Masters Thesis. Three-dimensional inversion of magnetic data in the presence of remanent magnetization

Transcription

1 CGEM Masters Thesis Colorado School of Mines Center for Gravity, Electrical & Magnetic Studies Three-dimensional inversion of magnetic data in the presence of remanent magnetization Sarah E. Shearer Department of Geophysics Colorado School of Mines Golden, CO 841

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3 CGEM Masters Thesis Colorado School of Mines Center for Gravity, Electrical & Magnetic Studies Three-dimensional inversion of magnetic data in the presence of remanent magnetization Sarah E. Shearer Defended: March 11, 25 Advisor: Committee Chair: Committee Members: Dr. Yaoguo Li (GP) Dr. Thomas M. Boyd (GP) Dr. Misac N. Nabighian (GP) Dr. VJS (Tien) Grauch (USGS) Department of Geophysics Colorado School of Mines Golden, CO 841

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5 THREE-DIMENSIONAL INVERSION OF MAGNETIC DATA IN THE PRESENCE OF REMANENT MAGNETIZATION by Sarah E. Shearer

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7 A thesis submitted to the Faculty and the Board of Trustee of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Master of Science (Geophysics). Golden, Colorado Date: April 14, 25 Signed: on original copy Sarah E. Shearer Approved: on original copy Dr. Yaoguo Li Thesis Advisor Golden, Colorado Date: April 14, 25 on original copy Dr. Terence K. Young Professor and Head Department of Geophysics ii

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9 ABSTRACT Magnetics is a common geophysical technique used to image subsurface structure though variations of magnetic properties. Three-dimensional inversion has been used successfully to achieve quantitative interpretation of magnetic data. However, a crucial parameter in this process is the direction of magnetization. The total magnetization is a vector sum of two components. Induced magnetization is well approximated by the inducing field direction; remanent magnetization is commonly unknown and can have a direction significantly different from that of the current field or a magnitude large enough to alter the direction of total magnetization. As a result, the magnetization direction becomes an unknown quantity and hampers inversion and interpretation of magnetic data. I present a general approach for inverting magnetic data in the presence of strong remanent magnetization. Two quantities, the amplitude of the anomalous magnetic field and the total gradient, defined as the magnitude of the gradient vector of magnetic anomaly data, are weakly dependent upon the magnetization direction in three dimensions. Therefore, I invert amplitude and total gradient data directly to recover the magnitude of magnetization without precise knowledge of its direction. Since amplitude and total gradient data depend nonlinearly upon magnetization, solution of a nonlinear inverse problem is required. Further nonlinearity is introduced by imposing a positivity constraint on the magnitude of magnetization. I formulate the inversion using Tikhonov regularization, impose positivity by using a logarithmic barrier method, and solve the resulting optimization by truncated Gauss-Newton method. iii

10 The ability to invert magnetic data with little information about the nature of remanent magnetization increases the areas in which three-dimensional inversion of magnetic data can be applied. In fact, it is now possible to invert any magnetic data to some extent. This newfound ability opens the door to quantitative interpretation of data in a variety of practical problems ranging from archaeological investigations, mineral and resource exploration, and crustal and planetary studies. iv

11 TABLE OF CONTENTS ABSTRACT... iii LIST OF FIGURES... ix LIST OF TABLES... xiii ACKNOWLEDGEMENTS... xiv DEDICATION...xv Chapter 1: INTRODUCTION...1 Chapter 2: BACKGROUND Magnetic Field Magnetization Induced Magnetization Magnetic Susceptibility Remanent Magnetization Occurrence of Remanent Magnetization and Its Challenges Current Methods for Magnetic Interpretation Depth Estimation Parametric Inversion Physical Property Inversion Influence of Remanent Magnetization Current Methods to Interpret Magnetic Data with Strong Remanent Magnetization Direction Estimation Prior to Inversion Use of Analytic Signal...27 v

12 2.5. Extension Toward 3D: Utilizing Quantities with Weak Dependence on Magnetization Direction Amplitude of the Anomalous Magnetic Field Total Gradient Summary...38 Chapter 3: INVERSION METHODOLOGY Introduction to Geophysical Inversion The Need for Inversion Tikhonov Regularization Forward Modeling Data with Weak Dependence on Magnetization Direction Model Representation Forward Modeling of Amplitude of Anomalous Magnetic Field Data Forward Modeling of Total Gradient Data Inversion Algorithm Data Misfit Model Objective Function Selection of Regularization Parameter Positivity Constraint Depth Weighting Numerical Solution of the Inverse Problem Bound Constraint Using Logarithmic Barrier Method Gauss-Newton Method Sensitivity Calculations Convergence Criteria Synthetic Examples Synthetic Cubic Model Results...7 vi

13 Synthetic Dipping Slab Results Summary...86 Chapter 4: PRACTICAL ASPECTS Limitation of Current Algorithms in the Presence of Remanent Magnetization Testing the Assumption Inversion Results Depth Weighting Obtaining Amplitude and Total Gradient Data Numerical Calculation Amplitude Calculations Total Gradient Calculations Direct Field Measurements Summary...11 Chapter 5: FIELD EXAMPLE Areas of Application Kimberlites and Diamond Exploration Formation of Kimberlites Composition of Kimberlites Occurrence of Kimberlites Geophysics of Kimberlites Field Example from Victoria Island, Northwest and Nunavut Territories, Canada Background Geophysics Inversion of Magnetic Data Inversion Results Summary vii

14 Chapter 6: CONCLUSIONS AND DISCUSSION Conclusions Discussion Future Work...14 REFERENCES CITED viii

15 LIST OF FIGURES Figure 2.1. Description of the orientation of the main magnetic field using inclination and declination...9 Figure 2.2. Elemental current loop used to represent the basic source of magnetic field....1 Figure 2.3. Total magnetization, J r is the vector sum of induced magnetization, J r i, and remanent magnetization, J r r...12 Figure 2.4. Magnetic susceptibility values of common types of rocks and magnetic minerals, hematite and magnetite...15 Figure 2.5. Conditions of the assumption on remanent magnetization...21 Figure 2.6. The amplitude of the anomalous magnetic field and total gradient of any component are independent of magnetization direction of the source. The rectangular source body is located between 75 m and 75 m and extends from 75 m to 225 m in depth...29 Figure 2.7. The amplitude of the anomalous magnetic field and total gradient of any component are independent of magnetization direction of the source. The larger, rectangular source body is located between 275 m and 275 m and extends from 75 m to 225 m in depth...31 Figure 2.8. Illustration of amplitude data with weak dependence on different magnetization directions over a simple cubic model (κ =.5 SI). The inducing field orientation is held constant at I = 45º and D = 45º, while the magnetization direction varies as labeled at the top of each column...35 Figure 2.9. Illustration of total gradient data with weak dependence on different magnetization directions over a simple cubic model (κ =.5 SI). The inducing field orientation is held constant at I = 45º and D = 45º, while the magnetization direction varies as labeled at the top of each column...37 ix

16 Figure 3.1. Forward modeling calculates a synthetic response (d pred ) from a representation of the subsurface (m) by using a forward operation (F) that describes the underlying physics...42 Figure 3.2. The inverse operator (F -1 ) estimates distributions of physical properties in the recovered, or inverted, subsurface model (m*) based on physical properties and mathematical parameters...42 Figure 3.3. In a three-dimensional model, the model region is represented as a series of cuboidal cells...49 Figure 3.4. Least squares fitting for depth weighting parameters, z and ξ using the actual signal decay and decay of the depth weighing function...58 Figure 3.5. Analytic center and trajectory path taken by the logarithmic barrier method...61 Figure 3.6. Step length reduction in logarithmic barrier method...64 Figure 3.7. The sensitivity matrix characterizes the change of the magnetic data at the i th observation point with respect to a change in the effective susceptibility value in the j th cell of the current model...66 Figure 3.8. Synthetic models used to illustrate inversion algorithm...71 Figure 3.9. Synthetic total field anomaly and calculated amplitude and total gradient data for tests over the cubic model...73 Figure 3.1. Curves of data misfit, model norm, and logarithmic barrier parameter values for the inversion of synthetic amplitude data over a cubic model (Test A)...74 Figure Inversion results from Test A with amplitude data over the cubic model at three depths: 5 m, 15 m, and 25 m...75 Figure Curves of data misfit, model norm, and logarithmic barrier parameter values for the inversion of synthetic total gradient data over a cubic model (Test B)...77 Figure Inversion results from Test B with total gradient data over the cubic model at three depths: 5 m, 15 m, and 25 m...78 x

17 Figure Curves of data misfit, model norm, and logarithmic barrier parameter values for the inversion of synthetic total gradient data over a cubic model (Test C)...8 Figure Inversion results from Test C with total gradient data over the cubic model at three depths: 5 m, 15 m, and 25 m...81 Figure Synthetic total field anomaly and calculated amplitude data for the test over a dipping slab...83 Figure Curves of data misfit, model norm, and logarithmic barrier parameter values for the inversion of synthetic amplitude data over a dipping slab model (Test D)...84 Figure Inversion results from Test D with amplitude over a dipping slab at three depths: 5 m, 25 m, and 4 m...85 Figure Volume rendered images of the true and recovered model for the inversion of amplitude data over the dipping slab (Test D)...87 Figure 3.2. Curves of data misfit, model norm, and logarithmic barrier parameter values for the inversion of synthetic amplitude data over a dipping slab model (Test E)...88 Figure Inversion results from Test E with amplitude over a dipping slab at three depths: 5 m, 25 m, and 4 m...89 Figure Volume rendered images of the true and recovered model for the inversion of amplitude data over the dipping slab (Test E)...9 Figure 4.1. Model norms for correct and incorrect inversions as a function of magnetization inclination (Test A)...95 Figure 4.2. Model norms for correct and incorrect inversions as a function of magnetization inclination (Test B)...96 Figure 4.3. Model norms for correct and incorrect inversions as a function of magnetization declination (Test C) and effective deviation. Negative effective deviations indicate that the declination of the magnetization direction is less than 45º...97 Figure 4.4. Cross-sections from Test A at various magnetization inclinations...99 Figure 4.5. Cross-sections from Test B at various magnetization inclinations...1 xi

18 Figure 4.6. Cross-sections from Test C at various magnetization declinations...11 Figure 4.7. Decay rates of total gradient over a dipole with three orientations in comparison to standard decays of 1/r 3, 1/r 4, and 1/r Figure 5.1. Kimberlite facies, zones and depths. The left figure shows the three zones (hypabyssal, diatreme and crater) and their corresponding depths. The right figure, not scaled, provides three-dimensional detail for each facies and geological details of the crater facies Figure 5.2. Location map and geology of Victoria Island, Canada. The red star indicates the approximate location of the Blue Ice kimberlite exploration project Figure 5.3. Magnetic map of Galaxy kimberlite trend in Blue Ice project, Victoria Island, Canada Figure 5.4. Rotated total field anomaly data over the target area, Blue Ice Project, Victoria Island, Canada Figure 5.5. Amplitude of anomalous magnetic field, calculated via linear transform from total field anomaly data over the target area, Blue Ice Project, Victoria Island, Canada Figure 5.6. Curves of data misfit ( φ d ) and model norm ( φ m ) values for inversion of magnetic data Figure 5.7. Plot of logarithmic barrier parameter (λ) as a function of iteration Figure 5.8. Volume rendered inversion results of the recovered distribution of the magnitude of magnetization with a translucent color display of the amplitude data Figure 5.9. Plan sections of inversion results at depths of m, 4 m, and 8 m Figure 5.1. Cross-sectional views of inversion results. (a) All anomalies, view from rotated S to N; (b) cross section through anomalies C, D, E and F at 137 m North; (c) cross section through anomaly C at 2 m East; (d) cross section through anomaly A at 32 m East xii

19 LIST OF TABLES Table 2.1. Magnetic susceptibility values for common minerals and rocks...14 Table 2.2. Descriptions of different types of remanent magnetization...17 Table 3.1. Test conditions for synthetic cubic model tests. The anomaly projection direction is the same for each test, (9,). Test A is conducted with amplitude data and a magnetization direction of (6, ). Tests B and C are conducted with total gradient data with magnetization directions of (75,) and (9,), respectively...72 Table 4.1. Test conditions for identifying the limitations on the current assumption in the presence of remanent magnetization. The first two tests (A and B) vary the magnetization inclination in the presence of a constant anomaly projection and magnetization declination. Test C varies the magnetization declination in the presence of a constant anomaly projection and magnetization inclination...94 xiii

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21 ACKNOWLEDGEMENTS My gratitude and appreciation is given to my advisor, Dr. Yaoguo Li, for his enthusiasm, patience, and guidance, both in research and within the bigger picture. I also want to thank my committee members, Dr. Tom Boyd, Dr. V.J.S. (Tien) Grauch, and Dr. Misac Nabighian, for their advice and support as I tried my hand at research. I would like to acknowledge Dr. Jules Lajoie, Teck Cominco, and Diamonds North Resources for their support and use of the magnetic field data from Canada. Further, the members and sponsors of the Center for Gravity, Electrical, and Magnetic Studies (CGEM) and the Gravity and Magnetics Research Consortium (GMRC) at the Colorado School of Mines (CSM) have provided research and financial support during my graduate studies. I will always be indebted to the National Science Foundation (NSF) for the opportunity to participate in the GK-12 Learning Partnerships Program at Hodgkins Middle School. It was truly an enlightening experience and has forever changed my life. Through the course of my undergraduate and graduate work, the faculty, staff, and students of the Geophysics department have had a tremendous impact on my education, career, and life. Thank you for being there to answer my questions, pose new problems, offer suggestions, or provide an ear or shoulder. With all of my love, I must thank my family and friends for supporting me throughout my entire education. I would not be the person I am or cleared the hurdles that I have without each of you in my corner. xiv

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23 To Russell xv

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25 CHAPTER 1: INTRODUCTION Magnetics is a commonly used geophysical technique to identify and image potential subsurface targets in a wide variety of applications, ranging from archaeological site investigations to global-scale studies. Data, particularly aeromagnetic data, can be collected rapidly and effectively over a large area. Magnetic anomalies reflect variations of magnetic properties in the subsurface, including distribution of magnetic minerals and magnetization. Interpretation of the shape and magnitude of these anomalies provides information about the size, distribution, and magnetization of the subsurface and can be used to draw conclusions about geology or be incorporated into further investigations. Interpretation of magnetic anomalies is a complex process due to the superposition of multiple magnetic sources, presence of geologic and cultural noise, and acquisition and positioning error. Mathematical nonuniqueness associated with magnetic data further complicates the issue. Furthermore, total magnetization of a material is the vector sum of two components: induced magnetization and remanent magnetization. Induced magnetization is caused by the current magnetic field and has a constant direction over the entire survey area under commonly used assumptions. Remanent magnetization is a function of subsurface materials, varies spatially, and can be acquired in several ways including chemical alteration, thermal cooling, or contact metamorphism. This property is characterized as the magnetic memory of the material, remembering the orientation of the magnetic field at the time of deposition or formation. The orientation of the magnetic field varies globally and experiences secular variation, causing the direction of remanent magnetization to be varied and typically unknown. The presence of remanent magnetization can pose severe challenges to the quantitative interpretation of 1

26 magnetic data by skewing or laterally shifting magnetic anomalies relative to the subsurface source (Haney and Li, 22). Since the beginning of the magnetic method, it has been hampered by the presence of remanent magnetization, a vector quantity that is frequently unknown and determined only at great expense. Although this problem has existed for a long time, it has not received much attention for two reasons. First, in the majority of exploration problems, the direction of the remanent magnetization is approximately collinear, that is, aligned or antiparallel, with the current inducing field or the strength is weak and therefore not of major consequence. Under these conditions, the total magnetization direction is similar to the orientation of the inducing field. Secondly, many earlier interpretation techniques, such as depth estimation techniques (e.g., Hartman et al., 1971; Reid et al., 199), seek semi-quantitative information and are not affected by a lack of known total magnetization direction. However, in applications such as archaeology, some mineral exploration, basement imaging in petroleum exploration, and crustal and planetary studies, remanent magnetization is often strong and cannot be disregarded. Quantitative interpretation critically relies upon the ability to forward model the predicted response from a given representation of geologic structure. Without knowing the remanent magnetization, a vital piece of information is missing and forward modeling required in interpretation can no longer be carried out. Thus, various interpretation techniques have been developed to handle the presence of remanent magnetization in different manners. One of these is inversion, which requires recovered causative bodies to reproduce the observed data and dictates that we must know the magnetization direction. Unrealistic distributions of magnetization and magnetic susceptibilities can result from inversion if incorrect magnetization direction is specified. With the problem of unknown magnetization direction, modeling and inversion techniques thus far have neglected this quantity by investigating values that are independent of magnetization 2

27 direction (e.g., Nabighian, 1972) or under the assumption that remanent magnetization does not contribute significantly to the total magnetization (e.g., Li and Oldenburg, 1996; Pilkington, 1997). As a result, inversion techniques have been limited from successfully contributing to the interpretation of magnetic data in the presence of strong remanent magnetization. Thus, there is a need to develop an interpretation technique that recovers the distribution of magnetic material in the subsurface when strong remanent magnetization exists. To achieve this, quantities that are independent of magnetization direction must be utilized. Nabighian (1972) introduced a two-dimensional quantity, analytic signal, whose amplitude is independent of magnetization direction and therefore is not affected by remanent magnetization. Although a similar quantity exists in three dimensions (Haney et al., 23), no method has been found to calculate it directly from magnetization. Empirically, it has been observed that the amplitude of the anomalous magnetic field and the total gradient, defined as the magnitude of the gradient vector of magnetic anomaly data, exhibit weak dependence on the total magnetization direction. Therefore, these quantities can lend themselves to interpretation in the presence of remanent magnetization. I develop a three-dimensional inversion algorithm that uses these two quantities with weak dependence on magnetization direction to construct the magnitude of the magnetization. The algorithm solves a nonlinear inverse problem based upon the relationship between data quantities and magnetization. I also impose a positivity constraint to ensure that the inverted magnitude of magnetization is nonnegative. The inversion algorithm is formulated using Tikhonov regularization and positivity is imposed using a logarithmic barrier method. The algorithm is tested using both synthetic models and a field example. The ability to invert magnetic data with little information about the nature of remanent magnetization expands the areas in which we can apply 3

28 three-dimensional inversion of magnetic data. It is now possible to invert any magnetic data to some extent. Assumptions about magnetic data and the model region are necessary in order to conduct inversion. It is assumed that there are no self-demagnetization effects on the magnetic data and that magnetic anomalies are occurring based on magnetization contrasts and the presence of remanent magnetization. The additional assumption of isotropic magnetic susceptibility is also made. There are six chapters in this thesis. The first chapter is this introduction. I provide a general summary of the problem encountered with the magnetic method and my approach to solving the issue of inversion in the presence of strong remanent magnetization. In Chapter 2, a general background of the magnetic field and magnetization is given, followed by an overview of previous work on magnetic interpretation techniques. Finally, I formally introduce amplitude of the anomalous magnetic field and total gradient and discuss their weak dependence on magnetization direction. Illustration of independence on magnetization direction in two dimensions is illustrated for both quantities. Additionally, through forward modeling, it is found that while both quantities have a weak dependence on magnetization direction, the amplitude has a weaker dependence than the total gradient. In Chapter 3, I describe the inversion algorithm developed as part of this work. I begin with a summary of the general inversion process and then discuss details incorporated into my three-dimensional algorithm that is applicable in the presence of remanent magnetization. At the end of the chapter, the results of testing the algorithm against two synthetic examples, a cubic model and a dipping slab, are given. 4

29 Interpretation of the results shows that inversions with either quantity are successful in recovering the shape and location of the true model. However, while amplitude has good ability to recover susceptibility at depth, total gradient does not. This is attributed to a lack of low frequency information in the gradient data. Following the development of theory and implementation of my inversion algorithm, practical aspects are discussed in Chapter 4. To find the limitation on the current assumption, I conduct a series of inversions using MAG3D (Li and Oldenburg, 1996). I observe that there is a deviation threshold of 1 to 15 between the true and assumed magnetization direction before the alignment assumption on remanent magnetization is violated. Past this threshold, inversion results are too poor to be useful. Furthermore, the selection of depth weighting parameters is investigated through numerical modeling. Finally, ways to obtain the amplitude of the anomalous field and total gradient are given. There are advantages to calculating the quantities from existing data as well as using data that is acquired by using the developing gradient and tensor technology. Chapter 5 focuses on a field example where my algorithm is applied to a field data set from kimberlite exploration in the Canadian Arctic. A general geologic and geophysical background of kimberlites is given. I then discuss the background details of the kimberlite exploration area. Application of my inversion algorithm to the field data follows. Results from the inversion of amplitude data over the kimberlite exploration area are presented. Finally, I draw conclusions about my inversion algorithm, quantities with weak dependence on magnetization direction, current interpretation and inversion techniques, and some of the operational aspects associated with such in Chapter 6. I present a 5

30 discussion of the strengths and weaknesses of my inversion algorithm. Lastly, suggestions for future work are given at the end of the chapter. 6

31 CHAPTER 2: BACKGROUND To understand the complications that remanent magnetization presents in the inversion and interpretation of magnetic data, a brief overview of the magnetic field and magnetization is given. Subsequently, previous work relevant to inversion and interpretation of magnetic data is presented and forms the groundwork for my threedimensional inversion in the presence of remanent magnetization Magnetic Field The source of magnetic induction, B r, is electric current. The measurements and behavior of this field are described by two of Maxwell s equations. B r = (2.1) v r B = µ C (2.2) The divergence of the magnetic induction, eq.(2.1), states that the net flux of the magnetic field over any closed surface is zero. That is, all field lines that go into a closed surface also leave the surface. The curl of the magnetic induction, eq.(2.2), is equal to a vector sum of the all the various forms of charges moving within a region (Blakely, 1996). µ is the magnetic permeability of free space and is equal to 4π x 1-7 H/m. The 7

32 vector sum is expressed in terms of current density, C r, and includes bound and conduction currents. The geomagnetic field of the Earth is comprised of three parts: the main field, the external field, and variations of the main field. The main field originates internally and is thought to arise from a self-sustaining dynamo in the outer, molten core of the Earth. This main field can be mathematically represented as a series of spherical harmonics from globally derived observations to yield the International Geomagnetic Reference Field (IGRF) (Merrill et al., 1996). The highly varying external field is, under normal conditions, a small fraction of the geomagnetic field, less than one percent, and originates from external sources such as electric currents in ionized layers of the outer atmosphere. Lastly, variations in the main field are caused by localized magnetic anomalies in the crust (Telford et al., 199). These areas of crustal magnetization are the targets of exploration geophysicists. The main magnetic field of the Earth at any point is a vector quantity and can be described using three parameters: inclination, declination, and strength. The orientation of the main field can be described using the first two parameters, as depicted in Figure 2.1. Inclination is the angle between the magnetic field and the local horizontal surface plane of the earth. Declination is the angle between the magnetic field projection to the Earth s surface and geographic north. Alternatively, the magnetic field vector can also be described using three components in a local Cartesian coordinate system. A right-handed coordinate system with x positive north and z positive down is commonly used. Because the parameters used to describe the magnetic field of the Earth vary globally and exhibit temporal variation known as secular variation, the response of and the information collected over the subsurface are a function of both location and time. 8

33 Figure 2.1. The orientation of the main magnetic field can be described using inclination (I) and declination (D). Inclination is the angle between the magnetic field vector and the local horizontal surface plane of the Earth. Declination is the angle between the magnetic field projection to the Earth s surface and geographic north. In exploration geophysics, bound currents distributed in magnetic material produce the anomalous magnetic field. In order to better describe the magnetic sources within subsurface geological units in a mathematical fashion, the elemental concept of current loops is used (Figure 2.2). Current loops are distributed within and bounded by the edges of a magnetic material. Further, it is assumed that the current loops are small with respect to the distance of observation, r, parallel, and of equal magnitude. Thus, neighboring loops cancel interior current in adjacent loops. As a result, magnetic sources can be approximated as the net effect of all elemental bounded electrical currents. Each current loop has an associated dipole with a dipole moment, m r, given by r 2 m = iπa nˆ (2.3) 9

34 r r nˆ a r r ' i Figure 2.2. Elemental current loop used to represent the basic source of a magnetic field. Current loops are assumed to be small with respect to distance of observation, r r, parallel to each other, and of equal magnitude. Current, i, flows around the loop and nˆ is the unit normal vector and abides by the right-hand rule. The radius of the loop is given by a. where i is the current, nˆ is the unit normal vector, and a is the radius of the loop. Dipole moment has units of Ampere meter 2 (Am 2 ) in the direction of nˆ which abides by the right-hand rule (Blakely, 1996; Moskowitz, 24). Similarly, the magnetic induction, B r, at a loop is a dipolar field and can be expressed as r µ r 1 B = m 4π r r' (2.4) with units of Tesla (T). Hence, the small current loops are termed magnetic dipoles. Thus, the source distribution of the magnetic field can be described in terms of dipole density, or magnetization. 1

35 2.2. Magnetization It was previously mentioned that the source distribution of the magnetic field can be described in terms of dipole density. The net volume density of magnetic dipole moments is magnetization, J r. Magnetization is a vector quantity defined as the sum of the individual dipole moments, m r i, divided by a volume, V. r 1 r (2.5) J = m i V i Magnetization is a function of location and varies from point to point (Blakely, 1996). In a material, magnetization consists of two parts. Referred to as total magnetization, J r is the vector sum of induced magnetization, J r i, and remanent magnetization, J r r, (Figure 2.3). r r r J = J i + J r (2.6) Induced Magnetization In the subsurface, magnetic domains in magnetically susceptible minerals and rocks act as a collection of small magnets, with each domain having a dipole moment. In the absence of an external magnetic field, the individual dipole will be randomly orientated. With this arbitrary orientation, the net magnetization is zero. 11

36 J r i J r r J r Figure 2.3. Total magnetization, J r, is the vector sum of induced magnetization, J r i, and remanent magnetization, J r r. However, exposing a magnetic material to an external magnetic field will cause the magnetic domains to align with the orientation of this field, creating induced magnetization, J r i. Induced magnetization is proportional to the magnitude of the external field and will approximately align in the direction of the field (Blakely, 1996). It is also a function of the quantity, composition, and size of magnetic mineral grains (Reynolds et al., 199) in the subsurface. Induced magnetization is not permanent and will decrease to zero once the external field is removed, neglecting the effect of hysteresis. Macroscopically, the ease with which a material can be magnetized is quantified by a physical property called magnetic susceptibility, κ. Induced magnetization is given by r J i r = κh (2.7) where H r is the total magnetic field in the material. H r is a quantity useful for describing the behavior of the field inside a medium. It is defined as 12

37 r H r B = µ r J. (2.8) It is important to note that equations (2.7) and (2.8) show that J r is a nonlinear function of κ. However, in the majority of exploration problems, κ is small (i.e. κ << 1. SI) and the Born approximation works well. Under this approximation, the magnetic field inside a medium is approximated by the external field. r r (2.9) B H µ Therefore, induced magnetization can be given as r r. (2.1) κb J i = µ Under this approximation, the induced magnetization is aligned with the inducing field Magnetic Susceptibility Magnetic susceptibility measures how easily a substance will become magnetized in the presence of a magnetic field based on a ratio of magnetization to magnetic field (Moskowitz, 24). Magnetic susceptibility is a dimensionless quantity in SI. Further, magnetic susceptibility depends on the composition, size, and domain properties of magnetic grains. Generally, larger grains have a greater susceptibility than smaller 13

38 magnetic grains (Reynolds et al., 199). The influence of remanent magnetization, discussed next, is dependent on magnetic susceptibility of the subsurface material. The higher the magnetic susceptibility of a mineral or rock, the larger the remanent magnetization may be. Table 2.1 lists the magnetic susceptibility values for common minerals and rocks. Figure 2.4 compares the magnetic susceptibility values of common rocks with the magnetic minerals, hematite, and magnetite. Many minerals and rocks have broad, overlapping ranges of magnetic susceptibility. Table 2.1. Magnetic susceptibility values for common mineral and rock types. Taken from Sharma (1997, pg. 74). Mineral or Rock Type Magnetic Susceptibility (κ x 1-6 SI) Granite (with magnetite) 2 4, Slates 1,2 Basalt 5 8, Oceanic basalts 3 36, Limestone (with magnetite) 1 25, Gneiss 3, Sandstone Hematite (ore) 42 1, Magnetite (ore) 7 x x 1 6 Magnetite (crystal) 15 x

39 15 Figure 2.4. Magnetic susceptibility values of common types of rocks and magnetic minerals, hematite and magnetite. Many magnetic minerals and rocks have broad, overlapping ranges in susceptibility values.

40 Remanent Magnetization The second component of magnetization is remanent magnetization, J r r. In general, remanent magnetization is the net magnetization present in a material in the absence of an external field (Merrill et al., 1996). This type of magnetization is relatively permanent and retained in magnetic materials. During the acquisition of remanent magnetization, via a variety of phenomena, the magnetic grains retain the direction of the external field, creating a magnetic memory. As a result, the total magnetization of a target will not necessarily be aligned with the current inducing field. Rather it will be the sum of the remanent magnetization and the induced magnetization produced by the current inducing field orientation. J r r is a function of quantity, atomic, crystallographic, chemical makeup, and grain size of the magnetic minerals. Small magnetic grains support strong, stable remanent magnetizations (Reynolds et al., 199; Blakely, 1996). It is also affected by the geologic, tectonic, and thermal history of the mineral or rock (Blakely, 1996). There are two types of remanent magnetization: primary and secondary. A summary of different types of remanent magnetization is given in Table 2.2. Primary remanent magnetization is acquired as the mineral or rock forms. It is aligned with the inducing field orientation at the time of deposition or formation. Since this magnetization is permanent, it can provide information about the orientation of the external magnetic field at the time of formation (Merrill et al., 1996). Thermoremanent magnetization (TRM) is a type of primary remanent magnetization found in igneous rocks. TRM is first acquired when minerals cool below a characteristic Curie temperature. In sedimentary rocks, primary remanent magnetization 16

41 Table 2.2. Descriptions of different types of remanent magnetization. Modified from (Merrill et al.,1996; Sheriff, 22; Moskowitz, 24). Remanent Magnetization Acronym Rock Types Description Natural NRM All Summation of all remanent magnetization components (primary and secondary). Thermal TRM Igneous, Metamorphic Primary remanent magnetization acquired during cooling from a temperature above the Curie temperature in the presence of an external magnetic field. Viscous VRM All Secondary remanent magnetization acquired over time, related to thermal agitation and causes decay of primary remanent magnetization. Depositional DRM Sedimentary Primary remanent magnetization acquired during deposition in the presence of an external field by the physical rotation of magnetic mineral particles. Usually occurs as grains settle out of water. Post-depositional PDRM Sedimentary Acquired during postdepositional retention of interstitial grains. Chemical CRM All Remanent magnetization acquired during growth of magnetic minerals in presence of an external field. Includes growth by nucleation or replacement. Isothermal IRM All Secondary remanent magnetization acquired over a short time at one temperature in a strong, external field. 17

42 exists as depositional remanent magnetization (DRM) or post-depositional remanent magnetization (post-drm) or, occasionally, chemical remanent magnetization (CRM) (Merrill et al., 1996). Secondary remanent magnetization is acquired subsequent to formation. Magnetic grains are susceptible to remagnetization by time, temperature, or chemical alteration (Moskowitz, 24). Natural remanent magnetization (NRM) is a sum of primary and secondary magnetizations (Merrill et al., 1996). Influenced by the physio-chemical environment (Reynolds et al., 199), NRM is the vector sum of the different types of magnetization acquired over the history of a rock (Moskowitz, 24) Occurrence of Remanent Magnetization and Its Challenges Having defined magnetization and its components, I present a brief discussion about the occurrence of remanent magnetization in minerals and rocks. The following discussion includes an outline of where remanent magnetization occurs and the challenges that the presence of remanent magnetization introduces, including two physical methods of measuring remanent magnetization. From archaeology investigations to planetary studies, remanent magnetization occurs in a range of locations and is associated with a variety of natural and man-induced events. Igneous rocks, such as basalts and volcanics, banded iron formations (BIF), pyrrhotite-bearing rocks and ores, magnetite skarns, and laterite (Clark and Emerson, 1991) can acquire remanent magnetization at the time of deposition or formation. Rocks, however, are rarely found in-situ and undisturbed. Over time, they are subject to tectonic 18

43 movement, structural deformation, uplift, and rotation, all processes that change the direction of remanent magnetization to a different, unknown orientation. Unfortunately, in many cases, it is difficult to determine from magnetic data how much induced and remanent magnetization is present in the response of subsurface targets. Thus, it would seem that remanent magnetization of minerals and rocks should be measured from orientated cores and active-source experiments. However, these approaches are not always feasible and have limitations. This method is costly and time consuming. Numerous orientated cores from widely separated collection sites need to be collected and analyzed in order to obtain an accurate characterization of the bulk magnetization direction and its variation (Butler, 1992). Active-source experiments attempt to separate the remanent and induced components of magnetization in-situ using geomagnetic micropulsations (Goldstein and Ward, 1966). Even in ideal field collection, to have a geologically correct interpretation, it is not only important to know the direction and magnitude of the induced and remanent magnetizations, but also the type of remanent magnetization (Reynolds et al., 199). In light of these complications, alternative, more data based approaches have been developed. These approaches make use of geophysical data to characterize both total magnetization and remanent magnetization in semi-quantitative and quantitative manners Current Methods for Magnetic Interpretation Given the difficulties associated with measuring remanent magnetization, a variety of numerical interpretation approaches have been developed. The following 19

44 discussion will focus on the successive development of depth estimation, parametric inversion, and physical property inversion methods. These methods overcome the obstacle of unknown remanent magnetization by working independently of this component or utilizing an assumption. One strength of these techniques is their applicability to a wide variety of problems and applications. Weaknesses, however, arise with these methods in the presence of strong remanent magnetization. Recall from eq.(2.6), that total magnetization is the vector sum of two components: induced and remanent. Some current techniques make an assumption on the remanent magnetization. It is assumed that the remanent magnetization is either (a) approximately collinear, that is parallel or antiparallel, with the current inducing field or (b) has weak strength. In either case, the remanent magnetization component will have a negligible influence on the total magnetization direction, as shown in Figure Depth Estimation Depth estimation techniques are useful in obtaining a semi-quantitative representation of the source locations. This type of technique initially presumes a generalized geometric body shape, such as contacts, dikes, plates, or cylinders, in order to solve nonlinear inversion problems. By assuming parameters, such as structural complexity, for subsurface targets, the number of unknown parameters is vastly reduced and the solution to an over-determined problem is found. When applied along magnetic profiles or maps, the semi-quantitative results provide estimates of parameters, such as depth. Groupings of solutions then guide interpretation. These methods are not influenced by the presence of unknown remanent magnetization because they do not rely on total magnetization direction. 2

45 (a) J r r J r i J r J r r J r J r i (b) J r r J r i J r Figure 2.5. Conditions of the assumption on remanent magnetization. It is assumed that the remanent magnetization, J r r, is either (a) approximately collinear, that is parallel or antiparallel, with the current inducing field, or (b) has weak strength. In either case, the remanent magnetization component will have a negligible influence on the total magnetization direction, J r. A number of automated depth estimation techniques have been developed. Naudy (1971) cross-correlates symmetric and asymmetric portions of profile-based anomalies with theoretical curves. Solutions are then compared on the structural hypotheses of dikes and plates and sampling intervals to find source depth. In 1972, O Brien introduces CompuDepth, a technique that uses Hilbert transform and frequency shifting to estimate the edges and depths of two-dimensional prismatic bodies. Methods involving Werner deconvolution (e.g. Hartman et al., 1971; Jain, 1976) utilize total field as well as vertical and horizontal derivative information to estimate the depth, dip, horizontal position, and susceptibility contrast of an assumed dike or interface 21

46 source body. Solutions to polynomials are found using windows of six to seven data points along each set of profile data. Phillips (1979) introduces ADEPT, a method that estimates source parameters from an autocorrelation of an evenly sampled magnetic anomaly profile with dike or contact models. Thompson (1982) presents a technique, EULDPH, which solves Euler s homogeneity relation (also known as Euler s equation) to find depth estimations based on a structural index and increasing window sizes. Without assuming a geologic model, depth solutions can be paired with structural indices to interpret depth estimates for a variety of geologic structures, including faults, magnetic contacts, dikes, and extrusives. Reid et al. (199) implements this technique in 3D, offering a technique for analyzing mapped magnetic data. Nabighian and Hansen (21) introduce an extended Euler deconvolution based on three-dimensional Hilbert transform. These techniques can be used regardless of the presence of remanent magnetization, and are therefore applicable in a variety of applications where the total magnetization direction is unknown. Methods involving Werner deconvolution (e.g. Hartman et al., 1971; Jain, 1976) are independent of remanent magnetization and the calculations are theoretically valid at any magnetic inclination. However, these techniques are limited based on assumptions of simple causative bodies and minimal resolution of neighboring or superposed bodies. Solutions are vulnerable to noise and interpreter bias and inexperience Parametric Inversion The second interpretation method is a quantitative inversion technique. Inversion techniques recover the simple geometry of causative bodies that reproduce observed data 22

47 within a certain tolerance of error. This method requires certain a priori information, such as a known magnetization direction. Using the wrong magnetization direction will produce inversion results that have an unrealistic distribution of magnetization or magnetic properties, such as susceptibility. This type of nonlinear inversion solves an over-determined problem and recovers parameters for simple bodies, such as prisms and dikes. Assumptions about the remanent magnetization are common with this type of technique. In an attempt to better characterize the magnetized region that fits the measured anomalous response of the subsurface, Bhattacharyya (198) presents a threedimensional iterative approach. Cells with constant magnetization comprise the region of interest. The horizontal dimensions of the rectangular blocks that comprise the model region are based on the height of the observation surface; the vertical extents of the blocks are adjusted to find a least squares fit between the observed and calculated field values. The method takes the remanent magnetization component into account by constraining the magnetization vector of each block to lie within a specified angle of the normal or reversed direction of the inducing field. Zeyen and Pous (1991) introduce a three-dimensional inversion algorithm that recovers parameters for the top and base of vertical rectangular prisms, susceptibility, and remanent magnetization intensity and direction. The algorithm includes a priori information of model parameters and is strongly dependent on an initial model. Fixing the remanent magnetization intensity or direction, or restricting the component to be the same direction for all prisms reduces the nonuniqueness of the problem. Nonuniqueness is further reduced between equivalent models by selecting the model that is compatible with the prior information. 23

48 This type of parametric inversion requires a large amount of prior information. Therefore, it is limited to areas where details about causative body geometry, parameter values, and property values are well known. Further, the initial parameterization restricts the solution to a subset of possible models and does not let the inversion recover a set of parameters that fall outside this model space. Wang and Hansen (199) present a three-dimensional extension of O Brien s CompuDepth (1972). Using complex polynomials, this technique determines a series of coefficients that describe the depth and location of polyhedral vertices. Although this technique does not depend on an initial model, it is assumed that the subsurface source bodies can be approximated by homogeneously magnetized polyhedrons. Although Wang and Hansen (199) improve upon other parametric inversion techniques, the algorithm is still limited in constructing causative bodies from discrete vertices Physical Property Inversion The second type of inversion is established to recover the subsurface distribution of a physical property, such as magnetic susceptibility (Li and Oldenburg, 1996). These techniques solve an underdetermined problem. Li and Oldenburg (1996) introduce a generalized magnetic inversion for distributions of susceptibility. Using surface magnetic data, this inversion methodology solves an underdetermined problem by minimizing a global objective function comprised of a model objective function and data misfit. Prior information can be incorporated into the model objective function to reduce the nonuniqueness of the solution. Further constraints, such as positivity and depth weighting, are also imposed to increase the geologic reasonability of the solution. This three-dimensional algorithm is useful in areas 24

49 of multiple anomalies on a variety of scales. Pilkington (1997) presents a similar threedimensional inversion algorithm to find a distribution of magnetic susceptibility. This method employs a preconditioned conjugate gradient method for computational efficiency. These three-dimensional methodologies assume that the surface magnetic data used in the inversion is produced by induced magnetization only. Under this assumption, the remanent magnetization component is small and there is no self-demagnetization, thus the magnetization is given by eq.(2.1). Therefore, in areas where the remanent magnetization strongly influences the total magnetization, the results will be unrealistic. A study of the error produced by using the three-dimensional inversion algorithm developed by Li and Oldenburg (1996) in the presence of deviated remanent magnetization is presented in Chapter Influence of Remanent Magnetization Given the flexibility and diversity in application of the physical property inversion method that Li and Oldenburg (1996) introduced, I focus on this methodology for further development. Expanding this approach to include remanent magnetization will increase the applicability of physical property inversion. The ability to invert magnetic data in the presence of remanent magnetization will provide great flexibility in practical applications. As a result, it will be possible to take any magnetic data and carry out interpretation by inversion. In light of this, a discussion of current methods which operate in the presence of remanent magnetization follows. 25

50 2.4. Current Methods to Interpret Magnetic Data with Strong Remanent Magnetization Although the previous methods are applicable in a variety of areas, there are numerous cases where the presence of strong remanent magnetization violates the alignment assumption. In those cases, different techniques must be utilized. There are two classes of methods that can be used to interpret magnetic data in the presence of strong remanent magnetization. These methods address the presence of remanent magnetization through direction estimation and exploitation of mathematical relationships, often utilizing quantities that have no or minimal dependence on magnetization direction Direction Estimation Prior to Inversion These methods overcome the presence of remanent magnetization in a two-step process. First, an estimation of the direction of total magnetization is calculated. Then, the estimation is used in interpretation of magnetic data, either through qualitative analysis or quantitative processing, such as inversion. Recall that the presence of remanent magnetization causes a lateral shift or skewness to magnetic anomalies (Haney and Li, 22). Thus, separation of source dip and remanent magnetization is important in aiding in interpretation of magnetic data, and is the focus of several of current direction estimation techniques. Utilizing the properties of the magnetic anomaly, Roest and Pilkington (1993) use the total gradient and pseudogravity to estimate the total magnetization direction and determine the location of source bodies. The authors compare these two quantities to show that the presence of remanent magnetization causes a skewness that does not affect the analytic signal, which is independent of magnetization direction in two dimensions, 26

51 but has a significant effect on the pseudogravity results. Additionally, the authors show that coupling these quantities provide enhanced detail, particularly when the inducing field is vertical. Reduction to pole (RTP), however, requires the use of total magnetization direction, which is unknown. By estimating an arbitrary set of source parameters and forward modeling the response of both quantities, the authors conduct an iterative investigation until agreement between the two sets of information is reached. Using continuous wavelet transforms (CWT), Haney and Li (22) extract information about the total magnetization direction and dip from total field measurements. Separation of the two quantities is important in the presence of remanent magnetization. The magnitude of the extrema of the CWT is dependent of the geometry (dip) of the source while the location of the extrema provides information about the intrinsic properties such as total magnetization. Estimation of total magnetization direction allows for inversion of magnetic data in the presence of remanent magnetization Use of Analytic Signal There are a number of approaches that involve working with quantities that are calculated from observed magnetic data and have no to minimal dependence on the magnetization direction. This is accomplished by exploiting mathematical relationships. The best-known approach of this type involves the analytic signal approach introduced by Nabighian (1972). It can be shown that by using Hilbert transforms, denoted as H, the vertical derivative of the magnetic field can be calculated from the horizontal derivative, allowing for a fast and accurate method of computing the vertical derivative from a given magnetic profile, 27

52 M M = H z x (2.11) where M is the magnetic anomaly data, for instance, total field anomaly. Further, these two quantities can be combined into a two-dimensional quantity known as the analytic signal (A), A ( z) = T( x) + it1 ( x) (2.12) where ( M ) T ( x) = x and T 1 ( ( M ). (2.13) x) = z The amplitude is defined as A ( z) + T 2 2 1/ 2 = ( T 1 ). (2.14) Further, Nabighian (1972) shows that the amplitude of the analytic signal, which is the amplitude of the two-dimensional gradient vector, is independent of magnetization direction. This is an important step in developing an inversion technique that is not dependent on unknown remanent magnetization direction. To illustrate the independence on magnetization direction, the two-dimensional envelope of the amplitude of the anomalous magnetic field and total gradient are given in Figure 2.6. The blue lines are the total field anomalies and the horizontal derivatives at different magnetizations directions for each quantity, respectively. The maximum of each magnetization is 28

53 (a) (b) Figure 2.6. The (a) amplitude of the anomalous magnetic field and (b) total gradient of any component are independent of magnetization direction of the source. The blue lines in the two panels are (a) total field anomalies and (b) horizontal derivatives computed over a rectangular source body at various magnetization directions. The red lines are amplitude and total gradient, respectively. The rectangular source body is located between 75 m and 75 m and extends from 75 m to 225 m in depth. 29

54 outlined in red, defining the amplitude and total gradient. The source is a rectangular body with upper vertices at x-y coordinates of ( 75 m, 75 m) and (75 m, 75 m) and extends from 75 m to a depth of 225 m. It is interesting to note that the maximum of the total gradient is slightly flatter than the maximum of the amplitude. Based on a second simulation over a larger body, with upper vertices located at ( 275 m, 75 m) and (275 m, 75 m), it is found that the peaks of the total gradient occur over the edges of the causative body (Figure 2.7). The amplitude over the same larger body shows two maximums that have a smaller amount of relief. The amplitude of the analytic signal over a simple contact is a symmetric bellshaped function; a combination of signals produces a plot in which each maxima appears over a corner of a polygonal cross section of the body. By examining profiles across a magnetic source, the analytic signal can be used in interpretation to provide indications of the edges of causative bodies, based on magnetization contrasts, yielding source depth estimation, locations, and geometry. In 1984, Nabighian further developed the relationship between horizontal and vertical derivatives. Using Hilbert transforms, the two-dimensional relationship between horizontal and vertical derivatives of the magnetic field is expanded to three dimensions. It is shown that the vertical derivative in three dimensions is a Hilbert transform of a combination of both horizontal, x and y, derivatives, in eq.(2.15) M z M M = H + H (2.15) x y x y 3

55 (a) (b) Figure 2.7. The (a) amplitude of the anomalous magnetic field and (b) total gradient of any component are independent of magnetization direction of the source. The blue lines in the two panels are (a) total field anomalies and (b) horizontal derivatives at various magnetization directions. The red lines are amplitude and total gradient, respectively. The larger, rectangular source body is located between 275 m and 275 m and extends from 75 m to 225 m in depth. Note that the peaks of the total gradient coincide with the edges of the source body where as the peaks of the amplitude do not have the same amount of relief. 31

56 where H x and H y are the x- and y- components of the three dimensional Hilbert transform ~ iω ~ iω x y and are defined in the wavenumber domain as H = and H =. x 2 2 y 2 2 ω + ω ω + ω Despite the advancements made in the expanding of Hilbert transforms, the dependence on magnetization direction is not investigated. Many authors have tried to extend this approach to three-dimensional problems, but to date no one has found a quantity that can be calculated directly from observed data and is entirely independent of magnetization direction. x y x y In an effort to invert the three-dimensional magnetic source distributions in the presence of remanent magnetization, Paine et al. (21) examine two related transform quantities. The first is the total gradient of the vertically integrated magnetic anomaly, termed ASVI by Paine et al. (21), and the second is the vertical integration of the total gradient, termed VIAS. Due to weak dependence on magnetization direction and both quantities possessing the dimensions of the magnetic field, nt, Paine et al. (21) treat the transforms as reduced to pole (RTP) fields and apply the three-dimensional inversion algorithm by Li and Oldenburg (1996). When compared to inverting total field anomaly data by assuming induced magnetization only, this approach produces somewhat better results. The approach taken by Paine et al. (21) is an interesting attempt to use the inversion algorithm developed by Li and Oldenburg (1996). Instead of ignoring the remanent magnetization, Paine et al. (21) uses a transformation of the data to incorporate both components of the total magnetization into a single cohesive signal. By using these computed quantities as RTP data, the authors show the significance of utilizing the mathematical relationships presented by Nabighian (1972; 1984). However, as Paine et al. (21) acknowledge, there are a number of theoretical difficulties 32

57 associated with this approach. The total gradient is not a Laplacian field, thus, computations of VIAS data in the Fourier domain are invalid. Paine et al. (21) account for the DC component of an infinite thin dike by removing it from the data. Further, the vertical integration of total gradient will always be positive; yet, the VIAS transform does not restrict the presence of negative values. The application of VIAS transforms amplifies low frequency noise and reduces the drop off rate of the signal. To compensate, results are broader and deeper than true sources Extension Toward 3D: Utilizing Quantities with Weak Dependence on Magnetization Direction Motivated by the analytic signal work by Nabighian (1972; 1984) and the introduction of the idea of quantities that have a weak dependence on remanent magnetization, my research improves upon current work by developing a threedimensional algorithm that directly inverts quantities that exhibit a weak dependence on magnetization direction. Such quantities include the amplitude of the anomalous magnetic field vector and the magnitude of the gradient vector of magnetic anomaly data, total gradient. My algorithm fully incorporates the nonlinear relationship between these quantities and subsurface magnetization. Despite various attempts, no one has found a quantity that can be calculated directly from the observed data and is entirely independent of magnetization direction. However, it has been demonstrated that the magnitude of the gradient vector in three dimensions, referred to here as total gradient and often erroneously termed the threedimensional analytic signal, has a weak dependence on magnetization direction (Roest et al., 1992; MacLeod et al., 1993; Paine et al., 21). Because of this property, total 33

58 gradient has been widely used for interpreting magnetic data acquired in both exploration and environmental problems. Further, it is noted that the magnitude of the anomalous magnetic vector has a similar property. The magnitude of the anomalous field vector in two dimensions is also independent of the magnetization direction. This arises from the fact that the horizontal and vertical derivatives of the magnetic potential form a Hilbert transform pair and the amplitude therefore forms an envelope of either component. The envelope is, by definition, independent of direction. Although it is not entirely direction independent in three dimensions, the amplitude can be demonstrated as exhibiting much less sensitivity to it. Therefore, this quantity can also be inverted in a manner similar to total gradient data Amplitude of the Anomalous Magnetic Field The amplitude is defined as that of the anomalous magnetic field vector, B a B a = B 2 x = r, (2.16) + B 2 y + B 2 z where B x, B y, and B z are the three components of the magnetic field in a three dimensional Cartesian coordinate system. Figure 2.8 illustrates the weak dependence of amplitude of magnetization direction. While the inducing field is held constant at I = 45 and D = 45, two different 34

59 (a) 1 I m = 65, D m = 25 I m = 25, D m = -65 (b) Northing (m) Easting (m) 1-5 nt 35 (c) (d) -5 nt 5 Figure 2.8. Illustration of amplitude data with weak dependence on different magnetization directions over a simple cubic model (κ =.5 SI). The inducing field orientation is held constant at I = 45º and D = 45º, while the magnetization direction varies as labeled at the top of each column. The total field anomalies (a and b) are distinctly different, but the corresponding amplitudes (c and d) are similar. The shape, size, and magnitude are approximately equal confirming a weak dependence on magnetization direction. 35

60 magnetization directions (I m = 65, D m = 25 and I m = 25, D m = -65 ) were tested over a simple cubic model with constant magnetic susceptibility of.5 SI. Although the total field anomalies are very different in shape, orientation, and magnitude under the two magnetization directions, the amplitude data are similar. The results have remarkably similar shape, size, and magnitude. Despite different magnetization directions, the results are almost indistinguishable Total Gradient The total gradient is the amplitude of the gradient of anomalous magnetic data, g = M, (2.17) = ( M / x) 2 + ( M / y) 2 + ( M / z) 2 where M is a given component of the anomalous field such as the total field anomaly or the vertical anomaly. Correspondingly, Figure 2.9 illustrates the weak dependence of total gradient of magnetization direction. While the inducing field is held constant at I = 45 and D = 45, two different magnetization directions (I m = 65, D m = 25 and I m = 25, D m = -65 ) were tested over a simple cubic model with a magnetic susceptibility of.5 SI. The total gradient results show some similar characteristics under the two magnetization directions. Both total gradient results have a rounded shape and are approximately equal in size. However, the results from the second magnetization direction are less symmetric and possess a smaller magnitude. Unlike the amplitude data, there are visible differences between the results under different magnetization directions. 36

61 (a) 1 I m = 65, D m = 25 I m = 25, D m = -65 (b) Northing (m) Easting (m) 1-5 nt 35 (c) (d) nt/m 4 Figure 2.9. Illustration of total gradient data with weak dependence on different magnetization directions over a simple cubic model (κ =.5 SI). The inducing field orientation is held constant at I = 45º and D = 45º, while the magnetization direction varies as labeled at the top of each column. The total field anomalies (a and b) are distinctly different, but the corresponding total gradients (c and d) are similar. The dependence on the magnetization direction is not as favorable as that of amplitude data, but the shapes and sizes are approximately the same. The magnitudes vary. 37

62 2.6. Summary Previous interpretation and inversion methods have utilized techniques that are either independent of magnetization direction or make an assumption on such. Although valid in a variety of applications, these methods are not applicable in the presence strong remanent magnetization in three dimensions. Thus, it is appropriate to develop a threedimensional algorithm that inverts magnetic data in the presence of strong remanent magnetization. Based on the results of Figures 2.8 and 2.9, it is confirmed that both amplitude and total gradient have a weak dependence on magnetization direction. Furthermore, the amplitude exhibits a significantly weaker dependence on magnetization direction than total gradient. With this insight, it is not expected that an assumed magnetization direction will drastically influence the inversion results. Motivated by the work on analytic signal by Nabighian (1972; 1984) and the introduction of the idea of quantities that have a weak dependence on remanent magnetization, my research improves upon current practice by developing a threedimensional algorithm that directly inverts quantities, such as amplitude of the anomalous magnetic field and the magnitude of the gradient vector of the magnetic anomaly data, that exhibit a weak dependence on the direction of the magnetization. 38

63 CHAPTER 3: INVERSION METHODOLOGY In the preceding chapter, I establish that the amplitude of the anomalous magnetic field and total gradient are quantities that exhibit a weak dependence on magnetization direction. Exploiting this property, I now proceed to develop an algorithm that inverts these quantities to recover a distribution of magnetic material in the subsurface. In this chapter, I first provide a general description of the geophysical inverse problem. I then introduce the inversion methodology developed to invert the quantities discussed previously. Lastly, I illustrate the algorithm with synthetic examples Introduction to Geophysical Inversion Geophysical data communicates information about the subsurface. They contain details about the distribution of physical properties as well as the dimensions and locations of natural and artificial objects and events. This information is acquired by measuring changes of a physical field, which are caused by variations in one or more physical properties. The scale of such variations can range from microscopic to regional and are rooted in geological events, such as faulting and mineralization, and man-induced activity. For example, information about the magnetization of rock units is contained within magnetic data. We would like to extract information about the geometry and extent of magnetic material from these data to aid in the determination of subsurface structure for applications such as archaeological investigations, mineral and resource exploration, and crustal and planetary studies. 39

64 Magnetic data acquired in the field are an integrated response over a volume of magnetic material. To obtain particular information about the subsurface, data must be interpreted. Interpretation is conducted using a variety of tools and techniques, which help separate different types of information contained in the data. One common interpretation tool is geophysical inversion. This technique provides a reconstructed image of the subsurface to aid in the interpretation of geologic structures and features The Need for Inversion For inversion to be a successful interpretation tool, it must provide a reliable reconstruction of the subsurface. To obtain such, basic inversion tests a sequence of subsurface structures, known as models. Each model is a possible representation of the subsurface that produced the observed data. The goal of inversion is to find the model that best reproduces the observed data and satisfies other constraints. Testing the sequence of models requires a process known as forward modeling. This is a critical component in any inversion. Forward modeling uses physical properties and mathematical parameters of the subsurface to calculate a synthetic response expected from a known representation of the subsurface. These predicted data are expressed as functionals of a model by F = pred ( m) d (3.1) where F is an operator based on physics and geometry, m is a subsurface model representing the distribution of physical properties and d pred is the data predicted based on 4

65 m (eq.3.1). Forward modeling is a link between subsurface models and measured data, as shown in Figure 3.1. The data predicted by the model, d pred, can then be compared to the measured, or observed data, d obs. If the two data sets differ by more than a specified tolerance, the model is modified and a new set of synthetic data is calculated. Unfortunately, there can be multiple, diverse models that produce similar agreement between the two sets of data. As a result, forward modeling must be coupled with other independent constraints that help indicate which model, among all that fit the observed data, is the most geologically plausible. Symbolically, the process of finding a model can be expressed as m 1 = F [ d ] (3.2) * obs where * m is the recovered, or inverted, model and 1 F is the inverse operator. This process is presented in Figure 3.2. The end product of inversion is a description of the subsurface distribution of a physical property. While this can, in principle, be conducted manually by trial and error, a practical, effective approach requires automation. Although eq.(3.2) has a simple form, inversion rarely exists this simply in practice. The reasoning is two fold. First, in general, we do not have a simple, closed form solution of the forward operator. Secondly, as mentioned previously, there are multiple models that will fit the observed data to the same degree. Therefore, the numerical solution of eq.(3.2) entails more complexity, including the incorporation of other information that is independent of the observed data. 41

66 m F ( m) = d pred Figure 3.1. Forward modeling calculates a synthetic response (d pred ) from a representation of the subsurface (m) by using a forward operation (F) that describes the underlying physics. m = F [ d * 1 obs ] m* Figure 3.2. The inverse operator (F -1 ) estimates distributions of physical properties in the recovered, or inverted, subsurface model (m*) based on physical properties and mathematical parameters. The process is complete when the forward modeled data (d pred ) approximates the observed data (d obs ). 42

67 There are several general classes of inversion techniques available depending on the amount and type of information available. A brief overview of these is given in Chapter 2. I will focus on inversion for a single physical property from a single set of geophysical data. There are generally two approaches to this type of problem: parametric inversion and physical property inversion. Parametric inversion recovers source body parameters for simple causative bodies, such as dikes and prisms. By assuming a subsurface geometry, the number of parameters used to describe the subsurface is reduced and this allows for solution of an over-determined problem. The solution provides information about various location parameters, such as depth. Parametric inversion is advantageous due to the reduced degrees of freedom in terms of calculation complexity. The second approach reconstructs a subsurface physical property distribution as a function of spatial position. Physical property inversion reconstructs subsurface structure based on the presence or absence of anomalous property values within the subsurface. Contrary to parametric inversion, this type of inversion does not impose a prior restriction on the geometry to be recovered. Consequently, the solution for a discrete set of observations is formulated as an underdetermined problem. It is necessary to introduce additional constraints in order to reduce the degrees of freedom. In magnetics, inversion is used to find distributions of a magnetic property, such as magnetic susceptibility, magnetic permeability, or magnetization as a function of position. The recovered property distribution helps characterize the geometry and extent of magnetic material present in the subsurface, thereby aiding the interpretation of geologic structure. 43

68 For this research, I have chosen to invert surface magnetic data for a distribution of a physical property as a function of position in three-dimensional space. Specifically, my inversion is formulated to recover a distribution of effective magnetic susceptibility. Inversion results will provide information about the geometry and distribution of magnetic material in the subsurface. Development of this type of technique will aid in interpretation over a wide range of applications, including archaeological investigations, mineral and resource exploration, and crustal and planetary studies Tikhonov Regularization As mentioned previously, finding a solution through inversion is not as simple as eq.(3.2) suggests. In order to recover accurate results and use inversion as an aid in the interpretation of observed magnetic data, several operational criteria must be addressed, including data misfit and assessment of the structural complexity of recovered models. Different aspects of each criterion are important to various types of geophysical inversion. Utilizing a combination of criteria helps evaluate and characterize the value of each model generated through the inversion process. The first criterion for inversion requires that the predicted data must fit, or agree with, the observed data. This condition requires the use of a quantitative measure of the agreement between data sets, known as data misfit. This function, φ d, is defined based on the fact that there is only a finite number of data available and each datum has certain error. Data misfit is given by φ = d W ( d d obs d pred ) 2 (3.3) 44

69 where W d is a square, diagonal weighting matrix. Elements along the diagonal are the reciprocal of the standard deviation of the data and will be discussed later in this chapter. Geophysical data are imperfect. Observed data and quantities computed from them are always contaminated by various errors and noise. Errors are deviations from the true value and originate from instrument or positioning errors in data acquisition or truncation errors in processing and data reduction (Sheriff, 22). Noise occurs due to near surface material that cannot be represented within the assumed model (Sheriff, 22). In most cases, the exact source, quality, and magnitude of noise present in data is unknown. To precisely match predicted data to observed data would be fitting error and noise. Thus, data misfit should never be zero for any inversion. Additionally, too small a data misfit means that the error and noise are being fit; too large a data misfit suggests that some information in the data is lost. Preferably, the data predicted by a solution model should agree with the observed data within an accepted range of values based on estimates and assumptions about the error and noise present in the data. Since the true character of the error and noise is unknown, it is common to assume that errors are uncorrelated, independent, and Gaussian with a zero mean. This assumption leads to the discrepancy principle, which is discussed later in this chapter. Further, a key problem in geophysical inversion is nonuniqueness, which I alluded to earlier in terms of multiple, diverse models that have similar agreement between observed and predicted data. Nonuniqueness is a concept that a particular set of observed data could be reproduced by an infinite number of models. This complication arises because there are a finite number of inaccurate data describing the subsurface (Parker, 1972; Menke, 1989; Parker, 1994; Scales et al., 1997). This leads to problems in inversion. 45

70 Presented with an infinite number of possible solutions that equally reproduce the observed data, an additional criterion must be introduced to distinguish the model that most likely represents the subsurface geology among the range of possible models. Measuring the structural complexity of a model serves as a good criterion for this purpose. To quantify this, a model objective function, also referred to as model norm, φ m, is introduced. This criterion is given by r r 2 φ m = L( m( r) m ( r)) (3.4) where L is a differential operator. The criteria of data misfit and model objective function place different, and competing, requirements on the models. The best model will be one that minimizes the model objective function, φ m, while fitting the data within an acceptable range of data misfit, φ d. Letting either aspect dominate will result in an unreasonable representation of the subsurface. If data misfit is allowed to dominate, the predicted and observed data will have good agreement, but the subsurface structure may be overly complex and geologically unrealistic. However, if the model objective function is strictly satisfied, the subsurface structure may be simple and smooth but the predicted and observed data may not agree. To harness the benefits of both criteria, it is necessary to determine how to control the contribution of each in order to obtain the best solution. The roles of data misfit and model objective function are balanced using Tikhonov regularization (Tikhonov and Arsenin, 1977) through a global objective function, φ, defined as 46

71 φ = φ d + βφ m (3.5) where β is a regularization parameter under the constraint that < β <. By using Tikhonov regularization, the inversion jointly optimizes a reduction in the data misfit and minimization of the model norm. The objective is to find a regularization parameter that balances the two components, providing a model that fits the observed data according to the error and noise level while minimizing the model complexity. A small value of β ( β ) allows the model to be dominated by data misfit and restricts the model objective function from regulating the complexity of the solution. A small regularization parameter value emphasizes the data misfit by overfitting the data. As a result, noise in the data will be reproduced by complex structure. However, a large value of β ( β ) can also lead to an unrealistic solution. The resulting model may have large disagreement between the observed and predicted data and structural information in the data may be lost Forward Modeling Data with Weak Dependence on Magnetization Direction As introduced earlier, forward modeling is a critical component in inversion. It uses physical properties and mathematical parameters of a subsurface model to calculate a synthetic response expected from a representation of the subsurface. In this section, I will begin with details of model representation through discretization. Then, I will present the forward modeling of amplitude and total gradient data, using discrete representation. 47

72 Model Representation The subsurface model is a representation of the distribution of a given physical property. To facilitate the numerical computation, I represent the three-dimensional subsurface by a set of cuboidal cells. Each cell has a constant physical property value and serves as a parameter in the description of the subsurface property distribution. The r T cells in the model region can be represented as a vector of cells, m = m, m,..., ), where M is the total number of cells. ( 1 2 m M Two issues arise when determining the size of the subsurface model and cell discretization. First, subsurface models need to be larger than the area of interest (i.e. the area directly over the target) in order to account for sources outside the data area. Second, features in the subsurface cannot be resolved on a scale smaller than that of the cell size. A finer model discretization will allow for better resolution of features. However, the larger number of parameters leads to an increase in computational cost. Thus, there is a tradeoff between the need for resolution and discretization and the cost of computation. For this inversion, a three-dimensional distribution of effective magnetic susceptibility is being recovered. The distribution of effective magnetic susceptibility r T values within the model is represented as κ = κ, κ,..., and shown in Figure 3.3. ( 1 2 κ M ) Within each cell, magnetization is given by the product of the effective magnetic susceptibility and the strength of the current inducing magnetic field. Magnetization is allowed to vary from one cell to another within the model region. A right-handed Cartesian coordinate system with x positive north and z positive down has been adopted. 48

73 Figure 3.3. In a three-dimensional model, the model region is represented as a series of cuboidal cells. Each cell has a single physical property parameter of effective r T magnetic susceptibility and the model region is represented as κ = ( κ1, κ 2,..., κ M ). For use with this model, a right-handed Cartesian coordinate system with x positive north and z positive down has been adopted. 49

74 Forward Modeling of Amplitude of Anomalous Magnetic Field Data Recall that the amplitude of anomalous magnetic field data is calculated using three components of the anomalous field B a B a = B 2 x = r, (3.6) + B 2 y + B 2 z where B x, B y, and B z are the three components of the magnetic field. Given such a discretization, the components of the magnetic field are linearly related to the magnetic susceptibility. Let r 1 2 N T B = ( B, B,..., B ), r 1 2 N T B = ( B, B,..., B ), and r B = ( B, B,..., B ) z 1 z 2 z N z are given in matrix form by T x x x x where N is the number of observation points. The linear relations y y y y r r r r r r B = G κ, B = G κ, B = G κ (3.7) x x y y z z where G x, G y, and G z are the sensitivity matrices for B x, B y, and B z, respectively. The elements of G x, G x ij, are the B x at the i th observation location produced by a unit susceptibility at the j th cell. A magnetization direction has been assumed, with the understanding, as presented in Chapter 2, that the latter will not drastically affect the amplitude. The amplitude data are then given by 5

75 2 2 B = + + i i i a B x B y Bzi 2 (3.8) where i = 1, 2,, N Forward Modeling of Total Gradient Data Similarly, recall that total gradient is the amplitude of the gradient of anomalous magnetic data g = M (3.9) = ( M / x) 2 + ( M / y) 2 + ( M / z) 2 where M is a given component of the anomalous field such as the total field anomaly or the vertical anomaly. Given the previous discretization, the components of the total gradient are linearly related to the susceptibility. Let r 1 2 N T M x = ( M / x, M / x,..., M / x ), N T M r 1 2 = ( M / y, M / y,..., M / y ), and r 1 2 N M = ( M / z, M / z,..., M / z ) T y where N is the number of observation points. The linear relations are given in matrix form by r ~ r r ~ r r ~ r M x = Gxκ, M y = Gyκ, M z = Gzκ (3.1) z 51

76 where G ~ x, G ~ y, and respectively. The elements of G ~ z are the sensitivity matrices for M / x, M / y, and M / z, G ~ x, G ~, are the M / x at the i th observation location x ij produced by a unit susceptibility at the j th cell. A magnetization direction has been assumed, with the understanding, as presented in Chapter 2, that the latter will not drastically affect the amplitude. The total gradient data are then given by g i = 2 ( M x ) + ( M / y ) + ( M / z ) 2 / i i i 2 (3.11) where i = 1, 2,, N. In light of the different sensitivity matrices, G and G ~, for amplitude and total gradient, respectively, I will use the sensitivity matrix for the amplitude for discussion in this thesis Inversion Algorithm I proceed to the details of the inversion algorithm developed to perform a threedimensional inversion of magnetic data in the presence of strong remanent magnetization. Discussion includes details about data misfit, model objective function, positivity constraints, and weighting functions. 52

77 Data Misfit As indicated earlier, data misfit is the first criterion that inversion must satisfy to obtain an inversion solution. Previously, the agreement between observed and predicted data was generally defined in eq.(3.3). For the inversion algorithm I present, data misfit is modified to be φ d = N r r obs pred d d i = 1 σ i 2 (3.12) where the diagonal elements of W d from eq.(3.3) are expressed as the standard deviation of the data at the i th datum, σ i, and N is the number of data Model Objective Function The model objective function is the second criterion used to find an appropriate solution to the inversion. This measure of structural complexity was generally defined in eq.(3.4). For this inversion, the model objective function, as applied to a discretized model region, is defined as r r 2 r r r r 2 ( κ κ ) φ ( κ, κ ) = α W ( κ κ ) dv + α W dv m o s s x x x V V r r 2 ( κ κ ) α W dv y + α y z y + V V r r ( κ κ Wz z ) 2 dv... (3.13) 53

78 where r κ is a reference model and α s, α x, α y, and α z are smoothing parameters, often held constant, for each component. W s, W x, W y, and W z are weighting matrices for each term. Prior information can be incorporated into the weighting matrices of the model objective function as information from other geophysical surveys, geological data and understanding of the geologic structure becomes available. Discretization, as discussed above, yields r r 2 φm = W m( κ κ ) (3.14) where W m is the model weighting matrix of the model objective function. This weighting function is the summation of weighting terms for each component of the model objective function and is given by T W m W m = α W W + α W W + α W W + α W W. (3.15) s T s s x T x x y T y y z T z z The matrix W T s W s controls the size of the cells while matrices W T x W x, W T y W y, and W T z W z control the rate of change in each direction. Each of these matrices has a series of coefficients. The width of the coefficients determines the rate of change in the model weighting. Corresponding to the indexing of the model cells, the fastest rates of change are permitted in the z-direction, followed by the y- and x-directions. For the purpose of this research, the weighting matrices, W s, W z, W y, and W z are held at unity (Li & Oldenburg, 1996) and roughness is not penalized in a direction dependent manner. The smoothing parameters, α x, α y, and α z, are all set to unity to give equal directional influence on the model weighting matrix, based on the values for α x, α y, and α z. W m. The value of α s is determined 54

79 As a result of more detailed data misfit and model objective functions, the global objective function is now defined as φ r r r r ( ( κ κ ) 2 obs pred = W d d ) + β W d m 2. (3.16) Selection of Regularization Parameter It is appropriate to briefly discuss techniques used to select the optimal regularization parameter. Choice of regularization parameter is crucial for obtaining good inversion results. Here, I summarize and comment on the discrepancy principle. The discrepancy principle, usually credited to Morozov, uses residual norms to define the regularization parameter (Hansen, 1992). In data misfit, the depth weighting matrix, W d, is defined by the standard deviation of the data at each datum. Based on the assumption that the error and noise in the observed data are uncorrelated, independent, and Gaussian with zero mean, φ d can be characterized as a χ 2 variable with N degrees of freedom. As a χ 2 variable, the expected value of the data misfit function, E( φ d ), is N, the number of data observations. This provides a target misfit for inversion. The optimal regularization parameter is chosen so that the data misfit is equal to N. This approach is used to find the regularization parameters in the synthetic model inversions at the end of this chapter. It is appropriate for these inversions because we know the standard deviation of the synthetically generated data sets and the singular source bodies are not contaminated with geological noise, as would be the case with observed field data. 55

80 Positivity Constraint The global objective function, as defined in eq.(3.16), does not mitigate the existence of negative susceptibility values (Li and Oldenburg, 1996). By formulating my inversion to solve for the magnitude of magnetization, it is required that the values of effective susceptibility in the subsurface model must be positive. This adds a condition on the global objective function. min. subject to φ = r κ r W ( d d obs r d pred ) 2 + β W m r r ( κ κ ) 2 (3.17) The positivity constraint is implemented by using a logarithmic barrier method, which adds a logarithmic barrier term to the global objective function φ = r r 2 obs pred r r 2 (3.18) W ( d d ) + β W ( κ κ ) 2λ lnκ d m j j where λ is the logarithmic parameter that controls the contribution of the logarithmic term. Numerical solution via the logarithmic barrier method is presented later in this chapter. However, it is noted that the logarithmic barrier term introduces additional nonlinearity. Thus, the optimization problem in eq.(3.18) must be solved iteratively starting from an initial model. 56

81 Depth Weighting Depth weighting completes the detailed statement of the inversion problem. This function is incorporated into the model objective function to distribute susceptibility values with depth. Depth weighting is important in inversion. The rapid decay of the data signal with depth causes the magnetic susceptibility distribution to be concentrated along and near the surface, which is an unrealistic solution. To counteract the decay, a depth weighting function is applied to the susceptibility distributions. By including such, the inversion algorithm places magnetically susceptible materials at different depths with equal probability. This eliminates a preference for shallow distributions (Li & Oldenburg, 1996). The depth weighting function is given by 1 w( z) = ξ ( z + z ) / 2 (3.19) where z is the center of the first cell of the model. Depth weighting parameters, z and ξ, are approximated based on a least squares fit between the actual signal decay and the decay of w 2 (z). The result of least squares fitting is displayed in Figure 3.4. The elongated valley of minimal values is the feature of interest when choosing the depth weighting parameters. For the synthetic examples that I present in this chapter, z is 46 and ξ is 4.8. To apply depth weighting to susceptibility distributions, the function is incorporated into the model objective function as 57

82 Z ξ Figure 3.4. Least squares fitting for depth weighting parameters, z and ξ using the actual signal decay and decay of the depth weighing function. The elongated valley of minimal values is the feature of interest when choosing the depth weighting parameters. For the synthetic examples presented in this chapter, z is 46 and ξ is

83 w T T φ ( m) = m~ W W m~ (3.2) m m m where m ~ = w( z ) r κ. The final weighted objective function is given as w r T T T r φ ( m) = m Z W W Zm (3.21) m m m where m ~ = Z r κ and is a M by M sparse matrix of cell weighting values Numerical Solution of the Inverse Problem I now present aspects of the numerical solution. Details of various numeric aspects include discussion of the logarithmic barrier and iterative Gauss Newton methods, model update, sensitivity calculations, conjugate gradient, and termination conditions Bound Constraint Using Logarithmic Barrier Method As introduced in eq.(3.18), a bound constraint term is added to the model objective function. Incorporation of the logarithmic barrier term restricts the magnitude of magnetization in the recovered models from being negative. By confining the feasible class of models nonuniqueness of the solution is reduced. This added constraint also consolidates the susceptibility at a reasonable depth and reduces the artifacts that may be present below the target in the inverted model (Oldenburg et al., 1999). 59

84 Logarithmic barrier methods are a type of interior point method (IPM) used in constrained optimization. Interior point methods are a class of tools for minimizing a function subject to a combination of equality and inequality constraints (Nocedal and Wright, 1999). For this work, a primal logarithmic barrier method is used to constrain the effective susceptibility values between zero and an upper bound. The logarithmic barrier process starts with an initial model and a large λ value, a position known as the analytical center. As iterations proceed and models change, decreases in λ can be tracked along a trajectory path. The inversion terminates when changes in the successive global objective functions are minimal and the logarithmic barrier term is sufficiently small. Analytically, the trajectory path would map smooth changes from the initial model to the solution model (Wright, 1998). Numerically, the path is not smooth because a full minimization at each λ is not completed. Rather, a single Gauss-Newton step is taken, resulting in a series of zigzag of steps that bound either side of the trajectory path. The difference between the analytic and numerical paths is shown in Figure Gauss-Newton Method Since both forward modeling mechanisms and the logarithmic barrier term are nonlinear, the inversion is solved iteratively. At a given iteration, n, the current r n κ. We seek to construct a new model, r n+1 κ, by susceptibility distribution is noted as finding a change to the current model, via a model perturbation r κ, which brings the susceptibility distribution closer to a solution. This process is given by v n+ 1 v n v κ = κ + κ (3.22) 6

85 r κ, final λ r κ, initial λ Figure 3.5. Analytic center and trajectory path taken by the logarithmic barrier method. The process starts with an initial model and a large logarithmic barrier parameter, λ, value at a position known as the analytical center, symbolized by the center dot. As iterations proceed, decreases in λ can be tracked along a trajectory path. The inversion terminates when changes in successive global objective functions are minimal and the log barrier term is sufficiently small. Analytically, the trajectory path would be smooth as indicated by the dashed line. But numerically, the changes occur as a zigzag course (solid line) in an area surrounding the trajectory path. 61

86 To solve this inversion iteratively and find the sought model perturbation, a Gauss-Newton method is used. To minimize the global objective function, defined in eq.(3.16), it is rewritten as φ = r obs r r r r r (3.23) W ( d F lnκ d 2 n n 2 [ κ + κ ]) + β W ( κ + κ κ ) 2λ 2 m 2 j j where predicted data are represented as a function (F) of the new model defined by r n r F κ + κ using a Taylor series expansion gives eq.(3.22). Expanding the function i d r r d n n M i [ κ + κ ] = F[ κ ] + κ H. O. T. pred = F + i i j j= 1 κ j r r. (3.24) Ignoring higher order terms (H.O.T.) yields d pred i d n i + M j=1 di r κ j κ j (3.25) where d pred i are the predicted data at the n th iteration. This can be expressed in a matrix format as r r r pred n d = d + G κ (3.26) where G is the sensitivity matrix of amplitude data. Discussion of the calculation of sensitivity is held until later in this chapter. 62

87 Substituting eq.(3.26) in eq.(3.23) gives a linearized objective function defined as φ = r r obs n r 2 r n r r 2. (3.27) W ( d d + G κ ) + β W ( κ + κ κ ) 2λ lnκ d 2 m 2 j j Finally, setting r κ φ = leads to T T T r ( G W W G + βw w + λχ 2 ) κ d d z z r T T n T r n = G W W δd βw W δκ λχ d d m m 1 v e (3.28) r r r n n obs r where δ d = d d, δκ n r r = κ n κ, and n d v and v n κ are the predicted data and model at the n th iteration. X is defined using the susceptibility values of the current model, X = diag κ, κ,..., κ } and e r is a vector of ones of length M. Both terms stem from the { 1 2 M presence of the logarithmic barrier term. This linearized equation is solved to find the model perturbation, κ r, using conjugate gradient. Once the model perturbation is found, the updated model is given by v n+ 1 v n v κ = κ + γα κ (3.29) where α is a maximum step length that can be taken before reaching the bound. A step reduction, γ, is utilized to scale back the step length, ensuring that the solution does not lie on the bound. Reduction of step length by logarithmic barrier method is illustrated in Figure 3.6. Based on Gauss-Newton methodology, the maximum step length is typically set to 1. The value of γ ranges from.9 to

88 κ v (n) α γα v ( n+1) κ Figure 3.6. Step length reduction in logarithmic barrier method. Starting at iteration n n, the current model, r κ, is perturbed by r κ in the direction of the dashed arrow. α is a maximum step length that can be taken before reaching the bound Further, a step reduction, γ, moves the change from the edge of the bounded region. Typical Gauss- Newton values for α is 1; reduction factor values range from.9 to κ v 64

89 The majority of the computational cost for inversion lies in solving the Gauss- Newton equation, eq.(3.28), and the calculation of the required sensitivity matrix, G. For this reason, sensitivity calculations are discussed next Sensitivity Calculations The sensitivity matrix, which has appeared in discussion of forward modeling and numerical solution to the inverse problem, is a critical concept. As shown in eq.(3.7) and eq.(3.1), the sensitivity matrices, G and G ~, aid in forward modeling of predicted data from susceptibility distributions for both amplitude and total gradient. Here, I will derive the sensitivity matrix for amplitude data. A similar derivation can be completed for the sensitivity matrix for the total gradient. For this work, the sensitivity matrix are implemented to characterize the change of the magnetic data at the i th observation point with respect to a change in the effective susceptibility value in the j th cell of the current model. This derivative is given by G ij di = κ j (3.3) and the spatial relationship between the locations is shown in Figure 3.7. A matrix representation of G is given by 65

90 i j Figure 3.7. The sensitivity matrix characterizes the change of the magnetic data at the i th observation point with respect to a change in the effective susceptibility value in the j th cell of the current model. 66

91 d1 κ1 G = M d N κ1 L O L d 1 κ M g M = M d g N N κ M 11 1 L O L g g 1M M NM. (3.31) where N and M are the number of observations points and cells, respectively. Direct differentiation of data with respect to susceptibility in a cell yields that the sensitivity is given by the inner product of two vectors for amplitude and total gradient data, eq.(3.32) and eq.(3.33), respectively. G ij = r B r B pred i pred i r b ij (3.32) In eq.(3.32) which defines the sensitivity for the amplitude data, B r pred i is the magnetic field produced by the current susceptibility model as observed at i th location and b r is the magnetic field produced by the susceptibility in the j th cell at the i th ij observation location. That is, the sensitivity of the amplitude with respect to a particular cell has a simple form: it is the inner product of the unit vector of the magnetic field at the i th location and the magnetic field produced by the susceptibility in the j th cell at the i th observation location. G ij = B B pred i pred i b ij (3.33) 67

92 In the sensitivity calculation for the total gradient, eq.(3.33), pred B i is the gradient vector of anomalous field predicted by the current magnetization at the i th observation location and bij is the magnetic gradient at the i th location due to unit susceptibility in the j th cell. That is, the sensitivity of the total gradient with respect to a particular cell has a simple form: it is the inner product of the unit directional vector of the gradient produced by the entire model and the gradient produced by the unit susceptibility in that cell. matrices It is noted that the components of G ~ x, G ~ y, G ~ z bij are the elements of sensitivity component in eq.(3.1). These three matrices are required in order to calculate both the forwarded modeled data as well as the sensitivity matrix at each iteration Convergence Criteria In order to define a suitable solution model, convergence criteria is established in terms of two stopping criterion. For my inversion algorithm, convergence is determined by the normalized difference between consecutive values of global objective function and the ratio of current logarithmic barrier term to current global objective function. Both conditions must be satisfied before the inversion is terminated. If the current model does not satisfy the established criteria, model perturbations continue until a suitable solution is found. As the model improves, changes between successive models, as measured by the first stopping criterion, will become smaller. The first termination condition, υ, is defined as the absolute value of the normalized difference between the current and past global objective function values. 68

93 n n 1 φ φ υ = (3.34) n φ This condition requires that there are no large fluctuations between successive global objective functions. Such fluctuations often occur during initial changes in susceptibility models, but the rate decreases as the number of iterations increase. This condition is satisfied when υ is less than one percent. The second termination condition, ψ, is used to measure the contribution of the logarithmic barrier term in the modified global objective function. n φlog ψ = (3.35) n φ M n n where φ = 2λ lnκ. This condition requires that the logarithmic barrier term is not log j = j 1 significantly contributing to the model objective function and ensures that the barrier term λ is sufficiently small. When ψ is less than one percent the termination condition is satisfied Synthetic Examples To test my inversion algorithm and illustrate the utility of the amplitude and total gradient quantities, I conduct a series of tests over two synthetic examples: a cubic model and a dipping slab. 69

94 A cubic model (Figure 3.8a) is used with both data quantities. The model is discretized in a region of 2 by 2 by 1 cells (4, cells). Cell dimensions are 5 m in each direction, making the entire model region 1, m by 1, m with 5 m in depth. Centered in the model region, the cubic model is 2 m by 2 m and located at a depth of 5 to 25 m. Within the model region, cells comprising the synthetic cube have a constant magnetic susceptibility of.5 SI while remaining cells have a background susceptibility of. SI. A second synthetic model, a dipping slab (Figure 3.8b), is only used with amplitude data. The top of the slab is located between 55 to 8 m easting and 35 to 65 m northing at a depth of 5 m. The model has a 45º dip to the west and extends to 4 m. Cell and model region dimensions as well as susceptibility distributions are the same as those for the cubic model. In each of the synthetic inversions, a zero-value reference model and an initial model distribution of.1 are included. No prior information is incorporated Synthetic Cubic Model Results I conducted three tests over the cubic model, Test A with amplitude data and Tests B and C with total gradient data. For each test, the inducing field is I = 9º and D = º, while the magnetization direction varies. For Test A, the magnetization direction is I m = 6º and D m = º. For Test B the magnetization direction is I m = 75º and D m = º. For the final test (C), the magnetization direction is aligned with the inducing field direction. Table 3.1 summarizes the tests. The various synthetic data sets, total field anomaly and calculated quantities, are calculated using the forward modeling equations 7

95 (a).5-5 (m) 1 (b).5-5 (m) 1 Figure 3.8. Synthetic models used to illustrate inversion algorithm. Both models have magnetic susceptibility of.5 SI with backgrounds of. SI (removed for illustration purposes). Cell dimensions are 5 m in each direction, with the entire region measuring 1, m by 1, m by 5 m in depth (2 x 2 x 1 cells). The cubic model (a) is tested with both amplitude and total gradient data. The cube is centered in the region, extends from 5 to 25 m in depth, and is 2 m in each direction. The dipping slab model (b) is only tested with amplitude data. The top of the slab is located between 55 and 8 m easting and 35 and 65 m northing at a depth of 5 m. The slab has a 45º dip to the west and extends to 4 m. 71

96 Table 3.1. Test conditions for synthetic cubic model tests. The anomaly projection direction is the same for each test, (9,). Test A is conducted with amplitude data and a magnetization direction of (6, ). Tests B and C are conducted with total gradient data with magnetization directions of (75,) and (9,), respectively. Anomaly Projection Magnetization Test Data Used Direction Direction (I, D ) (I m, D m ) A Amplitude (9, ) (6, ) B Total Gradient (9, ) (75, ) C Total Gradient (9, ) (9, ) introduced earlier and a 2% Gaussian distribution of noise plus an additional 2nT datum is added. Synthetic observed data for each test are given in Figure 3.9. Convergence curves for the inversion of amplitude data in Test A is shown in Figure 3.1. Both the data misfit and the model norm change rapidly during the first six iterations, then approach an asymptote. The rapid decrease in data misfit indicates that agreement between observed and predicted data improves with each iteration. The change in the model norm indicates that major structure is built within the first six iterations. The decreasing λ value corresponds to decreasing influence of the logarithmic barrier term. According to the discrepancy principle, a regularization parameter, β, of 6,6 is used. 42 iterations are completed to achieve the inversion results presented in Figure

97 (a) 1 5 (b) Northing (m) Easting (m) 1 (c) (d) (e) (f) Figure 3.9. Synthetic total field anomaly and calculated amplitude and total gradient data for tests over the cubic model. Panels a and b correspond to Test A, panels c and d to Test B, and panels e and f to Test C. The calculated quantities (panels b, d, and f) are contaminated with 2% plus 2 nt of Gaussian noise. 73

98 (a) 5 x 1 4 φ d Iteration 45 (b).5 φ m Iteration 45 (c) 1 3 λ 1-6 Iteration 42 Figure 3.1. Curves of (a) data misfit, (b) model norm, and (c) logarithmic barrier parameter values for the inversion of synthetic amplitude data over a cubic model (Test A). Data misfit and model norm fluctuate rapidly during the first six iterations, then become asymptotic. The change in the model norm indicates that major structure is built within the first six iterations. The decreasing lambda value corresponds to decreasing influence of the logarithmic barrier term of the model objective function. 74

99 (a) 1.5 (m) (b) 1 (m) 1.5 (m) (c) (m) (m) (m) 1 Figure Inversion results from Test A with amplitude data over the cubic model at three depths: (a) 5 m, (b) 15 m, and (c) 25 m. The shape, size, location, and susceptibility of the model at the various depths provide a good image of the true susceptibility model. 75

100 The results from the cubic model test using amplitude data (Figure 3.11) are presented in plan view for three depth slices, (a) 5 m, (b) 15 m, and (c) 25 m, to show the shape, size, location, and magnetic susceptibility distribution of the recovered model. The recovered body is compact, located in the correct position, both laterally and vertically, and has good recovered susceptibility at depth. The susceptibility values are approximately equal to that of the original model. Overall, the inversion results provide a good image of the true susceptibility distribution. Convergence curves for the inversion of total gradient data for Test B is shown in Figure Both the data misfit and the model norm change rapidly during the first eight iterations, then become asymptotic. The rapid change in data misfit indicates varying agreement between observed and predicted data initially but improves with an increasing number of iterations. The change in the model norm indicates that major structure is built within the first eight iterations. The decreasing λ value corresponds to decreasing influence of the logarithmic barrier term of the model objective function. In accordance with the discrepancy principle, a regularization parameter, β, of 9,374 is used. 36 iterations are completed to achieve the inversion results presented in Figure The results from the cubic model test using total gradient data with a magnetization direction of I m = 75º and D m = º (Test B, Figure 3.13) are presented in plan view for three depth slices, (a) 5 m, (b) 15 m, and (c) 25 m, to show the shape, size, location, and magnetic susceptibility distribution of the recovered model. At various depths, the recovered distribution of effective magnetic susceptibility has the general shape and location of the true model but lacks adequate depth of investigation. The size of the model at depth is too large. The effective magnetic susceptibility values are lower than the original model, but exhibit a behavior where larger values occur in the center of the model. Overall, the inversion results provide a good image of the true susceptibility distribution. 76

101 (a) 1 x 1 4 φ d Iteration 4 (b) 3.5 φ m Iteration 4 (c) 1 6 λ 1-4 Iteration 36 Figure Curves of (a) data misfit, (b) model norm, and (c) logarithmic barrier parameter values for Test B. Data misfit and model norm fluctuate rapidly during the first eight iterations, then become asymptotic. The change in the model norm indicates that major structure is built within the first eight iterations. The decreasing lambda value corresponds to decreasing influence of the logarithmic barrier term. 77

102 (a) 1.5 (m) (m) 1 (b) 1.5 (m) (m) 1 (c) 1.5 (m) (m) 1 Figure Inversion results from Test B with total gradient data over the cubic model at three depths: (a) 5 m, (b) 15 m, and (c) 25 m. The shape and location of the recovered model is good. But the size at depth is too large and the susceptibility values are high. 78

103 Convergence curves for the inversion of total gradient data for Test C is shown in Figure Both the data misfit and the model norm change rapidly during the first eight iterations, then become asymptotic. The rapid change in data misfit indicates varying agreement between observed and predicted data initially but improves with an increasing number of iterations. The change in the model norm indicates that major structure is built within the first eight iterations. The decreasing λ value corresponds to decreasing influence of the logarithmic barrier term of the model objective function. In accordance with the discrepancy principle, a regularization parameter, β, of 8,995 is used. 36 iterations are completed to achieve the inversion results presented in Figure The results from the cubic model test using total gradient data when magnetization direction is I m = 9º and D m = º (Test C, Figure 3.15) are presented in plan view for three depth slices, (a) 5 m, (b) 15 m, and (c) 25 m, to show the shape, size, location, and magnetic susceptibility distribution of the recovered model. At the various depths, the recovered distribution of effective magnetic susceptibility has the general shape and location of the true model but lacks adequate depth positioning and information. The effective magnetic susceptibility values are much higher than the original model, but exhibit a behavior where larger values occur in the center of the model. I find that the amplitude inversion generally produces better results than total gradient inversions. This difference is partially caused by the loss of low frequency content in total gradient data, which results in a lack of recovered susceptibility at depth. I also observe that total gradient inversion tends to image the edges of the causative body better and is sensitive to lateral changes in the subsurface susceptibility. This is especially true when the causative body has a large width compared to its depth and thickness. 79

104 (a) 1 x 1 4 φ d Iteration 4 (b) 3 φ m Iteration 4 (c) 1 6 λ 1-4 Iteration 36 Figure Curves of (a) data misfit, (b) model norm, and (c) logarithmic barrier parameter values for Test C. Data misfit and model norm fluctuate rapidly during the first eight iterations, then become asymptotic. The rapid change in data misfit indicates varying agreement between observed and predicted data initially but improves with an increasing number of iterations. The change in the model norm indicates that major structure is built within the first eight iterations. The decreasing lambda value corresponds to decreasing influence of the logarithmic barrier term. 8

105 (a) (m) (m) 1 (b) (m) (c) 1 (m) (m) (m) 1 Figure Inversion results from Test C with total gradient data over the cubic model at three depths: (a) 5 m, (b) 15 m, and (c) 25 m. The shape and location of the recovered model is good. But the size at depth is too large and the susceptibility values are extremely high. However, we are able to conclude that the total gradient is useful in detecting the edges of bodies. 81

106 Synthetic Dipping Slab Results For the dipping slab, I conducted two tests, referred to as Tests D and E, using amplitude data. The inducing field and magnetization direction are aligned at I = 65º and D = 25º for Test D. In Test E, the inducing field has an orientation of I = 65º and D = 25º while the magnetization direction is aligned at I m = -25º and D m = 25º. The various synthetic data sets, total field anomaly and amplitude, are calculated using the forward modeling equations introduced earlier and an assumed 2% Gaussian distribution of noise plus an additional 2 nt datum is added. Synthetic observed data for each test is given in Figure Convergence curves for the inversion of amplitude over the dipping slab with aligned inducing and magnetization directions (Test D) are shown in Figure Curves of data misfit, model norm, and logarithmic barrier parameter values are similar to those of previous synthetic tests. Both the data misfit and the model norm change rapidly during the first eight iterations, then become asymptotic. The decreasing λ value corresponds to decreasing influence of the logarithmic barrier term of the model objective function. In accordance with the discrepancy principle, a regularization parameter, β, of 5,68 is used. 39 iterations are completed to achieve the inversion results presented in Figure The recovered model for Test D is presented at three depth slices, (a) 5 m, (b) 25 m, and (c) 4 m (Figure 3.18), to show the shape, size, location, and magnetic susceptibility distribution of the recovered model. The dip of the slab is indicated as the location of the recovered results shifts with depth. There is good depth of investigation of the dipping slab. At 4 m depth, magnetically susceptible material is placed in the corners of the model opposite the dip of the body as a result of edge effects. These 82

107 (a) 1 5 (b) 6 Northing (m) nt nt/m Easting (m) (c) 2 (d) 6 nt nt/m -5 Figure Synthetic total field anomaly and calculated amplitude data for tests over the dipping slab model. Panels a and b correspond to Test D and panels c and d to Test E. The amplitude data (panels b and d) are contaminated with 2% plus 2 nt of Gaussian noise. 83

108 (a) 15 x 1 4 φ d Iteration 4 (b) 3 φ m Iteration 4 (c) 1 5 λ 1-6 Iteration 4 Figure Curves of (a) data misfit, (b) model norm, and (c) logarithmic barrier parameter values for the inversion of synthetic amplitude data over a dipping slab model (Test D). Both the data misfit and the model norm fluctuate rapidly during the first eight iterations, then become asymptotic. The rapid decrease in data misfit indicates that agreement between observed and predicted data improves with each iteration. The change in the model norm indicates that major structure is built within the first eight iterations. The decreasing lambda value corresponds to decreasing influence of the logarithmic barrier term of the model objective function. 84

109 (a) 1.5 (m) (m) 1 (b) 1.5 (m) (m) 1 (c) 1.5 (m) (m) 1 Figure Inversion results from Test D with amplitude over a dipping slab at three depths: (a) 5 m, (b) 25 m, and (c) 4 m. The shape, size, location, and susceptibility distributions are well recovered. The dip of the slab is seen as the location of the recovered results shifts with depth. There is good depth of investigation of the dipping slab. At 4 m depth, magnetically susceptible material is placed in the corners of the model opposite the dip of the body. 85

110 results agree with the shape, size, location, and depth of the true model. A volume rendered comparison of the true and recovered model is given in Figure Convergence curves for the inversion of amplitude over the dipping slab when inducing and magnetization directions differ (Test E) are shown in Figure 3.2. Curves of data misfit, model norm, and logarithmic barrier parameter values are similar to those of previous synthetic tests. Both the data misfit and the model norm change rapidly during the first eight iterations, then become asymptotic. The decreasing λ value corresponds to decreasing influence of the logarithmic barrier term of the model objective function. In accordance with the discrepancy principle, a regularization parameter, β, of 5,675 is used. 43 iterations are completed to achieve the inversion results presented in Figure The recovered model is presented at three depth slices, (a) 5 m, (b) 25 m, and (c) 4 m (Figure 3.21), to show the shape, size, location, and magnetic susceptibility distribution of the recovered model. The dip of the slab is indicated as the location of the recovered results shifts with depth. There is good depth of investigation of the dipping slab. At 4 m depth, magnetically susceptible material is placed in the corners of the model opposite the dip of the body as a result of edge effects. These results agree with shape, size, location, and depth of the true model. A volume rendered comparison of the true and recovered model is given in Figure Summary In this chapter, I have presented the inversion process used in the development of my algorithm for the 3D inversion of magnetic data in the presence of remanent magnetization. Beginning with a general discussion on inversion, details of specific functions and numerical solution techniques are discussed in order to address issues of nonuniqueness, geological feasibility, and nonlinear relationships between the physical 86

111 (a).5 5 (m) 1 (b).5 5 (m) 1 Figure Volume rendered images of the true and recovered model for the inversion of amplitude data over the dipping slab (Test D). The shape, size, location, and susceptibility distribution of the recovered model is in good agreement with the true model. The dip of the slab is also recovered. This illustrates that the inversion algorithm that I have developed is applicable to non-vertical, magnetic structures. 87

112 (a) 15 x 1 4 φ d Iteration 45 (b) 3 φ m Iteration 45 (c) 1 6 λ 1-6 Iteration 45 Figure 3.2. Curves of (a) data misfit, (b) model norm, and (c) logarithmic barrier parameter values for inversion over a dipping slab model (Test E). Both the data misfit and the model norm fluctuate rapidly during the first eight iterations, then become asymptotic. The change in the model norm indicates that major structure is built within the first eight iterations. The decreasing lambda value corresponds to decreasing influence of the logarithmic barrier term of the model objective function. 88

113 (a) 1.5 (m) (m) 1 (b) 1.5 (m) (m) 1 (c) 1.5 (m) (m) 1 Figure Inversion results from Test E with amplitude over a dipping slab at three depths: (a) 5 m, (b) 25 m, and (c) 4 m. The shape, size, location, and susceptibility distributions are well recovered. The dip of the slab is seen as the location of the recovered results shifts with depth. There is good depth of investigation of the dipping slab. At 4 m depth, magnetically susceptible material is placed in the corners of the model opposite the dip of the body. 89

114 (a).5 5 (m) 1 (b).5 5 (m) 1 Figure Volume rendered images of the true and recovered model for the inversion of amplitude data over the dipping slab (Test E). The shape, size, location, and susceptibility distribution of the recovered model is in good agreement with the true model. The dip of the slab is also recovered. This further illustrates that the inversion algorithm that I have developed is applicable to non-vertical, magnetic structures. 9

115 properties of the subsurface, calculated data, and various inversion elements including mechanisms for forward modeling and bound constraints. I test and illustrate my inversion algorithm with a series of inversions over synthetic models. While all of the inversions are reasonably successful in recovering general aspects of the true model, inversions of amplitude tend to produce better results than inversions of total gradient. Total gradient inversions appear to be more sensitive to lateral changes in the subsurface susceptibility and tend to image the edges of the causative body. Recovered susceptibility at depth is poor with total gradient results due to a loss of low frequency content in the data through derivative calculations. Haney et al. (23) discusses the statement by Roest et al. (1992) and demonstrates that for the magnitude of the gradient vector, termed total gradient, to be the envelope, the observed magnetic data has to be reduced to the pole (RTP). In order to complete reduction to the pole, however, the source magnetization parameters must be known. In the presence of remanent magnetization, this is problematic. Aligning the inducing parameters, which are known, to the pole, thus performing a half reduction to pole (HRTP) provides a corresponding total gradient that has an even weaker dependence on magnetization direction than total gradient of the original observed data. Quantities that are HRTP can then be used in inversion in the presence of strong remanent magnetization. This inversion algorithm improves on the current physical property inversion techniques available by working with magnetic data that are collected over areas with remanent magnetization. By utilizing the weak dependence that amplitude and total gradient data exhibit on magnetization direction, I can now invert any magnetic data and perform quantitative interpretation. 91

116

117 CHAPTER 4: PRACTICAL ASPECTS It is pertinent to discuss some of the practical aspects associated with magnetic data and their inversion. This chapter begins by identifying the limitation of current techniques that rely on the assumption of known magnetization direction. I then present a brief discussion on the operational issues of depth weighting parameters. Finally, discussion on means of obtaining amplitude and total gradient data for use with my algorithm conclude the chapter Limitation of Current Algorithms in the Presence of Remanent Magnetization As discussed in Chapter 2, many of the current techniques used in practice are based on the assumption of known magnetization direction. This assumption states that the remanent magnetization is either approximately collinear with the inducing field or has small magnitude. Thus, the total magnetization direction is approximately equivalent to that of the current inducing field. However, a violation of either circumstance will alter the magnetization direction and consequently invalidate this assumption. Although valid in a variety of applications, it is important to understand the conditions that violate this assumption. 92

118 Testing the Assumption The MAG3D program developed at the Geophysical Inversion Facility (GIF) at the University of British Columbia (UBC) is an inversion technique commonly used in mineral exploration and has its theoretic basis in Li and Oldenburg (1996). It utilizes an assumption that the surface magnetic data used in the inversion is produced by induced magnetization only in order to invert magnetic data to recover three-dimensional susceptibility models. It is expected to yield poor results when the assumption is violated. However, it is not clear what conditions cause inversion results to be too poor to be useful. I carry out a sequence of simulations to identify such. I conduct three tests using the cubic model introduced in Chapter 3. Synthetic total field data are generated by varying a magnetization direction parameter over a range of values while maintaining a constant inducing field direction. Data are contaminated with 2% Gaussian noise plus a minimum of 2 nt. To eliminate numerical errors that might be introduced by wavelet compression, I have chosen not to use it. Two of the tests (A and B) vary magnetization inclination while maintaining a constant magnetization declination. The third test (C) varies the magnetization declination in the presence of a constant magnetization inclination. A summary of test conditions is given in Table 4.1. Two sets of inversions are completed: correct inversions and incorrect inversions. Here, correct inversions refer to those conducted with correct inducing field and magnetization directions, as if the direction of total magnetization were known. These tests are conducted to yield the ideal result that can be produced by MAG3D. Incorrect inversions refer to those in which it is assumed that the magnetization direction is the same as that of the inducing field, complying with the current assumption. The greater the deviation between the true and assumed magnetization directions, the worse the 93

119 current assumption is violated. I seek to determine the maximum deviation in magnetization direction before MAG3D, and the associated assumption, breaks down. Table 4.1. Test conditions for identifying the limitations on the current assumption in the presence of remanent magnetization. The first two tests (A and B) vary the magnetization inclination in the presence of a constant inducing field and magnetization declination. Test C varies the magnetization declination in the presence of a constant inducing field and magnetization inclination. Test Inducing Field Orientation (Iº, Dº) Range of Magnetization Direction (Iº, Dº) Increment (º) A (9, ) (6, ) to (12, ) 5 B (45, 45) (15, 45) to (75, 45) 5 C (45, 45) (45, ) to (45, 9) Inversion Results Results of the inversions are presented in two ways. Figures 4.1, 4.2, and 4.3 display model norm as a function of the varying parameter, inclination or declination, for both correct and incorrect inversions for Tests A, B, and C, respectively. For Test C a second graph, displaying effective deviation, is also presented. Since the inducing field and magnetization direction are in two different planes for this test, plotting model norm against declination does not show the true deviation. Therefore, I chose to display the 94

120 1 φ m Magnetization Inclination ( ) Figure 4.1. Model norms for correct (circles) and incorrect (squares) inversions as a function of magnetization inclination (Test A). The correct inversions have a slight decrease in model norm due to a loss of signal at high magnetization deviations from the inducing field (9, ). The incorrect inversions have a symmetric increase in model norm, centered at (9, ). The increase in model norm indicates increasing structural complexity in resulting models that equally satisfy the observed data. 95

121 φ m Magnetization Inclination ( ) Figure 4.2. Model norms for correct (circles) and incorrect (squares) inversions as a function of magnetization inclination (Test B). The correct inversions show a gentle change in model norm. The incorrect inversions exhibit a rapid increase in model norm. The increase in model norm indicates increasing structural complexity in resulting models that equally satisfy the observed data. 96

122 (a) 1 φ m 1-1 (b) Magnetization Declination ( ) 1 φ m Effective Deviation ( ) Figure 4.3. Model norms for correct (circles) and incorrect (squares) inversions as a (a) function of magnetization declination (Test C) and (b) effective deviation. Negative effective deviations indicate a declination of magnetization direction less than 45º. Correct inversions have a slight increase in model norm at high deviations from the inducing field. The incorrect inversions have a symmetric increase in model norm, centered on an effective deviation of. 97

123 model norm as a function of effective deviation. The effective deviation is defined as the angle between the true and assumed magnetization directions in three dimensions. Quantitatively, model norm is an indication of the structural complexity of a model. Subject to all models equally fitting the data, a model with higher model norm is more complex and less desirable. In this case, the extra complexity is a direct result of using the wrong magnetization direction. Thus, the increase in model norm is an indirect indication of the error produced by violating the assumption about magnetization. Cross-sections of models produced by incorrect inversions are given in Figures 4.4, 4.5, and 4.6 for Tests A, B, and C, respectively. Changes in structural complexity are illustrated in the shape, size, location, and magnetic susceptibility values of the recovered models. By comparing the quantitative results of model norm and the visual results of volume renderings, I can comment on the conditions under which the magnetization assumption is violated and MAG3D cannot produce useable results. Given that the correct inversions had complete information about the inducing field and magnetization directions, the results have low model norm values, indicative of simple structure. A slight loss of signal creates limited relief or shape in the resulting model norm curves. Although not shown, the volume rendered results are all similar and represent the true model well. Results for the incorrect inversions exhibit an increase in model norm values as the deviation between the assumed and correct magnetization inclination increases. This increase in model norm indicates that the inverted models become structurally more complex with larger deviation. Beyond the level of 1º to 15º, the quality of the inversion results notably deteriorates. The shape and size of the recovered models begin 98

124 A.11 B I = 9º.5 5 (m) 1 C I = 95º D I = 1º E I = 15º F I = 11º G I = 115º H I = 12º Figure 4.4. Cross-sections from Test A at various magnetization inclinations. Panel A shows the true model. As inclination increases past a deviation threshold of 1º, the ability to recover model shape, size, and susceptibility at depth decreases. 99

125 A.11 B I = 45º.5 5 (m) 1 C I = 35º D I = 4º E I = 5º F I = 55º G I = 6º H I = 65º I I = 7º J I = 75º Figure 4.5. Cross-sections from Test B at various magnetization inclinations. Panel A shows the true model. Panels C and D are orientated opposite the other panels. As inclination increases past a deviation threshold of 1º, the ability to recover model shape, size, and susceptibility at depth decreases while corner susceptibility increases. 1

126 A.1 B D = 45º.5 5 (m) 1 C D = 5º D D = 55º E D = 6º F D = 65º G D = 7º.347 H D = 75º 2.89 Figure 4.6. Cross-sections from Test C at various magnetization declinations. Panel A shows the true model. As declination increases past a deviation threshold of 1º, the ability to recover model shape, size, and susceptibility at depth decreases. Note the corner cells with high susceptibility at depth as declination increases. 11

127 In Chapter 3, I use a depth weighting function that is applied to susceptibility distributions in order to prevent magnetically susceptible materials from concentrating near the surface (Li and Oldenburg, 1996). This function is expressed as 1 w( z) = ξ ( z + z ) / 2 (4.1) where z is the center of the first cell of the model and two adjustable parameters, z and ξ, define the details of the depth weighting. The ideal depth weighting, according to Li and Oldenburg (1998), is given by choices of z and ξ such that the actual kernel decay and the decay of w 2 (z) are matched. It is well known that the magnetic field due to a small, compact source decays as 1/r 3. Theoretically, it seems logical that a quantity comprised of gradients, such as total gradient, would decay as 1/r 4, corresponding to a ξ value equal to 4. Numerically, a dipole is modeled with three different field orientations: (a) (9, ); (b) (, 9); and (c) (45, 45). The decay of total gradient over each dipole is shown in comparison to standard decay rates of 1/r 3, 1/r 4, and 1/r 5 in Figure 4.7. Orientations (a and b) that model dipoles at the magnetic pole and equator, respectively, show decay rates slightly slower than 1/r 4. However, the third orientation (c) is approximately equal to a decay rate of 1/r 4. Thus, I found that a ξ value of four is a good characterization of total gradient decay in a range of cases. Current research on learned regularization (e.g. Haber and Tenorio, 23; Oldenburg et al., 24) may provide an alternative way for choosing depth weighting parameters. Prior information corresponding to desirable geologic characteristics is incorporated into a suite of training models. Parameters, such as those used in depth 13

128 to lose definition, particularly at depth. A deviation of 5º past this threshold results in MAG3D inversion solutions that are no longer interpretable. Inversion results from Tests B and C (Figures 4.5 and 4.6) place magnetically susceptible cells at depth when the anomaly projection is (45, 45). This feature occurs in the corner opposite of the inducing field and is an inversion artifact. Model norm results from Test B (Figure 4.2) have an interesting feature. In addition to an asymmetrically skewed model norm curve for the incorrect inversions, the model norm value for the correct inversion at 4º is greater than its incorrect inversion counterpart. Further, by examining incorrect inversion results from Test B in Figure 4.5, it is intriguing that results from equal deviations from the minima at 4º exhibit different shapes, symmetry, and susceptibility values of recovered models. Based on the results presented here, I conclude that when the deviation between the assumed and true magnetization directions exceeds a threshold of 1º, current techniques such as MAG3D that are based on the assumption of no remanent magnetization can no longer provide valid inversion results. Thus, magnetization direction must be estimated within this tolerance or an alternative algorithm, such as the one I have developed, should be utilized Depth Weighting I now discuss the operational aspects of depth weighting parameters. Choosing appropriate depth weighting parameters is an important aspect in inversion. Incorrect parameters will cause recovered models to be located at incorrect depths. 12

129 (a) (b) Depth Function Value (c) Figure 4.7. Decay rates of total gradient over a dipole with three orientations: (a) (9, ), (b) (, 9), (c) (45, 45) are shown in comparison to standard decays of 1/r 3 (green), 1/r 4 (red), and 1/r 5 (blue). The first two orientations, modeling dipoles at the pole and equator, respectively, show decay rates slower than 1/r 4. However, the third orientation is approximately equal to a decay rate of 1/r 4. It is found that a decay rate of 1/r 4 is a good characterization of total gradient in a range of cases. 14

130 weighting, will be chosen to improve the agreement between the general structure of the recovered model and the training models. The parameters chosen through this process may be better suited for specific applications, thus improving the inversion process and corresponding results. However, difficulties with using learned regularizations exist in the development the suite of training models because the choice of model parameters is dependent on the training models provided. From the numerical investigation I have conducted, it is found that the choice of depth weighting parameters is problem dependent and varies on a number of factors including the orientation of the field, cell dimensions, and interpreter subjectivity. Thus, it is inappropriate to conclude a hard rule about choosing values for depth weighting parameters, z o and ξ. For general application, however, a choice of ξ equal to four and a selection of z o based on least squares fitting are appropriate Obtaining Amplitude and Total Gradient Data One of the primary practical aspects concerning my algorithm is the use of amplitude of anomalous magnetic field and total gradient, which exhibit a weak dependence on magnetization direction. Thus, it is important to outline where such data can be obtained. There are two primary means: numerical calculation and direct measurement. I will outline the procedures for calculating these quantities from commonly available magnetic data in this section. A brief discussion on developing technologies that can yield the required data for these quantities is also given. 15

131 Numerical Calculation Both total field anomaly and vertical anomaly of magnetic data can be used to calculate amplitude and total gradient. The vector magnetic components or gradient vectors are calculated from available data via linear transformations. The modulus is then taken to obtain the amplitude and total gradient, respectively Amplitude Calculations Amplitude calculations using total field or vertical component data are conducted in the frequency domain. Data is converted from the spatial domain into the frequency domain via Fourier transform. Once in the frequency domain, there are two methods of calculation depending on the type of data available. Given total field data, total field anomaly data, T ~, is obtained by ~ r T = ( Bˆ )φ ~ K (4.2) where ˆB is a unit vector of the inducing field, ~ φ is the magnetic potential of a dipole, r and K = iω, iω, ω ). Using total field anomaly data, the three orthogonal components ( x y r in the x-, y- and z- directions are found by 16

132 17 ( ) ( ) ( ) T K B B T K B i B T K B i B r z y y x x ~ ˆ ~ ~ ˆ ~ ~ ˆ ~ = = = r r r ω ω ω (4.3) where ω x and ω y are wavenumbers in the x- and y- directions. The radial wavenumber, ω r, is defined as 2 2 y x r ω ω ω + =. (4.4) Otherwise, given the vertical component of the total field, z B ~, calculations of the horizontal components of the anomalous field are found by Z r y y Z r x x B i B B i B ~ ~ ~ ~ ω ω ω ω = =. (4.5) Having calculated the directional components in the frequency domain from either total field or vertical data, the quantities are converted back to the spatial domain to calculate the amplitude of the anomalous field, B a z y x a B B B B + + =. (4.6)

133 Calculations for amplitude data are stable from mid to high latitudes. Unfortunately, in the frequency domain, instability occurs at low latitudes. To use data from this region with my algorithm, a stable reduction to pole is required Total Gradient Calculations Both total field and vertical component data can also be used in total gradient calculations using a Hilbert transform based approach. It has been established that the vertical and horizontal derivatives of a potential field are Hilbert transforms of each other, first in two dimensions (O Brien, 1971; Nabighian, 1972; Nabighian, 1974) and then in three dimensions (Nabighian, 1984). The use of three-dimensional Hilbert transforms applies to the calculation of the three partial derivatives for total gradient, g. Calculation of the total gradient by Hilbert transformation begins with computing the horizontal derivatives, B/ x and B/ y, in the spatial domain. The derivatives are then converted into the frequency domain using Fourier transform. Two well-behaved operators, H 1 and H 2, are applied to the transformed horizontal derivatives, respectively. H x iω ω x = (4.7) r H y iω y = (4.8) ω r The vertical derivative, B/ z, is the sum of the transformed derivative terms, given by 18

134 B B B = H + H. (4.9) x y z x y The vertical derivative is then converted back to the spatial domain where all three directional gradients are used to calculate the total gradient by ( B / x) 2 + ( B / y) 2 + ( B / z) 2 g =. (4.1) Numerical calculation of these quantities is advantageous and widely applicable due to the volume of and extent of total field data that is readily available. Thus, implementation of my inversion algorithm is not restricted by costly acquisition Direct Field Measurements Quantities with minimal dependence on magnetization direction can also be obtained from data directly measured using developing instrument technology. The development, processing, and application of gradient and tensor equipment are current areas of research with potentially large impacts on the use of magnetic data. Other researchers have also noted that such technologies are particularly beneficial in areas where the magnetic response is affected by the presence of remanent magnetization (Schmidt and Clark, 2). Development of magnetic gradient platforms, such as the 3D-GM (Mushayandebvu et al., 24), TRIAX-4 sensor gradient system, and MIDAS highresolution gradient system (Wooldridge, 24), has led to the introduction of new magnetic field measurements. Gradient measurements are advantageous due to a 19

135 reasonable insensitivity to sensor orientation, suppression of regional, deep sourced anomalies, increased resolution of shallow targets of interest, and reduced need for base stations, diurnal, and regional corrections (Schmidt and Clark, 2; Mushayandebvu et al., 24; Reid, 24). Further, magnetic vector data measures information about directional variations of the total magnetic field. By measuring the three components of the magnetic field, magnetic vector data is beneficial in areas of strong remanent magnetization and in low magnetic latitudes (Christensen and Dransfield, 22). With the developments of both gradient and vector technologies, it will be possible to obtain the amplitude and total gradient quantities used in my inversion algorithm directly, eliminating the need for numerical calculation. Further, developments associated with these technologies will improve our knowledge of the subsurface and provide information that will improve and enhance inversion results and interpretation of magnetic data Summary Understanding the practical aspects of inversion is important in determining the conditions and parameters under which certain methods are applicable. An investigation of the current assumption has led to the recommendation that if the assumed and true magnetization directions differ by more than 1 to 15 the recovered magnetic susceptibility distributions found using the common technique MAG3D may not accurately reflect the true structure of the subsurface. The choice of depth weighting parameters is examined due to dependence on field orientation. Lastly, the quality of inversion results can only be as good as the quality of the input data. Numerical calculation of the components of the amplitude and total gradient involves linear transformation and can be applied widely due to the current availability of total field 11

136 anomaly data. With developing gradient and tensor technologies, the components of the quantities will be measured directly with improved quality. Thus, it is not only important to understand how to obtain the quantities used in this algorithm, but also the advantages and disadvantages of each means. 111

137 CHAPTER 5: FIELD EXAMPLE In Chapter 3, I develop an algorithm for the three-dimensional inversion of magnetic data in the presence of remanent magnetization. Tests with synthetic models illustrate the algorithm as well as the utility of the amplitude of the anomalous magnetic field and total gradient data, which are weakly dependent on magnetization direction. To complete testing, I apply the algorithm to a field example in this chapter. The field example illustrates the utility of inversion of magnetic data over kimberlitic areas in diamond exploration. A general geologic and geophysical background of kimberlites is first introduced. Then specific details relating to the field area and inversion will be discussed. Inversion results, with confirmation, are presented at the end of the chapter Areas of Application Previous inversion methods have been restricted from being applied to certain applications due to the lack of knowledge of remanent magnetization. With the ability to invert magnetic data with little information about the magnetization direction it is now possible to take any magnetic data and carry out a three-dimensional inversion to some extent. This ability extends magnetic inversion to a wide range of problems on a variety of scales including archaeological investigations, mineral and resource exploration, and crustal and planetary studies, cases of which were previously inversion failures. 112

138 Magnetics is frequently used for intrasite archeological mapping to locate buried stone foundations, fired rocks, pottery, forges, kilns or hearths, and soils magnetically altered by campfires (Gibson, 1986; Wynn, 1986). These targets often acquire strong, stable remanent magnetization at the time of building, formation, or last firing (Gibson, 1986; Tarling et al., 1986; Wynn, 1986). Due to the large time span between acquisition and investigation, the remanent magnetization directions of targets and the current inducing field direction can deviate significantly, making archaeological investigations a potential application for my inversion algorithm. Within mineral exploration, magnetics is one of the primary geophysical techniques used. However, the interpretation of magnetic data is difficult due, in part, to the presence of remanent magnetization in subsurface targets. Full three-dimensional inversion is of value in steep terrain or at low magnetic latitudes, where traditional interpretation methods are difficult to use (Nabighian and Asten, 22). Mineral and ore bodies often have a remanent magnetization direction that is different than that of the current inducing field direction. Remanent magnetization arises at the time of formation due to mineralization, serpentinization, or oxidation and reduction of magnetic minerals. Magnetics can also be used to search for non-magnetic minerals, such as diamonds, in a magnetic host rock, such as kimberlites. Due to the magnetic susceptibility contrast between kimberlites and surrounding host rock, magnetic surveys are a major component of geophysical investigation in diamond exploration areas (Hitzman, 24; Power et al., 24). Strong remanent magnetization in kimberlites promotes a magnetic response, thus making this application ideal for testing my inversion algorithm, as discussed later in this chapter. Current use of magnetic data in resource exploration is primarily directed towards identifying lateral structure in map view. From this, information about lateral separation and the relative shape of bodies can be interpreted. But little information about depth, 113

139 size, or physical properties of subsurface bodies can be determined. To extract threedimensional structure, inversion of these data are necessary. Three-dimensional inversion of magnetic data has seen increased use over the last decade, especially at the deposit or district scale. The presence of strong and variable remanent magnetization requires inversion algorithms, such as mine, that are insensitive to magnetization direction. Magnetics is a useful tool for resource exploration within the petroleum industry and in crustal studies based on differences in magnetization, particularly remanent magnetization. Imaging of igneous basement rock beneath sedimentary basins is important in structural mapping, particularly in areas of limited outcrop and seismic information. Variations in magnetic field observations are caused by variations in magnetization of basement rock, overlain by weakly magnetic sediments (Pilkington, 22). In crustal studies, the diverse origins of continental and oceanic crusts hold clues, based on physical properties, about the tectonic evolution of the Earth. Differences in remanent magnetization direction expressed by magnetic anomalies help establish paleolocations of continents, rifts, the assembly of accreted terrain, and timelines for various tectonic events. Magnetics can also delineate intrusives and shallow volcanics, which possess contrasting magnetic properties, including various orientations of magnetization direction (Cannon, 22; Foss, 24). Lastly, magnetics is an important aspect in planetary geophysics. By examining crustal magnetization, scientists can develop models and theories about the thermal history, evolution, and composition of a planet (Acuña et al., 1998; Acuña, 23). Data collected by the Mars Global Surveyor (MGS) has allowed scientists to establish that Mars does not possess a significant global magnetic field due to the lack of an internal dynamo (Acuña et al., 1998; Acuña et al., 1999). However, the presence of strong crustal magnetic sources in ancient, heavily cratered terrain south of the dichotomy boundary 114

140 indicates that an active dynamo has existed in the past (Acuña et al., 1999). Thus, magnetic measurements arise from remanent magnetization in iron-rich concentrations in the Martian crust, which acquired orientation before the cessation of the dynamo approximately four billion years ago (Acuña et al., 1998; Acuña et al., 1999). Inversion to recover the distribution of magnetic materials can offer complementary information in understanding the crustal structure of Mars. With only the presence of remanent magnetization, my new inversion method seems to be ideal Kimberlites and Diamond Exploration As mentioned previously, kimberlites are important in diamond exploration. When diamonds, which cannot be directly detected using geophysics, originate in the region or along the path of kimberlites, they are transported to the surface as inclusions in kimberlitic intrusions. Although the occurrence of kimberlites does not guarantee the presence of diamonds, kimberlites are the most common host rock in which diamonds occur (Hitzman, 24). Thus, the primary approach to diamond exploration is to search for kimberlites. A general background on the formation, composition, and occurrence of kimberlites is given here to help characterize these exploration targets Formation of Kimberlites The process in which kimberlites rise to the surface is complex and not completely understood. Generally, kimberlites erupt quickly at rates between 1 to 3 kilometers per hour (km/hr), with eruptions lasting between five and 15 hours. These rapid velocities allow diamonds to be carried to the surface without being converted to graphite (Hitzman, 24). 115

141 Kimberlite emplacement has several stages and is associated with three texturally distinct facies: hypabyssal, diatreme, and crater. Figure 5.1 displays the different zones, depths, and basic geologic characteristics of each facies. Not all three facies are found within each kimberlite. Thus, exploration is dependent on the presence, location, and properties of the facies present. Hypabyssal facies occur from rapid, intrusive advancement of upper mantle material from the asthenosphere that pushes kimberlitic magmas through numerous feeder dikes (Guilbert and Park, 1986; CGS, 1999). Structurally, kimberlites have been associated with diabase dikes, regional faults, geologic contacts, and areas of local weakness in a stable craton crust. These conduits offer paths of lower resistance for kimberlitic intrusion (Hitzman, 24; Power et al., 24). A root zone lies in the transitional area between the feeder dikes of the hypabyssal facies and the overlying diatreme facies. At a critical depth of two to three kilometers, an explosion of degassed material forms a carrot-shaped, diatreme facies (Guilbert and Park, 1986) with a marginal dip of 75º to 85º. Significant cooling during gas expansion lowers the temperature of kimberlitic magma, confirmed by a lack of evidence of substantial thermal reaction with country rock (CGS, 1999; Hitzman, 24). This cooling prevents the acquisition of secondary remanent magnetization in the country rock. Diatremes facies contain the largest volume of kimberlites and consequently are exploration targets (Hitzman, 24). Lastly, within the upper 3 m of the subsurface, crater facies erupt as a mixture of solid rock fragments and gases (Hitzman, 24) and can be classified as pyroclastic fallback breccias or epiclastic water-lain material (Guilbert & Park, 1986). Although all 116

142 Zone Depth (km) Crater.5 1. Diatreme Hypabyssal m Figure 5.1. Kimberlite facies, zones, and depths. The left figure shows the three zones (hypabyssal, diatreme and crater) and their corresponding depths. The right figure, not scaled, provides three-dimensional detail for each facies and geological details of the crater facies. (modified from Hitzman, 24). 117

143 kimberlite facies are susceptible to weathering, alteration, and serpentinization, the crater facies is typically the first portion to be eroded (Hitzman, 24) Composition of Kimberlites In general, kimberlites have an unusual chemistry and petrology. They are described as volatile-rich, potassic, ultramafic igneous rocks, dominated by olivine and containing subordinate minerals of mantle derivation. Since all kimberlites contain olivine, mineralogical classification is based on groundmass, products in kimberlitic magma, and incorporates less universal minerals such as monticellite (calcium-rich olivine), phlogopite (magnesium-rich biotite), diopside (Ca-Mg clinopyroxene), serpentine, calcite, apatite, magnetite, chromite, garnet, and other upper mantle minerals (Guilbert and Park, 1986; Hitzman, 24). Compositionally, kimberlites tend to have a higher proportion of magnetite (Fe 3 O 4 ) than most rocks into which it intrudes. Magnetite is a ferrimagnetic mineral with magnetic susceptibility of 1.5 x 1 3 SI units (Sharma, 1997; Moskowitz, 24). Rocks, such as kimberlites, which contain a significant quantity of magnetite tend to have a strong remanent magnetization and often show an enhanced magnetic signature. Magnetite in kimberlites is not distributed evenly between or within various facies (J. Lajoie, personal communication, 25). Figure 2.4 shows the magnetic susceptibilities of magnetite and several types of common host rocks. 118

144 Occurrence of Kimberlites Kimberlites are derived from various sources and depths in the mantle, as established by the inclusion of diamonds of differing ages (Hitzman, 24). Although intrusive kimberlitic events range from middle Precambrian through Mesozoic and Eocene (CGS, 1999; Power et al., 24), it is common for kimberlites to contain diamonds that formed from 3,3 million years ago (Ma) to 99 Ma during the Precambrian (CGS, 1999). Dikes and pipes are common forms of kimberlite intrusions. Most kimberlites occur as multiple intrusive events and form a cluster of discrete intrusions; a typical cluster contains six to 4 kimberlites pipes (CGS, 1999; Hitzman, 24). Diamonds and kimberlites have been discovered on every continent except Antarctica (Guilbert and Park, 1986) and discoveries tend to coincide with Precambrian shield portions of continents (Hitzman, 24). However, diamond exploration in Canada is also influenced by regional advance and retreat patterns of successive ice sheets (The Atlas of Canada, 24) Geophysics of Kimberlites Potential diamondiferous kimberlitic targets are identified using various techniques from different geoscience disciplines. Geology, geophysics, geochemistry, and sample collection through drilling (Guilbert and Park, 1986) provide information about the structure and the physical and chemical properties of the subsurface, offering a broad, integrated understanding of the area of investigation. A variety of geophysical methods are applicable to diamond exploration due to the contrast of physical properties between kimberlites and host rocks. The shape and size of anomalies provide additional 119

145 information (Power et al., 24). The shallowest, crater facies tends to exhibit the highest contrast to the surrounding rock and is most easily detected by geophysical techniques. However, many rock types and features have broad, overlapping ranges of physical properties making correlations between geophysical responses and lithologies difficult. Due to their ultramafic nature and high magnetic susceptibility (1 to 8 x 1-3 SI units), magnetic responses of most kimberlites tend to be higher relative to surrounding country rock. The presence of strong remanent magnetization enhances this contrast (Power et al., 24). Magnetic surveys can be used to outline kimberlite deposits and indicate structural features (Hitzman, 24; Power et al., 24). Clusters of kimberlite pipes tend to have similar remanent magnetization orientation indicative of a brief duration for cluster emplacement; but different distributions of magnetite and different orientations within close proximity have also been found (Power et al., 24; J. Lajoie, personal communication, 25). The range of possible magnetic responses varies from positive to negative to no magnetic signature based on the size, shape, and orientation of the kimberlite (Power et al., 24). Additional geophysical data can provide further information about the subsurface. An electrical resistivity contrast is generally observed between kimberlites and host rocks (Hitzman, 24) due to serpentization and alteration of the crater facies (Power et al., 24). Gravity response of kimberlites is dependent on the density contrast between different facies and surrounding host rock, making this a secondary geophysical technique to magnetics and electromagnetics (Power et al., 24). Contrasts in dielectric permittivity and electrical resistivity between kimberlites and surrounding rocks can be detected with the use of ground penetrating radar (GPR) due to alteration processes. But, limited depth penetration forces GPR to be used following magnetic, gravity, electrical, and drilling results (Power et al., 24). Seismic refraction surveys can screen for targets 12

146 in granitic rock, but are complicated by overlapping and anisotropic velocity (Power et al., 24). Consequently, although a suite of geophysical techniques is available for imaging kimberlites in diamond exploration, magnetics is the dominant method based on physical properties, effectiveness, and efficiency Field Example from Victoria Island, Northwest and Nunavut Territories, Canada Teck Cominco and Diamonds North Resources provided the magnetic data used in testing my inversion algorithm. The data are located in an active kimberlite exploration area named the Blue Ice Property, on Victoria Island, Northwest and Nunavut Territories, Canada (Diamond News, 23; Diamonds North Resources, 24). A brief description of the location geology and magnetic survey details are provided. The inversion algorithm is applied and the results are presented Background The Blue Ice Property is located in the central portion of Victoria Island (Figure 5.2) and includes 45, acres of staked claims (Diamond News, 23; Diamonds North Resources, 24). This project area is underlain by an Archean aged Slave craton (Diamonds North Resources, 24), which extends under the eastern half of Victoria Island and has been host to a number of diamondiferous kimberlite discoveries since 1991 (Kolebaba et al., 23; Power et al., 24). The granitic basement is overlain by a Proterozoic sedimentary sequence with minor volcanics (J. Lajoie, personal communication, 24). Buff-colored Cambrian to Devonian flat-lying carbonate rocks host several hypabyssal kimberlite intrusions (InfoMine, 24; J. Lajoie, personal communication, 24). Due to regional advance and retreat glacial activity in the 121

147 Figure 5.2. Location map and geology of Victoria Island, Canada. The red star indicates the approximate location of the Blue Ice kimberlite exploration project. (Kolebaba et al., 23, Northwest Territories in Wikipedia, 25). Canadian arctic, this exploration area is mired with down ice depositional features, which makes exploration and interpretation of the area difficult (The Atlas of Canada, 24). In 22, Diamonds North Resources discovered a 2 kilometer trend of semicontinuous kimberlites (Figure 5.3; Diamonds North Resources, 25). This discovery, known as the Galaxy structure, has a northwest trend. The southeast extent of the feature contains kimberlite dikes and blows (root zones). Several large kimberlitic structures, 122

148 identified on the northwest part of the Galaxy trend, were drilled in 23. At least eight diamondiferous kimberlites have been identified in the area (Diamonds North Resources, 24). Figure 5.3. Magnetic map of Galaxy kimberlite trend in Blue Ice project, Victoria Island, Canada. Individual kimberlites are identified. The area of investigation is located on the eastern section of the trend. (Diamonds North Resources, 25) Geophysics At this stage in exploration of the Galaxy trend, geoscience applications have been restricted to structural interpretation, magnetic surveys, geochemical analysis, and drilling. Several magnetic surveys have been flown over the Blue Ice Property (Diamonds North Resources, 24). This field example will focus on one set of aeromagnetic data over a single kimberlite area. Kimberlites along the Galaxy trend are generally characterized by nearly symmetric, positive or negative magnetic anomalies with varying intensity (Kolebaba, et al., 23; InfoMine, 24). The discrete, higher frequency signatures of Galaxy kimberlites are prominent against the magnetic signature of the Archean basement due to noise reduction by the overlying, non-magnetic carbonate cover (Kolebaba et al., 23). 123

149 Various processing techniques indicate that the remanent magnetization component of various kimberlites is almost exactly opposite that of the inducing field (J. Lajoie, personal communication, 24). Due to large deviation between the inducing and remanent magnetizations, the alignment assumption used in the current inversion technique (e.g. Li and Oldenburg, 1996) is clearly violated, making it difficult to use existing methods. Further, there are areas of positive anomalies within the general trend. It is thought that these areas coincide with areas where magnetic susceptibility overtakes remanent magnetization (J. Lajoie, personal communication, 24). Therefore, magnetic data from this area are ideal for testing the algorithm I have developed Inversion of Magnetic Data This investigation focuses on a survey that is approximately 6 m by 1, m. The survey was flown using a high-resolution stinger mounted helicopter system at a nominal height of 2 m above ground with 25 m line spacing. The data grid has been rotated 34º counter clockwise so the grid is parallel to the dominant trend of the anomalies (Figure 5.4). A linear regional trend has been removed from the data. Measured data from 6,161 observation points were first gridded at 5 m intervals, but downsampled to 1 m for inversion. In the rotated data, the inducing field has an inclination of 86.7º and a nominal declination of -7.7º. A trend of strong negative magnetic anomalies aligns along the main Galaxy trend with a compact response off trend to the southwest. Within the data, several smaller, more compact anomalies surround areas of broader features. There are a number of dipolar anomalies with different orientations. 124

150 nt Figure 5.4. Rotated total field anomaly data over the target area, Blue Ice Project, Victoria Island, Canada. The 6 m by 1, m data have been rotated 34º in the counter clockwise direction and regridded at a 1 m spacing. The rotated data has an inducing field of 86.7º and a nominal declination of -7.7º. There are a number of dipolar anomalies with differing orientations throughout the survey area. Two types of anomalies dominate the data: several smaller, more compact, high frequency anomalies surround areas of broad, lower frequency. 125

151 According to Teck Cominco, negative magnetic anomalies in Figure 5.4 are caused by hypabyssal kimberlite dykes located on Victoria Island in Canada s north. Two drill holes intersected three kimberlite dikes with the main body measuring 1.35 meters true width. On Victoria Island, the Archean granitic basement is overlain by a Proterozoic sedimentary sequence with minor volcanics, capped by a flat-lying Cambrian-to-Devonian carbonate rocks. The host cover rocks are largely non-magnetic, and the kimberlite bodies stand out in the magnetic surveys as distinct sharp anomalies indicative of a shallow body, compared with the more rounded anomalies caused by deep features in the basement or within the sedimentary rocks. Kimberlite intrusions are associated with both positive and negative magnetic signatures. The ages of the kimberlite intrusions are in the 25 to 3 million year range (J. Lajoie, personal communication, 24). Based on the results from the synthetic inversions, I have chosen to use amplitude data, calculated from total field anomaly data (Gunn, 1975) over the target area (Figure 5.5). The negative anomalies in the total field data now correspond to positive anomalies in the amplitude data. The anomalies also exhibit a more rounded shape. To set up the inversion, a mesh of 168,858 cells (16 cells east-west x 59 cells north-south x 27 cells top-bottom) is established for the model region. The smallest cell is 1 x 1 x 1 in the central portion of the model. A zero reference model and a constant initial model of.1 are used. Using a regularization parameter of.1 x 1 7, an inversion with 29 iterations took 1:42:43 to complete on a 1.6 GHz CPU. Figures 5.6 and 5.7 display the behaviors of the data misfit, φ d, model norm, φ m, and the logarithmic parameter, λ, respectively, for the inversion of magnetic data over the target area. As the inversion progresses, the data misfit decreases, indicating improved fit between observed and predicted data. The model norm increases rapidly during the first 1 iterations but asymptotes to a plateau, indicating that major structure is built up within the first few 126

152 nt Figure 5.5. Amplitude of anomalous magnetic field, calculated via linear transform from total field anomaly data over the target area, Blue Ice Project, Victoria Island, Canada. The 6 m by 1, m displays a number of more rounded anomalies, with positive anomalies corresponding to negative anomalies from Figure

153 (a) 2x1 7 φ d Iteration 3 (b).35 φ m Iteration 3 Figure 5.6. Curves of (a) data misfit ( φ d ) and (b) model norm ( φ m ) values for inversion of magnetic data. The data misfit decreases as the number of iterations progresses while the model norm increases slightly over during the inversion process. A decreasing data misfit is indicative of better agreement between observed and predicted data. The increase in model norm signifies the need for some complexity in the model. The balance between the two functions leads to convergence of the inversion. 128

154 14 λ Iteration 3 Figure 5.7. Plot of logarithmic barrier parameter (λ) as a function of iteration. A large value is initially given and successive values are reduced as the inversion progresses. 129

155 iterations. A solution is converged upon within 29 iterations. A large initial value of λ decreases as iterations progress and the influence of the logarithmic barrier term is reduced Inversion Results The effective susceptibility recovered from the inversion is shown in Figures 5.8, 5.9, and 5.1. The model is displayed as a volume rendered image with an overlain translucent color display of the amplitude data (Figure 5.8). There are five main anomalous bodies of high susceptibility (A, B, C, D, and E). A sixth body (F) is due to edge effects. There is good depth of investigation for each of the bodies. It should be noted that the bottom of the bodies that appear in the inversion results are not necessarily the true geologic termination but rather the depth of investigation below which the data are not longer sensitive to magnetic material. Bodies B, C, and E are compact and vertically orientated whereas body D is a longer, more linear concentration of magnetic material. Body A seems to be a combination of the two types of bodies. The southern end of body A is compact and vertically orientated, similar to bodies B, C, and E. However, the north-south trending linear extension of this anomalous body is more similar to body D. It is interesting to note that body D and the north-south portion of body A extend deeper in depth than the compact bodies B, C, and E. Generally, concentrations of magnetic material coincide laterally with the magnetic anomalies in Figure 5.8. From cross-sections through the bodies (Figure 5.1), it is found that the magnetic susceptibility values of bodies C and E are higher than those of body B. These bodies coincide with more compact, higher magnitude anomalies in the 13

156 B C D E A F South 25 5 Figure 5.8. Volume rendered inversion results of the recovered distribution of the magnitude of magnetization with a translucent color display of the amplitude data (Figure 5.5). There are five main anomalies (A, B, C, D and E) and one exterior anomaly (F). Assemblies of magnetic material generally coincide with magnetic anomalies, even along the edges of the model region (F). Further, there are two types of general structures in the inversion results. The broad, lower frequency anomaly (D) coincides with a long, linear structure that resembles a kimberlite dike. The short, compact anomalies (B, C and E) correspond to more vertically oriented kimberlite pipes. Anomaly A appears to be a combination of a kimberlite pipe at the southern end attached to N-S oriented kimberlite dike. 131

157 (a) West (m) 14 (b) -5 South (m) 5 (c) Figure 5.9. Plan sections of inversion results at depths of (a) m, (b) 4 m and (c) 8 m. As depth increases, the magnetic susceptibility values of the anomalies A, B, C, and E decrease. The values at anomaly D increase with depth until 8 m and then decrease until the body terminates at 14 m. 132

158 (a) (b) -2-5 South (m) 5 (c) (d) Figure 5.1. Cross-sectional views of inversion results. (a) All anomalies, view from rotated S to N; (b) cross section through anomalies C, D, E, and F at 137 m North; (c) cross section through anomaly C at 2 m East; (d) cross section through anomaly A at 32 m East. Note the different distributions of susceptibility between kimberlite dikes and pipes. Same scale as Figure