Inverse Theory; Geomagnetism; Gravity; Electrical sounding methods.
1996 Backus, G., Parker, R., Constable, C. Foundations of Geomagnetism, Cambridge University Press, to appear Spring, 1996.
1996 Parker, R. L., and Booker, J. R., Optimal One-Dimensional Inversion and Bounding of Magnetotelluric Apparent Resistivity and Phase Measurements, to appear in Physics of the Earth & Planetary Interiors, summer 1996.
1995 Parker, R. L., Improved Fourier Terrain Correction , I , Geophysics, 60, No. 4, 1007-1017.
1994 Parker, R. L. Geophysical Inverse Theory. 386 pp, Princeton Univ. Press.
Robert L. Parker, 0225 IGPP
Scripps Institution of Oceanography
University of California, San Diego
La Jolla, CA 92093-0225
tel: 619/534-2475
internet: rlparker@ucsd.edu
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Summary
The subject matter of {\it Foundations of Geomagnetism} is the mathematical and physical basis of the science of geomagnetism; graduate students in the earth sciences are its intended audience. George Backus has always been passionately concerned with the logical foundation of scientific argument and mathematical rigor in quantitative developments. Thus when he taught about the decomposition of a magnetic field into its poloidal and toroidal parts, he would never begin, ``It can be shown that''; a proper account must start with the demonstration that a unique decomposition of this kind is always possible. Chapter 5 opens on this point. To build the foundations of geomagnetism, George calls upon some unusual mathematical tools, some of his own invention. The earth is nearly spherical, and geophysicists continually need to treat vectors and solve differential equations in spherical geometry. George shows how it is possible to maintain the elegance of a coordinate-free notation and at the same time to preserve the simplicity and familiarity of a Gibbs-like vector calculus for operators on a spherical surface. In this way he avoids the heavy baggage of the currently fashionable coordinate-free differential geometry \`a la Cartan. George's notation is intuitively right for the problem.
The final form of the book consists of seven chapters. The first is a brief overview of the phenomena that are of interest in geomagnetism. It includes a sketch of the history of the subject, followed by a description of the geomagnetic field and its variability in time and space. Chapter 2 concerns the classical theory of electromagnetism based on Maxwell's equations. We cover the physical and mathematical ideas of sources for the electromagnetic field and how the vacuum form of Maxwell's equations is adapted for polarizable media. We discuss the mathematical basis for the practice of neglecting small terms in an equation, and in particular we justify the neglect in geomagnetic work of the displacement current in Maxwell's equations. The discussion of sources introduces the concept of the separation of the magnetic field into parts of internal and external origin.
Chapter 3 is devoted to spherical harmonics. The aim is to develop from first principles all the standard results. The perspective and methods used are not the traditional ones, however. In particular, the spherical harmonics of degree $\ell$ are exhibited as homogeneous harmonic polynomials in $x$, $y$, and $z$, which are treated as members of a finite-dimensional vector space. We introduce an inner product on the space and then consider an orthogonal basis for it. Various linear operators mapping this space onto itself are used to explore the symmetries of the system, and thereby to discover its properties, such as the Addition Theorem, and the existence of a self-reproducing kernel. Up to this point, the results have been independent of the particular coordinate system. Once a special axis system is chosen, we can develop explicit expressions for the traditional set of orthonormal functions. We investigate the asymptotic properties of the functions, derive recurrence relations among them, and describe a scheme for their practical computation.
Chapter 4 gives the application of spherical harmonics in the description of the main geomagnetic field. We study the question of the uniqueness of the coefficients in a spherical harmonic expansion containing internal and external parts. Other topics include the geomagnetic power spectrum, downward continuation to the core, and how little information about the sources is contained in the Gauss coefficients.
The subject of Chapter 5 is the Mie representation, which is the expression of a solenoidal field as a sum of poloidal and toroidal parts. Again the theory is developed from first principles, beginning with the Helmholtz representation for tangent vector fields on a spherical surface, a useful representation in its own right. The Mie representation is then applied to a variety of geomagnetic problems, including the generalization of Gauss' separation to regions containing sources, outward diffusion of the toroidal magnetic field of the core, geomagnetic sounding, and the free decay of magnetic fields in a stationary core.
The material of the preceding chapters is brought to bear in Chapter 6, where we consider the physical processes taking place in the core of the earth. After a quick look at a simplified dynamo model with only two degrees of freedom, we derive the full system of partial differential equations governing the interaction of a moving conducting fluid with an embedded magnetic field. To obtain the version of Ohm's law needed in a moving conductor, we must appeal to relativistic physics. The two descriptions of a continuum, Lagrangian and Eulerian, are discussed and their relationship exposed. Two limiting problems are solved exactly: zero velocity and infinite conductivity. The latter is shown to be a useful approximation in the earth for time scales less than 100 years, and therefore of considerable interest in the interpretation of the secular variation. We derive the ``frozen-flux'' condition of Roberts and Scott, which, if valid, permits us to deduce from the magnetic field what the fluid velocity is on the lines where the radial field vanishes. The chapter closes with a sample of dynamo theory, a mixture of brief qualitative summaries of some important results together with a few topics laid out in mathematical detail, including Cowling's antidynamo theorem and a glimpse at mean field dynamos, currently so popular.
The Mie representation and, to a lesser extent, spherical harmonics are dependent on vector calculus on the surface of a sphere. Chapter 7 is a compendium of mathematical theorems and results needed elsewhere in the book. The chapter provides a general treatment of linear operators that act on scalar and vector fields. The general theory is specialized to the case of a spherical surface, and the properties of those most useful operators, surface gradient and surface curl, are then developed in some detail. Corresponding results for more general surfaces are touched upon. Other topics in Chapter 7 include surface forms of the integral theorems of Stokes and Gauss, and inclusion of jump discontinuities in those theorems and how this affects the Helmholtz Theorem.
The two junior authors have both learned an enormous amount by going over this material in detail. We can only hope some of the craftsmanship and the intellectual discipline demonstrated by this work has rubbed off on us. We are grateful to Philip Stark for suggesting the project. We would also like to express our thanks to Elaine Blackmore, who translated the hand written notes into TEX with great speed and accuracy, all the more amazing given that this was her first experience with TEX. We wish to express our gratitude to the Director of Scripps Institution of Oceanography for the financial help he provided as we were getting started. Cambridge University Press and its editors deserve our gratitude for their patience and understanding, as well as their enthusiasm for the idea. Once again, we want to thank the senior author, George Backus, for his example as a great scientist and a warm human being.
The properties of the log of the admittance in the complex frequency plane lead to an integral representation for one- dimensional magnetotelluric (MT) apparent resistivity and impedance phase similar to that found previously for complex admittance. The inverse problem of finding a one-dimensional model for MT data can then be solved using the same tech- niques as for complex admittance, with similar results. For instance, the one-dimensional conductivity model that minim- izes the $chi sup 2$ misfit statistic for noisy apparent resistivity and phase is a series of delta functions.
One of the most important applications of the delta function solution to the inverse problem for complex admittance has been answering the question of whether or not a given set of measurements is consistent with the modeling assumption of one-dimensionality. The new solution allows this test to be performed directly on standard MT data. Recently, it has been shown that induction data must pass the same one- dimensional consistency test if they correspond to the polarization in which the electric field is perpendicular to the strike of two-dimensional structure. This greatly magni- fies the utility of the consistency test.
The new solution also allows one to compute the upper and lower bounds permitted on phase or apparent resistivity at any frequency given a collection of MT data. Applications include testing the mutual consistency of apparent resis- tivity and phase data and placing bounds on missing phase or resistivity data. Examples presented demonstrate detection and correction of equipment and processing problems and verification of compatibility with two-dimensional B- polarization for MT data after impedance tensor decomposi- tion and for continuous electromagnetic profiling data.
This paper describes a new Fourier technique for calcu- lating the gravitational attraction of a layer with an irregular top surface, for application in the terrain correction of marine gravity surveys in shallow water. An earlier Fourier-based algorithm fails or becomes inaccurate when the peaks of the topography approach the sea surface too closely. The new approach divides the attraction into two parts: a local contribution from the material within a cylinder around each observation point and the attraction from the matter outside the cylinder. A special quadrature rule, optimized for the actual data distribution, evaluates the local contribution. The calculation of the exterior component represents the bulk of the numerical effort. For- tunately, the exterior integral possesses an expansion as a series of convolutions; by evaluating these in the Fourier domain the procedure can take advantage of the efficiency of the fast Fourier transform. Chebychev economization of the convolution series provides further significant improvements in computational speed. Two examples, one artificial the other based on a survey around Guadalupe Island, illustrate the application of the new technique. Estimates of the errors from computation sources and from inadequacies of the topographic model con- firm the general accuracy of the approach, except in regions of very steep terrain.
ROBERT L. PARKER
PRINCETON UNIVERSITY PRESS
Preface
Geophysical Inverse Theory is an expanded version of my lec- ture notes for a one-quarter graduate-level class, which I have been teaching for twenty years at the Institute for Geophysics and Planetary Physics at La Jolla, California. I have organized the subject around a central idea: almost every problem of geophysical inverse theory can be posed as an optimization problem, where a function must be minimized subject to various constraints. The emphasis throughout is on mathematically sound results whenever possible. This has meant that the level of discussion must be raised above that of undergraduate linear algebra, a level on which the material is often presented. The theory can be set out quite simply in the setting of abstract linear spaces but to do this it is desirable to provide the students with an introduction to elementary functional analysis. In this book, as in the class, the introduction is an informal sur- vey of essential definitions and results, mostly without proof. Fortunately for the students and the author, the classic text by Luenberger, Optimization by Vector Space Methods, covers all the necessary technical material and much more besides. Not only in the introduction, but throughout the book, I have normally omitted the proofs of standard mathematical results, unless the proof is short.
In geophysics, inverse theory must provide the tools for learning about the distribution in space of material properties within the earth, based on measurements made at the surface. The notion that optimization lies at the heart of much geophysical inverse theory arises from the observa- tion that problems based on actual measurements can never have unique solutions-there just is not enough information in finitely many measurements to prescribe an unknown func- tion of position. Nonetheless, we may elect to construct a particular model, which means choosing one from the infinite set of candidates. Alternatively, we can seek common pro- perties valid for every member of the set. In the first case the choice should be deliberately made and designed to avoid misleading artificial features: this leads to the minimization of complexity, an optimization problem involv- ing a suitable function, such as root-mean-square gradient. The second course also points to an optimization: if we wish to find an upper or lower bound on a property of the earth, like the average value in a given region, we should calcu- late the maximum or minimum value over the set of all models agreeing with the measurements. These two approaches will occupy most of our attention.
I believe in the value of concrete and realistic illus- trations for learning. This is particularly important when we must translate a result from an abstract setting to a practical computation. Therefore, I have made a practice of working through a few examples in some detail; the examples here are all rather small computationally, but realistic in the sense that they are, with one exception, based on actual measurements in geomagnetism, seismology, gravity, and elec- tromagnetic sounding. The observations used in the examples are tabulated, unless the number of data is too great. The same problem, for instance, one on seismic attenuation, appears in several different contexts, illustrating numeri- cal instability, the application of smooth model construc- tion, resolution studies, and the bounding of linear func- tionals. And of course, students learn best by practice so there are lots of exercises, some of them involving computer work. Computer calculations, when I started this work, required the students to program in Fortran which for most of them resulted in nothing but frustration and missed dead- lines. Therefore I wrote a numerical linear-algebra com- puter language to alleviate these difficulties but, in the last few years, personal computers and wonderful software packages like Matlab and Mathematica have completely solved the problem and I have been able to retire my program. In fact, all my own calculations for chapter 5 were performed in Matlab.
The scope of the book is certainly not intended to be comprehensive. I hope to give my readers the principles from which they can solve the problems they encounter. I have avoided analytic theories that treat arbitrarily accu- rate and extensive data, and concentrated instead on the more practical side of inverse theory. I have devoted only a small amount of space to strongly statistical approaches, first because I am unconvinced that one can place much con- fidence in results that depend on so many unverifiable assumptions. Second, Albert Tarantola's book Inverse Prob- lem Theory: Methods for Fitting and Model Parameter Estima- tion (1987) covers these methods in great detail and it would be redundant to attempt to reproduce it even in part.
It is time for a brief tour of the contents. There are five fairly long chapters, each divided into named sections, which vary in length from one to seventen pages. As I have already noted, we must have a survey of the results and language of abstract linear vector spaces and some optimiza- tion theory. This is chapter 1. It also includes a section on numerical linear algebra, mainly to introduce the QR fac- torization. The material covers the standard ideas of vec- tor spaces up to Hilbert spaces; the Decomposition and Pro- jection Theorems and the Riesz Representation Theorem each play a part later on because Hilbert space is the normal setting for the theory. Some of the material here, for example sections 1.08 and 1.09 on completion of normed spaces, is perhaps peripheral to geophysical interests, but useful if the student is ever to dip into the applied mathematical literature.
Chapter 2 discusses the simplified situation in which the data are error free and the problem is linear. We define linear inverse problems to be those in which the functional describing the measurement is linear in the model. Linear problems are much more fully understood than nonlinear ones; we see in chapter 5 that the only general theory for the nonlinear systems involves approximating them linearly. Thus most of the book is about linear inverse problems. In chapter 2 two general problems are discussed: finding the smallest model in the norm and constructing the smallest model when a finite-dimensional subspace is excluded from penalization (the latter is an example of sem- inorm minimization). We introduce a highly simplified, one-dimensional marine magnetization problem as the first illustration. More realistic examples follow, including interpolation of irregularly spaced data on a sphere.
In chapter 3 we continue to examine the issue of build- ing model solutions, but now from uncertain data. Once more we cast the problem as one in optimization: finding the smallest model subject to the constraint that the data and the predictions of the linear model are sufficiently close in the sense of the norm. The calculations for the minimum norm model are identical with those arising from what applied mathematicians call regularization of unstable prob- lems; our perspective provides a different and, in my view, clearer motivation for these calculations.
So far we have focused on the question of finding a simple model compatible with the observations; in chapter 4 we turn to the matter of drawing conclusions independent of any particular model. We briefly discuss the idea of resolution-that the incompleteness of our measurements smears out the true solution to an extent that can be calcu- lated. To obtain more quantitative inferences we find it is necessary to add to the observations additional constraints in the model space, like a bound on a norm, or the condition of model positivity. These cases are examined, and to enable calculations to be made with positivity conditions, linear and quadratic programming methods are described. The theory of ideal bodies is introduced as an illustration of the use of a non-Hilbert space norm, in this case the uni- form norm. Finally, we give our only example of an applica- tion a strong statistical theory, where the model is itself the realization of a statistical process, in this case sea- floor magnetization treated as a stationary stochastic pro- cess.
The last chapter deals with nonlinear problems, those with nonlinear relationships between observation and unknown model. As mentioned earlier, general results are essen- tially nonexistent: we must form approximations that draw upon the linear theory. To do this we need an analog of differentiation for abstract vector spaces: we describe Gateaux and Fr'echet differentiation. The general idea con- tinues to be that of minimization of the complexity function subject to the constraint of a good match to observation. Now the search for the minimum is iterative and numerical. We describe a once-popular scheme given by Backus and Gil- bert, and a more recent and efficient method called Occam's process. By changing the parameterization we can convert the linear gravity ideal body problem considered in chapter 4 into a nonlinear problem, which we solve by the iterative methods. The other major example of this chapter is that of magnetotelluric sounding, which has the distinction of being one of the few for which the question of existence of solu- tions can be rigorously decided. In nonlinear systems it is by no means guaranteed that there are any models capable of fitting the data and in general the only way to proceed is to allow the iterative minimization process to search. At the end of the chapter we give the analytic theory for existence and construction of models from magnetotelluric data and compare the results with those found by iteration.
This book has been over ten years in the writing. I have received continuous encouragement from my friends and colleagues at IGPP, particularly Cathy and Steve Constable and Guy Masters. I am also most grateful to the many stu- dents who have read early versions and offered valuable advice, notably Loren Shure, Philip Stark, and Mary Kappus. When she was a student Loren Shure and I wrote the graphics computer program plotxy which generated all but one of the figures in this book; unlike my matrix arithmetic program, plotxy has not yet retired. John Booker provided numerous helpful suggestions and corrections, for which I am grate- ful. Sara Van Rheenen at Princeton University Press has been ever sympathetic and helpful to this neophyte printer. I would like to thank the Office of Naval Research, which provided a small grant to get this project started. Finally, I wish to express my heart-felt appreciation to my wife, Joan, for her support and understanding and for giving me the final impetus to complete this task.
La Jolla, California