E L S E V I E R Joumal of Applied Geophysics 36 (1996) 155-156

Book review

Potential Theory in Gravity and Magnetic Applica- tions, R.J. Blakely, Cambridge University Press, Hardcover, xiv +441 pp. Price English Pounds 40.00. ISBN 0-521-41508-X.

This is an excellent book. I found it most informa- tive and very readable. This work is clearly aimed as a textbook at the final year undergraduate student in geophysics or for the graduate student undertaking a course on potential theory. There is also a demon- strable need for the book because of the renewed interest in gravity work and especially in magnetic work. The magnetic work has assumed much greater importance because of the conjunction of the vastly improve instrumentation such as the optically pumped magnetometers, together with the improved methods of interpretation and the ready availability and power of modern computers.

The book is divided into twelve chapters. The first six chapters deal with the fundamental topics of potential theory. The subject matter covered is along the lines of the classical textbooks on potential (Kel- log, Ramsay) widely used by geophysics students in the past. The book begins with a clear definition of equipotential surfaces, harmonic functions, force fields, Laplace's equation. The work, is thorough, clear and in no way tedious. The book moves on through the consequences of Laplace's equation through Green's identities and the theorems of Gauss and Helmholtz. All of this work is treated from the point of view of the potential being a gravitational potential. Chapter 3 is the next natural topic. It is on The Newtonian potential of mass distributions. The equations for the potential and the consequent force fields for many of the classically shaped bodies are derived and given. The simplifications in their

0926-9851/96/$15.00 Published by Elsevier Science B.V. PII S 0 9 2 6 - 9 8 5 1 ( 9 6 ) 0 0 0 3 9 - 0

derivation provided by Gauss's Law is shown. Stu- dents with a good undergraduate background in Physics will have covered the material from the point of view of electrostatics.

Chapter 4 follows with a well presented discus- sion on the more difficult subject of magnetic poten- tial (both scalar and vector) dipole moments and the consequential flux density. The text is illustrated with clear informative diagrams. Chapter 5 deals with Magnetization (uniform and non-uniform) sus- ceptibility and the magnetic field with the distinction between the Flux density (in Tesla) and the Intensity (Ampere per meter). Poisson's relationship is intro- duced and by building on the earlier chapters where the gravitational attraction for various bodies was derived the magnetic potential for the self-same bod- ies is also derived. The work is written with the interest of the geophysicists in mind. In the next chapter (pp. 100-127), the heavily mathematical treatment of spherical harmonic analysis follows. The chapter concludes with a useful discussion for the uninitiated, on the significance of each term in the harmonic expansion for idealized masses such as total mass, dipole, and quadruple etc.

The next two chapters cover Regional gravity fields and The geomagnetic field. These chapters are naturally dependent on an understanding of spheri- cal harmonics. The chapter on gravity includes the standard treatment of the shape of the Earth and the subsequent gravity corrections to be carried out on raw gravity measurements. The gravity anomalies (ie signal) that occur over large tectonic features are treated with attention to the numerical calculation of the amplitude and pattern of these features. In this chapter the methods of filter theory (Fourier trans- form) are introduced and these method are fully

156 Book reciew

discussed in a later chapter. Mention must be made here to the excellent package of twenty-nine com- puter subroutines (fully written out and in FOR- TRAN) in an Appendix in the book. The chapter on The geomagnetic field utilizes the theory on spheri- cal harmonics developed previously in the Chapter 6. The Earth field, the dipole field, the non-dipole field, and the harmonics up to order 10 are discussed, as is secular variation, crustal magnetic anomalies and the significance of the signal measured by a total field magnetometer.

From the start of Chapter 9 on (p. 182), the next 176 pages the book is heavily into the calculation of the signal recorded over a wide ranges of variously shaped bodies, the use of filter theory in interpreting magnetic and gravity signals to come up with the structure of the causative body, and the enhancement and display of the data. In the Appendix are a number of computer programs for the calculation of the signal from non-standard shaped bodies. Further- more, the programs provide for the signal to be calculated both in the space and the spectral domain. The signal over three dimensional polygons is de- rived and a computer subroutine is given in the Appendix. Chapter 10 (pp. 214-257) is an excellent chapter on the Inverse method. The non-uniqueness of the inversion is analyzed. The difficulties caused by noise, and non-uniformity in the Magnetization of the source-body is examined. Chapter 11 is a short compact chapter on Fourier-domain modelling. One and two dimensional transforms are given along with all the usual theorems of symmetry, linearity, scal- ing, shifting, differentiation, convolution, power and Parseval's theorem. There is also a short concise and useful treatment of Hankel and Hilbert Transforms. Over a range of classical bodies the theoretical ex- pression for the signal in terms of the spectral do- main is derived. The special feature of the difficult convolutions in the space domain collapsing into simple multiplicative operations. Furthermore, in the development of the expressions in the spectral do- main, the ease with which data may be transformed from one component to another (eg vertical to total

field) or from any latitude to the pole by means of a simple filter becomes apparent. The useful method of upward continuation is given in full. All these filters are derived. Wherever a filter can carry out an operation (a convolution in the space domain; a multiplication or division in the spectral domain) there is an inverse filter; but the inverse filter is not always stable. The difficulties with some inverse filters are shown in Fig. 11.7. Some geophysicists have had difficulty in understanding or appreciating the remarkable properties (and its limitations) of radial-power-density-spectrum. Chapter 12 (pp. 311- 358) looks at a range of transformations and their application to real data. Figures and contour maps show the consequences of the filtering operation of downward continuation and the uncontrolled amplifi- cation of noise. Transforming data to the pole has special problems of filter instability from low lati- tudes, noise and in the case where permanent magne- tization is present of singularities in the filter for some circumstances. The more straight forward oper- ations such as the identification of structural contacts by filtering in the spectral domain is given in theory, diagram and example-map.

The book is enhance by sets of questions follow- ing each chapter. Four appendices; A review ~( uector calculus, Computer subroutines (29 in total), Review of sampling theoo,, and Concersion of units. There is a extensive bibliography of 297 useful and readily accessible reference papers and a fairly thin index of five pages. On the other hand the book is well organized, logically put together and the mate- rial is easy to find in the text. The style is clear and easy to read. For practicing geophysicists, they will find that everyday operations they use in processing their data derived and discussed and also the book provides a direct line to many of the original articles. Finally, as they say, nothing in this world is perfect: one quibble I do have is that the definition by the author of magnetic scalar potential is in units of Weber m ~.

Ronald Green.