October 4, 2018

October 4, 2018

The material on this website is freely available for educational purposes.

Requests for re-use of digital images: contact the UC Press.

Proper citation of the material found in these pages is:

Tauxe, L, Banerjee, S.K., Butler, R.F. and van der Voo R, Essentials of Paleomagnetism, 5th Web Edition, 2018.

The printed version of this book appeared January, 2010. Order a printed version. (cheaper than printing it yourself!)

This book is intended to work with the companion software package described in PmagPy Cookbook.

This material is based upon work supported by the National Science Foundation.

Contents

1 The physics of magnetism

1.1 What is a magnetic field?

1.2 Magnetic moment

1.3 Magnetic flux

1.4 Magnetic energy

1.5 Magnetization and magnetic susceptibility

1.6 Relationship of B and H

1.7 A brief tour of magnetic units in the cgs system

1.8 The magnetic potential

1.9 Origin of the geomagnetic field

1.10 Problems

2 The geomagnetic field

2.1 Components of magnetic vectors

2.2 Reference magnetic field

2.3 Geocentric axial dipole (GAD) and other poles

2.4 Plotting magnetic directional data

2.5 Problems

3 Induced and remanent magnetism

3.1 Magnetism at the atomic level

3.2 Induced magnetization

3.3 Ferromagnetism

3.4 Problems

4 Magnetic anisotropy and domains

4.1 The magnetic energy of particles

4.2 Magnetic domains

4.3 Thermal energy

4.4 Putting it all together

4.5 Problems

5 Magnetic hysteresis

5.1 The “flipping” field

5.2 Hysteresis loops

5.3 Hysteresis of mixtures of SP, SD and MD grains

5.4 First order reversal curves

5.5 Problems

6 Magnetic mineralogy

6.1 Iron-oxides

6.2 Iron-oxyhydroxides and iron-sulfides

6.3 FeTi oxides in igneous rocks

6.4 Magnetic mineralogy of soils and sediments

6.5 Problems

7 How rocks get and stay magnetized

7.1 The concept of dynamic equilibrium

7.2 Essential N�el theory

7.3 Viscous remanent magnetization

7.4 Thermal remanent magnetization

7.5 Chemical remanent magnetization

7.6 Detrital remanent magnetization

7.7 Isothermal remanent magnetization

7.8 Thermo-viscous remanent magnetization

7.9 Natural remanent magnetization

7.10 Artificial remanences

7.11 Problems

8 Applied rock (environmental) magnetism

8.1 Images

8.2 Critical temperatures

8.3 Magnetic susceptibility

8.4 Magnetization

8.5 Hysteresis parameters

8.6 Trends in parameters with grain size

8.7 Ratios

8.8 Applications of rock magnetism

8.9 Concluding remarks

8.10 Problems

9 Getting a paleomagnetic direction

9.1 Paleomagnetic sampling

9.2 Measurement of magnetic remanence

9.3 Changing coordinate systems

9.4 Demagnetization techniques

9.5 Estimating directions from demagnetization data

9.6 Vector difference sum

9.7 Best-fit lines and planes

9.8 Field strategies

9.9 Problems

10 Paleointensity

10.1 Paleointensity with TRMs

10.2 Paleointensity with DRMs

10.3 Problems

11 Fisher statistics

11.1 The normal distribution

11.2 Statistics of vectors

11.3 Significance Tests

11.4 Inclination only data

11.5 Is a given data set Fisher distributed?

11.6 Problems

12 Beyond Fisher statistics

12.1 Non-Fisherian parametric approaches

12.2 The simple (na�ve) bootstrap

12.3 The parametric bootstrap

12.4 When are two data sets distinct?

12.5 Application to the “reversals test”

12.6 Application to the “fold test”

12.7 Problems

13 Paleomagnetic tensors

13.1 Anisotropy of magnetic susceptibility

13.2 Hext Statistics

13.3 Limitations of Hext statistics

13.4 Bootstrap confidence ellipses

13.5 Comparing mean eigenvectors with other axes

13.6 Shape

13.7 Anisotropy of magnetic remanence

13.8 Problems

14 The ancient geomagnetic field

14.1 Historical measurements

14.2 Archaeo- and paleomagnetic records

14.3 Time series of paleomagnetic data

14.4 Geomagnetic polarity time scale – a first look

14.5 The time averaged field

14.6 Long term changes in paleointensity

14.7 Statistical models of paleosecular variation

14.8 Problems

15 The GPTS and magnetostratigraphy

15.1 Early efforts in defining the GPTS

15.2 Current status of the geological time scale

15.3 Applications

15.4 Problems

16 Tectonic applications of paleomagnetism

16.1 Essentials of plate tectonic theory

16.2 Poles and apparent polar wander

16.3 The Gondwana APWP

16.4 Inclination shallowing and GAD

16.5 Paleomagnetism and plate reconstructions

16.6 Discordant poles and displaced terranes

16.7 Inclination only data and APWPs

16.8 Concluding remarks

16.9 Problems

17 References

Appendices

A Definitions, derivations and tricks

A.1 Definitions

A.2 Derivations

A.3 Useful tricks

B Plots useful in paleomagnetism

B.1 Equal area projections

C Paleomagnetic statistics and parameter estimation

C.1 Hysteresis Parameters

C.2 Directional statistics

C.3 Paleointensity statistics

D Anisotropy in paleomagnetism

D.1 The 15 measurement protocol

1 The physics of magnetism

1.1 What is a magnetic field?

1.2 Magnetic moment

1.3 Magnetic flux

1.4 Magnetic energy

1.5 Magnetization and magnetic susceptibility

1.6 Relationship of B and H

1.7 A brief tour of magnetic units in the cgs system

1.8 The magnetic potential

1.9 Origin of the geomagnetic field

1.10 Problems

2 The geomagnetic field

2.1 Components of magnetic vectors

2.2 Reference magnetic field

2.3 Geocentric axial dipole (GAD) and other poles

2.4 Plotting magnetic directional data

2.5 Problems

3 Induced and remanent magnetism

3.1 Magnetism at the atomic level

3.2 Induced magnetization

3.3 Ferromagnetism

3.4 Problems

4 Magnetic anisotropy and domains

4.1 The magnetic energy of particles

4.2 Magnetic domains

4.3 Thermal energy

4.4 Putting it all together

4.5 Problems

5 Magnetic hysteresis

5.1 The “flipping” field

5.2 Hysteresis loops

5.3 Hysteresis of mixtures of SP, SD and MD grains

5.4 First order reversal curves

5.5 Problems

6 Magnetic mineralogy

6.1 Iron-oxides

6.2 Iron-oxyhydroxides and iron-sulfides

6.3 FeTi oxides in igneous rocks

6.4 Magnetic mineralogy of soils and sediments

6.5 Problems

7 How rocks get and stay magnetized

7.1 The concept of dynamic equilibrium

7.2 Essential N�el theory

7.3 Viscous remanent magnetization

7.4 Thermal remanent magnetization

7.5 Chemical remanent magnetization

7.6 Detrital remanent magnetization

7.7 Isothermal remanent magnetization

7.8 Thermo-viscous remanent magnetization

7.9 Natural remanent magnetization

7.10 Artificial remanences

7.11 Problems

8 Applied rock (environmental) magnetism

8.1 Images

8.2 Critical temperatures

8.3 Magnetic susceptibility

8.4 Magnetization

8.5 Hysteresis parameters

8.6 Trends in parameters with grain size

8.7 Ratios

8.8 Applications of rock magnetism

8.9 Concluding remarks

8.10 Problems

9 Getting a paleomagnetic direction

9.1 Paleomagnetic sampling

9.2 Measurement of magnetic remanence

9.3 Changing coordinate systems

9.4 Demagnetization techniques

9.5 Estimating directions from demagnetization data

9.6 Vector difference sum

9.7 Best-fit lines and planes

9.8 Field strategies

9.9 Problems

10 Paleointensity

10.1 Paleointensity with TRMs

10.2 Paleointensity with DRMs

10.3 Problems

11 Fisher statistics

11.1 The normal distribution

11.2 Statistics of vectors

11.3 Significance Tests

11.4 Inclination only data

11.5 Is a given data set Fisher distributed?

11.6 Problems

12 Beyond Fisher statistics

12.1 Non-Fisherian parametric approaches

12.2 The simple (na�ve) bootstrap

12.3 The parametric bootstrap

12.4 When are two data sets distinct?

12.5 Application to the “reversals test”

12.6 Application to the “fold test”

12.7 Problems

13 Paleomagnetic tensors

13.1 Anisotropy of magnetic susceptibility

13.2 Hext Statistics

13.3 Limitations of Hext statistics

13.4 Bootstrap confidence ellipses

13.5 Comparing mean eigenvectors with other axes

13.6 Shape

13.7 Anisotropy of magnetic remanence

13.8 Problems

14 The ancient geomagnetic field

14.1 Historical measurements

14.2 Archaeo- and paleomagnetic records

14.3 Time series of paleomagnetic data

14.4 Geomagnetic polarity time scale – a first look

14.5 The time averaged field

14.6 Long term changes in paleointensity

14.7 Statistical models of paleosecular variation

14.8 Problems

15 The GPTS and magnetostratigraphy

15.1 Early efforts in defining the GPTS

15.2 Current status of the geological time scale

15.3 Applications

15.4 Problems

16 Tectonic applications of paleomagnetism

16.1 Essentials of plate tectonic theory

16.2 Poles and apparent polar wander

16.3 The Gondwana APWP

16.4 Inclination shallowing and GAD

16.5 Paleomagnetism and plate reconstructions

16.6 Discordant poles and displaced terranes

16.7 Inclination only data and APWPs

16.8 Concluding remarks

16.9 Problems

17 References

Appendices

A Definitions, derivations and tricks

A.1 Definitions

A.2 Derivations

A.3 Useful tricks

B Plots useful in paleomagnetism

B.1 Equal area projections

C Paleomagnetic statistics and parameter estimation

C.1 Hysteresis Parameters

C.2 Directional statistics

C.3 Paleointensity statistics

D Anisotropy in paleomagnetism

D.1 The 15 measurement protocol

1 Purpose of the book

The geomagnetic field acts both as an umbrella, shielding us from cosmic radiation and as a window, offering one of the few glimpses of the inner workings of the Earth. Ancient records of the geomagnetic field can inform us about geodynamics of the early Earth and changes in boundary conditions through time. Thanks to its essentially dipolar nature, the geomagnetic field has acted as a guide, pointing to the axis of rotation thereby providing latitudinal information for both explorers and geologists.

Human measurements of the geomagnetic field date to about a millenium and are quite sparse prior to about 400 years ago. Knowledge of what the field has done in the past relies on accidental records carried by geological and archaeological materials. Teasing out meaningful information from such materials requires an understanding of the fields of rock magnetism and paleomagnetism, the subjects of this book. Rock and paleomagnetic data are useful in many applications in Earth Science in addition to the study of the ancient geomagnetic field. This book attempts to draw together essential rock magnetic theory and useful paleomagnetic techniques in a consistent and up-to-date manner. It was written for several categories of readers:

- Earth scientists who use paleomagnetic data in their research,
- students taking a class with paleomagnetic content,
- other professionals with an interest in evaluating or using paleomagnetic data, and
- anyone with at least college level chemistry, physics and a cursory knowledge of Earth science with an interest in magnetism in the Earth.

There are a number of excellent references on paleomagnetism and on the related specialties (rock magnetism and geomagnetism). The ever popular but now out of print text by Butler (1992) has largely been incorporated into the present text. For in-depth coverage of rock magnetism, we recommend Dunlop and �zdemir (1997). Similarly for geomagnetism, please see Backus et al. [1996]. A rigorous analysis of the statistics of spherical data is given by Fisher et al. (1987). The details of paleomagnetic poles are covered in van der Voo (1993) and magnetostratigraphy is covered in depth by Opdyke and Channell (1996). The Treatise in Geophysics, vol. 5 (edited by Kono, 2007) and The Encyclopedia of Geomagnetism and Paleomagnetism (edited by Gubbins and Herrero-Bervera, 2007) have up to date reviews of many topics covered in this book. The present book is intended to augment or distill information from the broad field of paleomagnetism, complementing the existing body of literature.

An important part of the problems in this book is to teach students to write simple computer programs themselves and use programs that are supplied as a companion set of software (PmagPy). The programming language chosen for this is Python because it is free, cross platform, open source and well supported. There are excellent online tutorials for Python and many open source modules which make software development cheaper and easier than any other programming environment. The appendix provides a brief introduction to programming and using Python. The reader is well advised to peruse the PmagPy Cookbook for further help in gaining necessary skills with a computer. Also, students should have access to a relatively new computer (Windows, Mac OS 10.4 or higher are supported, but other computers may also work.) Software installation is described at: magician.ucsd.edu/Software/PmagPy.

2 What is in the book

This book is a collaborative effort with contributions from R.F. Butler (Chapters 1, 3, 4, 6, 7, 9, 11 and the Appendix), S.K Banerjee (Chapter 8) and R. van der Voo (Chapter 16). The MagIC database team designed and deployed the MagIC database which we have made liberal use of in providing data for problem sets and in the writing of the PmagPy Cookbook, so there were significant contributions to this book project from C.G. Constable, A.A.P. Koppers and Rupert Minnett.

At the beginning of most chapters, there are recommended readings which will help fill in background knowledge. There are also suggested readings at the end of most chapters that will allow students to pursue the subject matter in more depth.

The chapters themselves contain the essential theory required to understand paleomagnetic research as well as illustrative applications. Each chapter is followed by a set of practical problems that challenge the student’s understanding of the material. Many problems use real data and encourage students to analyze the data themselves. [Solutions to the problems may be obtained from LT by instructors of classes using this book as a text.] The appendix contains detailed derivations, assorted techniques, useful tables and a comprehensive explanation of the PmagPy set of programs.

Chapter 1 begins with a review of the physics of magnetic fields. Maxwell’s equations are introduced where appropriate and the magnetic units are derived from first principles. The conversion of units between cgs and SI conventions is also discussed and summarized in a handy table.

Chapter 2 reviews essential aspects of the Earth’s magnetic field, discussing the geomagnetic potential, geomagnetic elements, and the geomagnetic reference fields. The various magnetic poles of the Earth are also introduced.

Chaptes 3-8 deal with rock and mineral magnetism. The most important aspect of rock magnetism to the working paleomagnetist is how rocks can become magnetized and how they can stay that way. In order to understand this, Chapter 3 presents a discussion of the origin of magnetism in crystals, including induced and remanent magnetism. Chapter 4 continues with an explanation of anisotropy energy, magnetic domains and superparamagnetism. Magnetic hysteresis is covered in Chapter 5. Chapter 6 deals with specific magnetic minerals and their properties, leading up to the origin of magnetic remanence in rocks, the topic of Chapter 7. Finally Chapter 8 deals with applied rock magnetism and environmental magnetism.

Chapters 9-13 delve into the nuts and bolts of paleomagnetic data acquisition and analysis. Chapter 9 suggests ways of sampling rocks in the field and methods for treating them in the laboratory to obtain a paleomagnetic direction. Various techniques for obtaining paleointensities are described in Chapter 10. Once the data are in hand, Chapters 11 and 12 deal with statistical methods for analyzing magnetic vectors. Paleomagnetic tensors are introduced in Chapter 13, which explains measurement and treatment of anisotropy data.

Chapters 14-16 illustrate diverse applications of paleomagnetic data. Chapter 14 shows how they are used to study the geomagnetic field. Chapter 15 describes the development of the geomagnetic polarity time scale and various applications of magnetostratigraphy. Chapter 16 focuses on apparent polar wander and tectonic applications.

The appendix contains more detailed information, included for supplemental background or useful techniques. It is divided into several sections: Appendix A summarizes various definitions and detailed derivations including various mathematical tricks such as vector and tensor operations. Appendix B.1 describes some plots commonly employed by paleomagnetists. Appendix C.2 collects together methods and tables useful in directional statistics. Appendix D describes techniques specific to the measurement and analysis of anisotropy data. The PmagPy Cookbook provides an introduction to the Magnetics Information Consortium (MagIC) database, the current repository for rock and paleomagnetic data and summarizes essential computer skills including basic Unix commands, an introduction to Python programming and extensive examples of programs in the PmagPy software package used in the problems at the end of each chapter.

3 How to use the book

Each chapter builds on the principles outlined in the previous chapters, so the reader is encouraged to work through the book sequentially. There are recommended readings before and after every chapter selected to provide backgound information and supplemental reading for the motivated reader respectively. These are meant to be optional.

The reader is encouraged to study Chapter 6 in the PmagPy Cookbook before beginning to work on the problems at the end of each chapter. The utility of the book will be greatly enhanced by successfully installing and using the programs referred to in the problems. By conscientiously trying them out as they are mentioned, the reader will not only gain familiarity with PmagPy software package, but also with the concepts discussed in the chapters.

We have attempted to maintain a consistent notation throughout the book. Vectors and tensors are in bold face; other parameters, including vector components, are in italics. The most important physical and paleomagnetic parameters, acronyms and statistics are listed in Appendix A.

The problems in this book are intended to be solved with Jupyter Notebooks in conjunction with the PmagPy software of Tauxe et al., (2016). To get ready for this, you must do the following steps:

- Install Anaconda Python 3, following the instructions in the PmagPy Cookbook in the Section on Full PmagPy Install. Be sure to include the non-default Python packages.
- Install PmagPy following the instructions for pip install from the command line.
- Move the data files to a directory convenient to you (e.g., your Desktop) following the instructions in the cookbook.
- The data files used in the problems are in the PmagPy subdirectory data_files/notebooks/Essentials/, separated by directory.
- Navigate to the PmagPy subdirectory data_files/notebooks/Essentials/ and launch the jupyter notebook environment. Make a new notebook for a problem set at the end of each chapter.
- To see the Jupyter notebook solutions, navigate to the PmagPy subdirectory data_files/notebooks/Essentials/ and launch the jupyter notebook environment. Open each chapter’s solution in turn.

The problems at the end of each chapter assume proficiency in both Python programming and the use of Jupyter notebooks. See Python Programming for Earth Scientists for a complete course.

For examples on how to use PmagPy in a Jupyter notebook, see PmagPy.ipynb in your data_files directory.

4 Acknowledgements

LT is the primary author of this book who bears sole responsibility for all mistakes. There are significant contributions by RFB, SKB and RvdV. We are indebted to many people for assistance great and small. This book began life as a set of lecture notes based loosely on the earlier book by Tauxe (1998). Many pairs of eyes hunted down errors in the text and the programs each time the course was given. The course was also occasionally co-taught with Cathy Constable and Jeff Gee who contributed significantly to the development of the manuscript and the proof-reading there-of. Thanks go to the many “live” and “online” students who patiently worked through various drafts. Special thanks go to Kenneth Yuan, Liu Cy , Maxwell Brown and Michael Wack who provided many detailed comments and helpful suggestions. Reviews by Ken Kodama, Brad Clement, Scott Bogue and Cor Langereis improved the book substantially. Also, careful proof-reading by Newlon Tauxe of the first few chapters is greatly appreciated. And of course, I am deeply grateful to my mentors, Dennis V. Kent and Neil D. Opdyke who taught me how to do science.

I owe a debt of gratitude to the many sources of public domain software that ended up in the package PmagPy, including contributions by Cathy Constable, Monika Korte, Jeff Gee, Peter Selkin, Ron Shaar, Nick Swanson-Hysell, Ritayan Mitra and especially Lori Jonestrask, as well as the many dedicated contributors to the Numpy, Matplotlib, Basemap, Cartopy, and Pandas Python modules used extensively by PmagPy. Also, many illustrations were prepared with the excellent programs Magmap, Contour and Plotxy by Robert L. Parker, to whom I remain deeply grateful. I gratefully acknowledge the authors of many earlier books, too many to name but included in the Bibliography, which both educated and inspired me.

Finally, I am grateful to my husband, Hubert Staudigel, and my children, Philip and Daniel Staudigel who have long tolerated my obsession with paleomagnetism with grace and good humor and frequently good advice.

Lisa Tauxe

Scripps Institution of Oceanography

La Jolla, CA 92093-0220

U.S.A.

Scripps Institution of Oceanography

La Jolla, CA 92093-0220

U.S.A.