14) where . Equation 4.5 will be changed to

. (4.15) Substituting and instead of and correspondingly in equation 4.7 and taking the derivatives with respect to and setting the result to zero, the model vector will be estimated by a modified form of equation 4.8 as (Kalscheuer, 2008)

(4.16) or by modified form of equation 4.9 (4.17) Then in a similar way in the next iteration can be calculated from

(Siripunvaraporn and Egbert, 2000). Throughout this derivation, it has

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been assumed that 0⁄ (Kamm, 2014). The process will be contin-ued until a certain criterion is met. The Lagrange multiplier can be changed in iterations. This method is known as Gauss-Newton non-linear least-squares. Another famous method is non-linear conjugate gradient technique which I do not discuss here, but it has been implemented for geophysical inverse problems by many authors, e.g. Mackie and Madden (1993), Newman and Alumbaugh (2000), Rodi and Mackie (2001) and Kamm (2014).

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5. Geology of the study area

All of the papers in this dissertation are case studies from an area located in northern Sweden, i.e. in Norrbotten county close to the Finnish border (see Figure 5.1). The area is a part of the northern Fennoscandian shield which hosts Europe’s largest mineral resources (Grigull et al.,2014, Hellström and Jönsson,2015).

There are several noticeable reviews about the geology and metallogeny of northern Sweden like Ödman (1957), Bergman et al. (2001), Weihed et al. (2005) and Martinsson and Wanhainen (2013). The bedrock has been formed during the Precambrian era with an Archean granitoid-gneiss basement (called Råstojaure complex with age of 3.2–2.6 Ga) and is overlain by Prote-rozoic volcanic and sedimentary rocks. At the lowest stratigraphic level, metavolcanic rocks with Karelian age, i.e. 2.5–2.0 Ga, can be seen together with sedimentary rocks and greenstones. Calc-alkaline andesite porphyries of Svecofennian successions (1.9 Ga) overlie the Karelian metavolcanic rocks. Younger clastic sedimentary rocks are at the uppermost stratigraphic level. One typical characteristic for the greenstones is the presence of banded iron formations associated with dolomites and graphitic schists (Martinsson and Wanhainen,2013, Hellström and Jönsson,2015). There are stratiform lead, zinc and copper deposits in the greenstones, whereas apatite iron ores are associated with the porphyries (Martinsson and Wanhainen,2013, Hellström and Jönsson,2015).

Directions of geological structures vary from NNE to NNW with more in-tensive degree of deformation and metamorphisms (Bergman et al.,2001, Hellström and Jönsson,2015). The oldest deformation and metamorphism is of Archean age (Bergman et al.,2001, Martinsson and Wanhainen,2013, Hellström and Jönsson,2015). The younger metamorphisms and defor-mations have ages of 1.86–1.85 Ga and 1.80 Ga (Bergman et al.,2001, Hellström and Jönsson,2015)

A large portion of the study area is occupied by Lina granites, the name indicates the 1.80–1.79 Ga old granites and pegmatites (Öhlander et al.,1987). Theses rocks are part of the Transcandinavian igneous belt which extends about 1500 km in NS direction (Hellström and Jönsson,2015).

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Figure 5.1. Geological map of the study area in northern Sweden together with the settlements and infrastructure. The coordinate system is SWEREF 99 TM. The geological map is produced by SGU. The map of roads and villages is based on the GSD-Översiktskartan-vektor © Lantmäteriet. The red dashed polygon, blue dashed square and black dashed square are the study areas of Paper I, II and III respectively.

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6. Summary of papers

This dissertation is a comprehensive summary of three case study papers about VLF, gravity gradient and magnetic data. In this chapter, these papers are briefly described.

6.1. Paper I Extracting geoelectrical maps from vintage very-low-frequency air-borne data, tipper inversion, and interpretation: A case study from northern Sweden The quantitative interpretation of VLF data needs computation of the tippers as described in chapter 2. However, there are vintage data from decades ago when VLF measurements were used only for qualitative interpretations. For instance, there are data sets in northern Sweden, measured in 1985 by the company LKAB that only contains the total and the vertical magnetic field. The aim of Paper I is to extract tensor VLF data from one of such data sets for quantitative interpretation.

6.1.1. Summary The data was surveyed with east-west directed fight lines with nominal land clearance of 30 m and 200 m line spacing. The point spacing on each flight line is about 30 m. Data is dual frequency by selecting the two transmitters GBR (15.7 kHz) and NAA (24 kHz). The average of the total magnetic field magnitudes | | , was scaled to 1000 along each line. The vertical magnetic field component was scaled by the same factor. The anomalies are gener-ally oriented toward the corresponding transmitters’ directions. Infrastruc-tures like powerlines and roads have considerable influence on the imaginary part of .

In the first step, short wavelength effects were eliminated using a Gaussi-an low-pass filter on each flight line with cut-off wavelength of 600 m, which is more than twice of the spacing between the flight lines, but which reduces the effects of powerlines considerably.

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The second step was the computation of anomalous parts of the horizontal magnetic field components from vertical magnetic components using a Hil-bert transform in wavenumber domain (Becken and Pedersen, 2003).

In order to compute normal parts of the horizontal magnetic components from each transmitter, the coordinate system was rotated so that the horizon-tal axes were along and orthogonal to the transmitter direction. In the new coordinate system, the normal part of horizontal magnetic field is only to be found on the axis perpendicular to the transmitter direction, while the axis along the transmitter direction has only an anomalous component of the magnetic field. Then it was shown that the normal part of horizontal magnet-ic field in this coordinate system is a function of total and anomalous mag-netic field components.

Having the anomalous and normal parts of the magnetic field components for two transmitters, tippers were computed using equation 2.1 (Pedersen and Oskooi, 2004). Here I assumed that the transmitters had an average fre-quency of √15.7 24 kHz = 19.4 kHz.

I also computed the horizontal electric field components in order to calcu-late the impedance tensor (Berdichevsky, 1960) and the apparent resistivity and phase of the square root of the determinant of the impedance tensor (Pedersen and Engels, 2005). Here I used a relationship by Becken and Pedersen (2003) to compute the anomalous part of the electric field from the vertical magnetic field of the TE mode in the wavenumber domain. The normal part of the electric field components was calculated assuming an average background resistivity of 5000 and using the impedance of the corresponding homogenous half-space (Pedersen and Oskooi, 2004).

Figure 6.1 shows the computed apparent resistivity together with topog-raphy and fracture zones. The areas with high topography are generally cor-related with high apparent resistivity zones while the fractures and valleys are related to the zones of lower resistivity.

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Figure 6.1. Map of computed apparent resistivity together with topography contour lines. Thick black lines are fracture zones shown in Figure 5.1.

Having calculated the tippers, a three dimensional resistivity model of the study area was estimated by a 3D inversion algorithm developed by Kamm and Pedersen (2014). The model domain was discretized by 50 5025 blocks and was extended down to 600 m depth. However, the inver-sion model contains five layers with interfaces at 0, 25, 75, 150, 300 and 600 m. The problem was solved by an iterative nonlinear conjugate gradient technique and the model was estimated at the 34th iteration with an rms mis-fit of 0.0221, which is chosen as the best model, since the model is becom-ing increasingly oscillatory with further increasing number of iterations. Figure 6.2 shows the estimated resistivity model in the depth range 150-300 m together with the geology.

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Figure 6.2. The estimated resistivity model in the layer between 150 m to 300 m depths. The model is underlain by the geology map of Figure 5.1. Surface waters are also shown.

Generally speaking, the areas with lower resistivity are correlated with fracture zones. The lakes are also related to decreased resistivity. Probable mineralized zones around the basaltic parts in the center of the map display lower resistivity. This interpretation is supported by the presence of metal traces like iron oxide, sulfides and gold in the area.

6.1.2. Discussions In order to extract tensor VLF data from a vintage data set, several assump-tions have been made. For example, the effect of topography variations was disregarded in the entire process, since there are only few points with slopes greater than 5% and the effect of the topography in VLF data is generally small. However, it would have been better if the airplane altitude and topog-raphy were included in the model. Another assumption was that the normal fields produced by the transmitters are constant and the influence of the Earth-ionosphere waveguide is negligible. If this assumption is not valid then the measured total magnetic field might be affected. This possible error would then also be found in the vertical magnetic field since it was normal-

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ized by the total magnetic field magnitude. Thus, this error would be propa-gated into the anomalous field.

6.2. Paper II Consistency investigation, vertical gravity estimation and inversion of airborne gravity gradient data – A case study from northern Sweden

The challenge of airborne gravity gradient data is high frequency noise where the intrinsic noise of the instrument is combined with noise from the aircraft or “noise” from shallow density variations. One of advantages of full tensor gravity gradient data (FTG) is that all of the components can be de-rived from the same potential, and hence it is possible to check the internal consistency between data components. In Paper 2, all of the data compo-nents of a FTG data set are retrieved from a common potential function and estimated data components are compared with the original data.

6.2.1. Summary The FTG data set used in this paper was acquired by the company Bell Geo-space in 2013 in a large area but I selected an area of 12.5 km by 12.5 km. The flightlines were east-west directed with a 500 m line spacing. However, Bell Geospace had processed the data and delivered the data on a 50 m by 50 m grid (Bell Geospace Ltd, 2013). The data was acquired on a draped sur-face that varied in altitude between 462 m to 789 m ASL. The nominal land clearance was 80 m, but actually varied between 78 m and 340 m.

In this paper, the FTG data was transformed into the wavenumber domain where a potential function was defined so that all six components of the FTG data were second derivatives of that potential function (that is a function of data coordinates) similar to the technique developed by Henderson and Cordell (1971). In order to calculate theses derivatives, some coefficients appear. I considered those coefficients as model parameters, FTG data com-ponents as data vector and Fourier expansion based on sine and cosine func-tions as a linear data kernel. Then the coefficients were computed by data space inversion using equation 4.9 (I assumed here that and ). The standard deviation of the data was assumed to be 5 eötvös. Using the trade-off curve and finding the optimum Lagrange multi-plier, an rms misfit of 5.4 eötvös was reached.

Using the computed coefficients, the FTG components were estimated and compared with the original data components via histograms (see Figure 6.3).

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Figure 6.3. Histograms of the difference between estimated and original data for six components of FTG. Displayed quantities are in eötvös

The prediction errors in Figure 6.3 have mean values close to zero and standard deviation close to 5 eötvös which was the initial guess for standard deviation.

Bell Geospace also had computed terrain corrected data assuming a varia-ble terrain density (Bell Geospace Ltd, 2013). I computed the Bouguer gravi-ty field, on the topography by downward continuation of the common potential i.e. replacing the aircraft altitudes in the data kernel with the vary-ing topography level. Here I calculated the coefficients for the common po-tential for the terrain corrected data, and then I computed by taking the first spatial derivatives of the potential function in the vertical direction. Then the computed Bouguer anomaly was compared with some Bouguer gravity measurements by SGU (see Figure 6.4). Here I shifted the zero level of the computed by -39 mGal to fit the predicted from FTG to the measured on the ground. Both of them show similar trends but the fit is not perfect, since the SGU terrain correction was based on a fixed density of 2670 kg/m3 , whereas Bell Geospace used a variable density for terrain cor-rection using a wavelet approach (Bell Geospace Ltd, 2013). Another reason for this discrepancy is that long wavelengths are partly subdued in the FTG data and we only use an area of 12.5 km by 12.5 km to compute the Bouguer anomaly.

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Figure 6.4. Estimated Bouguer gravity from FTG data shifted by -39 mGal. The colored circles are measured Bouguer gravity anomalies on the ground.

Upward continuation of data to an arbitrary level is possible by substitut-ing the altitude of the aircraft in the data kernel with a fixed level above the highest point of topography. Thereby it becomes possible to use a 3D inver-sion algorithm developed by Kamm et al. (2015) that assumes the data is measured on a plane surface. Here, the non-terrain corrected data was up-ward continued to 750 m ASL and the 3D density model estimated from non-terrain corrected FTG data had an rms misfit of 5 eötvös. The reference model is assumed to be a zero density contrast homogeneous halfspace.

Figure 6.5 show plan sections of the estimated density model. The plan sections are at 300 m ASL i.e. below the minimum topography, 1300 m BSL, 3300 m BSL and 7300 m BSL. The density model sections at 1300, 3300 and 7300 m BSL are similar and the model becomes smoother with increasing depth. However, at the shallow depth of 300 m ASL, the model shows many short wave length features. This can be the effect of using depth weighting which forces the anomalous structure in the model to deeper lev-els.

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Figure 6.5. Plan sections of density model estimated from 3D inversion of non-terrain corrected data. Data is upward continued to 750 m ASL.

6.2.2. Discussion The largest difference between the observed FTG data and the predicted FTG data is over the areas where the land clearance is small (which is not shown in this chapter for, but the reader is referred to the paper instead). It means that the processing of airborne gravity gradient data should take the height dependence into account.

The method used in this paper is based on the general Fourier transform, so it can be applied in the cases where data grids are not uniform or data is arbitrarily located. However, in my case the data grid is uniform.

Applying depth weighting forces the density contrast to be located at deeper parts and hence short wavelength features appear in the density mod-el at shallow depths. To overcome this problem, I suggest to employ a better reference model.

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6.3. Paper III A new reference model for 3D inversion of airborne magnetic data in hilly terrain – a case study from northern Sweden

Non-uniqueness is a substantial problem in potential field methods. Theoret-ically, this problem can be reduced by adding extra information like using a reference model in data inversion. In addition, depth weighting in potential field data inversion can pushes the model structures to deeper levels. There-fore the estimated model at shallow depths differs from rock sample meas-urements taken from samples collected from the Earth’s surface. This paper attempts to find a good reference model for 3D inversion of airborne mag-netic data.

6.3.1. Summary Similar to Paper II, airborne total magnetic field data (TMI) was collected in November 2013 by Bell Geospace. Here the study area is a 25 km by 25 km square with flightlines along the east-west direction with a 500 m line spac-ing. The data was represented on a 50 m by 50 m grid. Aircraft altitude, to-pography and land clearance information is identical to Paper II. Again, the data was upward continued to 750 m ASL using the method in Paper II. Then I could use the code developed by Kamm et al. (2015) that assumes the data to be measured on a flat level (Abtahi et al., 2014). The data was inter-polated to a 200 m by 200 m grid to reduce the computation time.

Forward modeling is based on the formula given by Bhattacharyya (1964) with an inducing field 53208 nT and a unit vector of

0.2130,0.0339,0.9765 so the regional magnetic field is almost vertical (NOAA, 2016).

In order to define a reference model for 3D data inversion with the magnetization vector assumed to change freely, I first inverted the data using a 2D model confined to the depth between highest and low-est topography and with the magnetization assumed to vary only later-ally. Here, the reference model is a half-space with zero magnetiza-tion. The bounds of the apparent susceptibility were chosen to be from -1 to 3 and imposed using the logarithmic transform proposed by Pedersen and Rasmussen (1989) used to sustain the susceptibility val-ues in this range (which is a source of nonlinearity in the inverse prob-lem). The 2D inversion process was continued until the 6th iteration, where the model length variation saturates to a stable level. In order to emphasize the effect of topography in the objective function, data er-rors were multiplied by the ratio ‖ ‖ / ‖ ‖ . Here ‖ ‖ is the norm of the gravity gradient data at the observation point estimated

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for the topography of the area assuming constant density (Pedersen et al., 2015). The standard deviation for the data weighting matrix was assumed to be 10 nT. This value has been found by several experi-ments to avoid unrealistic models.

The predicted 2D inverse model was extended to a 3D model in order to be used as reference model in a general 3D data inversion. The extension function is an exponential one with an average depth 43 m ASL using the approach proposed by Spector and Grant (1970) and Pedersen (1978). Then, a model domain was constructed with 30 layers down to 5 km depth. The first 18 layers within the topography have a uniform thickness of 20 m and the thickness of the other 12 layers have an increasing thickness by a geo-metric progression with scale factor of 1.26. After the 6th iteration the rms misfit of 9 nT was reached. Figure 6.6 shows a comparison between the pre-dicted apparent susceptibility (i.e. the normalized modulus of magnetization with factor 40 A/m) for two cross sections. The reference model is also dis-played.

Figure 6.6. a) and d) are cross sections of the reference model. b) and e) are cross sections of the 3D inversion model computed using the reference model. c) and f) are cross sections of the 3D inversion result computed without the reference model. The left and the right columns display sections located at 7583 km and 7565 km northing coordinates, respectively.

In Figure 6.6, the shallow depth structures are obviously similar, but not identical to the reference model while the high magnetization zones are pushed down by the depth weighting if the reference model is not used.

Figure 6.7 shows a comparison between plan sections of the estimated models at 400 m ASL, with and without the reference model. Generally speaking high magnetization zones show better correlation with the meas-ured high remanent magnetizations from rock samples in the area (samples were collected and analyzed by SGU). However, for the low remanent mag-

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netization samples, the model without the reference model shows better cor-relation.

Figure 6.7. Plan sections of magnetization amplitudes for estimated model with (left) and without (right) reference model at 400 m ASL. The colored circles indicat-ed the measured remanent magnetization from rock sample on the surface.

In order to compare the inversion results with geology, I superimposed the zones with 25% of the highest total magnetizations onto the geology map from Figure 5.1 (See Figure 6.8). The high magnetization amplitudes are generally correlated with gabbros and volcanic (meta)sedimentary rocks (Ambros and Henkel, 1980).

Figure 6.8. The zones of upper 25% of estimated magnetizations at 400 m ASL superimposed on the geological of map the area.

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The final step was studying the direction of estimated magnetizations. The estimated magnetization vectors show unrealistic radial behavior around high magnetization zones, but as illustrated in the appendix of the paper, this is an inevitable effect caused by allowing the magnetization vector to vary freely and also has been shown (but not discussed) in a paper by Lelièvre and Oldenburg (2009).

6.3.2. Discussion The nonlinearity of the inverse problem discussed in this paper has relatively small effects. From Figure 6.6 it can be deduced that the data inversion pro-cedure actually deals with the perturbed model and the reference model seems to be untouched and just has been added to the final perturbed model. The reason is that the estimated apparent susceptibilities are far from the predefined bounds of apparent susceptibility.

The magnetization directions in the estimated model are artificial. In the appendix of the paper, a comparison of two synthetic models is shown with magnetic responses inverted for the cases where the magnetization vector is fixed or allowed to vary freely. In the latter case the norm of the estimated model is decreased and hence gives a smoother magnetization amplitude, but the unrealistic radial behavior of the magnetization vector is a price to be paid.

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Acknowledgments

At the end of this dissertation, I would like to sincerely thank all of the peo-ple who have helped me during my studies as a doctoral student in Uppsala University.

First I should give thanks to Laust Pedersen, my supervisor who accepted me as a PhD student and helped me in various stages of my studies. I will forever remember his patience and encouragements especially when my sturdy direction changed. I will be always in debt to him for his full supervi-sion, deep discussions and detailed revisions.

I should also thank Jochen Kamm, as an intelligent researcher, very help-ful colleague and finally as a co-supervisor whom I had a lot of vivid discus-sions with. In all of the three papers, I used inversion codes developed by him.

I am also in debt to Thomas Kalscheuer for his encouragements and his strict and precise revisions on the papers which helped me a lot to improve the papers.

During my studies, there are many other people who have helped me in different ways. Thanks to Alireza Malehmir and his wife Mahdieh Dehghan-nejad who helped me in the first step and found a proper housing for me in first few months. I also express my gratitude to Mehrdad Bastani who was one of my co-supervisors in the first years and prepared some data from SGU. Thanks to Omid Ahmadi, Hamzeh Sadeghi and Saeid Cheraghi who friendly helped me in my private life in Uppsala. Thanks to Christopher Juh-lin, Hossein Shomali, Fengjiao Zhang, Ari Tryggvason, Roland Roberts, Chunling Shan, Zeinab Jeddi, Majid Beiki, Ping Yang, Suman Mehta, Mag-nus Andersson, Muhammad Umar Saleem, Jawwad Ashraf and all other colleagues and students who helped me during these years. I would also like to thank Taher Mazloomian and Tomas Nord for their practical and technical helps.

I also like to appreciate Iranian Ministry of Researches, Sciences and Technology and Isfahan University of Technology for their financial support during my PhD studies.

I am mostly grateful to my parents who encouraged me to continue my studies at the PhD level and I would like to dedicate this thesis to them.

All the praises and thanks be to God for all of his gifts.

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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1394

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A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally throughthe series Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.(Prior to January, 2005, the series was published under thetitle “Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology”.)

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