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POTENTIAL THEORY IN GRAVITY AND MAGNETIC APPLICATIONS The Stanford-Cambridge Program is an innovative publishing venture resulting from the. resourceone.info Access. PDF; Export citation. 4 - Magnetic Potential. pp · resourceone.info Page 1. From Richard Blakely. Potential Theory in Gravity & Magnetic. Applications. Cambridge University Press For class use only please. (due to.


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Download Citation on ResearchGate | Potential Theory in Gravity & Magnetic Applications | This book bridges the gap between the classic texts on potential. Richard J. Blakely-Potential Theory in Gravity and Magnetic Applications ( Stanford-Cambridge Program) ()(1).pdf - Ebook download as PDF File .pdf) , Text. Trove: Find and get Australian resources. Books, images, historic newspapers, maps, archives and more.

Potential theory in gravity and magnetic applications by Richard J. Potential theory in gravity. Potential Theory in Gravity and Magnetic Applications. Potential By Richard J. From Richard Blakely. Potential Theory in Gravity Magnetic. Cambridge University Press For class use only please. New- ton correctly interpreted the discrepancy between these two measure- ments as reflecting the oblate shape of the earth.

The Ecuador expedition was led by several prominent French scientists, among them Pierre Bouguer, sometimes credited for the first careful observations of the shape of the earth and for whom the "Bouguer anomaly" is named. The reversible pendulum was constructed by H. Kater in , thereby facilitating absolute measurements of gravity. Near the end of the same century, R. Sterneck of Austria reported the first pendulum instrument and used it to measure gravity in Europe.

Other types of pendulum in- struments followed, including the first shipborne instrument developed by F. Vening Meinesz of The Netherlands in , and soon gravity measurements were being recorded worldwide. The Hungarian geode- sist, Roland von Eotvos, constructed the first torsional balance in Many gravity meters of various types were developed and patented dur- ing to as U.

Most modern instruments suitable for field studies, such as the LaCoste and Romberg gravity meter and the Worden instrument, in- volve astatic principles in measuring the vertical displacement of a small mass suspended from a system of delicate springs and beams. Various models of the LaCoste and Romberg gravity meter are commonly used in land-based and shipborne studies and, more recently, in airborne surveys e.

The application of gravity measurements to geological problems can be traced back to the rival hypotheses of John Pratt and George Airy published between and concerning the isostatic support of topography. They noted that plumb lines near the Himalayas were de- flected from the vertical by amounts less than predicted by the topo- graphic mass of the mountain range. Both Airy and Pratt argued that in the absence of forces other than gravity, the rigid part of the crust and mantle "floats" on a mobile, denser substratum, so the total mass in any vertical column down to some depth of compensation must balance from place to place.

Elevated regions, therefore, must be compensated at depth by mass deficiencies, whereas topographic depressions are un- derlain by mass excesses. Pratt explained this observation in terms of lateral variations in density; that is, the Himalayas are elevated because they are less dense than surrounding crust.

The gravity method also has played a key role in exploration geo- physics. Hugo V. Boeckh used an Eotvos balance to measure gravity over anticlines and domes and explained his observations in terms of the densities of rocks that form the structures. He thus was apparently the first to recognize the application of the gravity method in the ex- ploration for petroleum Jakosky [].

Indeed the first oil discovered in the United States by geophysical methods was located in using gravity measurements Jakosky []. About This Book Considering this long and august history of the gravity and magnetic methods, it might well be asked as I certainly have done during the waning stages of this writing why a new textbook on potential theory is needed now. I believe, however, that this book will fill a significant gap.

As a graduate student at Stanford University, I quickly found my- self involved in a thesis topic that required a firm foundation in potential theory. It seemed to me then, and I find it true today as a professional geophysicist, that no single textbook is available covering the topic of po- tential theory while emphasizing applications to geophysical problems.

The classic texts on potential theory published during the middle of this century are still available today, notably those by Kellogg [] and by Ramsey [] which no serious student of potential theory should be without.

These books deal thoroughly with the fundamentals of po- tential theory, but they are not concerned particularly with geophysical applications.

On the other hand, several good texts are available on the broad topics of applied geophysics e. These books cover the wide range of geophysical methodologies, such as seismology, electro- magnetism, and so forth, and typically devote a few chapters to gravity and magnetic methods; of necessity they do not delve deeply into the underlying theory.

This book attempts to fill the gap by first exploring the principles of potential theory and then applying the theory to problems of crustal and lithospheric geophysics. I have attempted to do this by structuring the book into essentially two parts. The first six chapters build the founda- tions of potential theory, relying heavily on Kellogg [], Ramsey [], and Chapman and Bartels [56].

Special at- tention is given therein to the all-important Green's identities, Green's functions, and Helmholtz theorem. Chapter 3 focuses these theoretical principles on Newtonian potential, that is, the gravitational potential of mass distributions in both two and three dimensions. Chapters 4 and 5 expand these discussions to magnetic fields caused by distributions of magnetic media. Chapter 6 then formulates the theory on a spherical surface, a topic of obvious importance to global representations of the earth's gravity and magnetic fields.

The last six chapters apply the foregoing principles of potential theory to gravity and magnetic studies of the crust and lithosphere. Chapters 7 and 8 examine the gravity and magnetic fields of the earth on a global and regional scale and describe the calculations and underlying theory by which measurements are transformed into "anomalies.

These schemes are divided into the forward method Chapter 9 , the inverse method Chapter 10 , inverse and forward ma- nipulations in the Fourier domain Chapter 11 , and methods of data en- hancement Chapter Here I have concentrated on the mathematical rather than the technical side of the methodology, neglecting such topics as the nuts-and-bolts operations of gravity meters and magnetometers and the proper strategies in designing gravity or magnetic surveys.

Some of the methods discussed in Chapters 9 through 12 are accom- panied by computer subroutines in Appendix B. I am responsible for the programming therein user beware , but the methodologies behind the algorithms are from the literature. They include some of the "classic" techniques, such as the so-called Talwani method discussed in Chapter 9, and several more modern methods, such as the horizontal-gradient calcu- lation first discussed by Cordell [66].

Those readers wishing to make use of these subroutines should remember that the programming is designed to instruct rather than to be particularly efficient or "elegant.

During alone, Geophysics the technical journal of the U. Instead, my approach has been to describe the various methodologies with key examples from the literature, including both classic algorithms and promising new tech- niques, and with apologies to all of my colleagues not sufficiently cited!

Acknowledgments The seeds of this book began in graduate-level classes that I prepared and taught at Oregon State University and Stanford University be- tween and The final scope of the book, however, is partly a reflection of interactions and discussions with many friends and col- leagues. Foremost are my former professors at Stanford University dur- ing my graduate studies, especially Allan Cox, George Thompson, and Jon Claerbout, who introduced me to geological applications of poten- tial theory and time-series analysis.

My colleagues at the U. I am grateful to Richard Saltus and Gregory Schreiber for carefully checking and critiquing all chapters, and to William Hinze, Tiki Ravat, Robert Langel, and Robert Jachens for reviewing and proofread- ing various parts of early versions of this manuscript.

I am especially grateful to Lauren Cowles, my chief contact and editor at Cambridge University Press, for her patience, assistance, and flexible deadlines. Finally, but at the top of the list, I thank my wife, Diane, and children, Tammy and Jason, for their unwavering support and encouragement, not just during the writing of this book, but throughout my career.

This book is dedicated to Diane, who could care less about geophysics but always recognized its importance to me. Richard J. No other single equation has so many deep and diverse mathematical relationships and physical applications. Duff and D. Naylor Every arrow that flies feels the attraction of the earth. Henry Wadsworth Longfellow Two events in the history of science were of particular significance to the discussions throughout this book. Each particle of matter in the uni- verse attracts all others with a force directly proportional to its mass and inversely proportional to the square of its distance of separation.

Nearly a century later, Pierre Simon, Marquis de Laplace, showed that gravitational attraction obeys a simple differential equation, an equation that now bears his name. These two hallmarks have subsequently devel- oped into a body of mathematics called potential theory that describes not only gravitational attraction but also a large class of phenomena, including magnetostatic and electrostatic fields, fields generated by uni- form electrical currents, steady transfer of heat through homogeneous media, steady flow of ideal fluids, the behavior of elastic solids, prob- ability density in random-walk problems, unsteady water-wave motion, and the theory of complex functions and confermal mapping.

The first few chapters of this book describe some general aspects of potential theory of most interest to practical geophysics. This chapter defines the meaning of a potential field and how it relates to Laplace's equation. Chapter 2 will delve into some of the consequences of this relationship, and Chapters 3, 4, and 5 will apply the principles of po- tential theory to gravity and magnetic fields specifically.

We begin by building an understanding of the general term field and, more specifically, potential field. The cartesian coordinate system will be used in the following devel- opment, but any orthogonal coordinate system would provide the same results.

Appendix A describes the vector notation employed throughout this text. We will be concerned primarily with two kinds of fields. Materialfieldsdescribe some physical property of a material at each point of the material and at a given time. Density, porosity, magnetization, and temperature are examples of material fields. A force field describes the forces that act at each point of space at a given time.

The gravitational attraction of the earth and the magnetic field induced by electrical currents are examples of force fields. Fields also can be classed as either scalar or vector. A scalar field is a single function of space and time; displacement of a stretched string, temperature of a volume of gas, and density within a volume of rock are scalar fields.

A vector field, such as flow of heat, velocity of a fluid, and gravitational attraction, must be characterized by three functions of space and time, namely, the components of the field in three orthogonal directions. Gravitational and magnetic attraction will be the principal focus of later chapters.

Both are vector fields, of course, but geophysical instru- ments generally measure just one component of the vector, and that single component constitutes a scalar field.

In later discussions, we often will drop the distinction between scalar and vector fields. For example, gravity meters used in geophysical surveys measure the vertical compo- nent gz a scalar field of the acceleration of gravity g a vector field , but we will apply the word "field" to both g and gz interchangeably.

A vector field can be characterized by its field lines also known as lines offlowor lines of force , lines that are tangent at every point to the vector field.

Small displacements along a field line must have x,? Let Q be at the origin and use equation 1. A set of points refers to a group of points in space satisfying some condition. Generally, we will be dealing with infinite sets, sets that consist of a continuum of points which are infinite in number even though the entire set may fit within a finite volume. A set of points is bounded if all points of the set fit within a sphere of finite radius.

A limit point does not necessarily belong to the set. The distinction between boundary and frontier is a fine one but will be an issue in one derivation in Chapter 2. A set of points is closed if it contains all of its limit points and open if it contains only interior points.

A domain is an open set of points such that any two points of the set can be connected by a finite set of connected line segments composed entirely of interior points. A region is a domain with or without some part of its boundary, and a closed region is a region that includes its entire boundary.

The test particle could be a small mass m acted upon by the gravitational field of some larger body or an electric charge moving under the influence of an electric field. Such physical associations are not considered until later chapters, so the present discussion is restricted to general force and energy relationships.

The kinetic energy expended by the force field in moving the particle from one point to another is defined as the work done by the force field. While under the influence of forcefieldF, a particle of mass m leaves point Po at time to and moves by an arbitrary route to point P, arriving at time t. If the particle moves from point Po to P during time interval to to t Figure 1. Equation 1. In general, the work required to move the particle from Po to P differs depending on the path taken by the particle.

A vector field is said to be conservative in the special case that work is independent of the path of the particle. We assume now that the field is conservative and move the particle an additional small distance Ax parallel to the x axis, as shown in Figure 1. A particle of mass travels through a conservative field; the particle moves first from PQ to P, then parallel to the x axis an additional distance Ax.

As Ax becomes arbitrarily small, we have dW 1. The vector force field F is completely specified by the scalar field W, which we call the work function of F Kellogg []. We have shown, therefore, that a conservative field is given by the gradient of its work function.

With equations 1. Consequently, any vector field that has a work function with continuous derivatives as described in equation 1. A corollary to equation 1. Kellogg [] summarizes these conventions as follows: If particles of like sign attract each other e. If particles of like sign repel each other e.

It follows that field lines at any point are always perpendicular to their equipotential surfaces and, conversely, any surface that is everywhere perpendicular to all field lines must be an equipotential surface. Hence, no work is done in moving a test particle along an equipotential surface. Only one equipotential surface can exist at any point in space.

The distance between equipotential surfaces is a measure of the density of field lines; that is, a forcefieldwill have greatest intensity in regions where its equipotential surfaces are separated by smallest distances. Exercise 1. In the following we discuss another property of potential fields: Several surprising and illustrative results follow from this statement.

We start by discussing the physical meaning of Laplace's equation, first with the trivial one-dimensional case and then the general equation. The stretched rubber band has no curvature in the absence of external forces.

This is simply the one-dimensional case of Laplace's equation, but it il- lustrates an important property of harmonic functions that will extend to two- and three-dimensional cases. Laplace's equation is not satisfied along any part of the band containing a local minimum or maximum. Displacement in the y direction of a stretched rubber band due to an applied force F x. Stretched membrane attached to an uneven loop of wire. Note that the membrane reaches maximum and minimum values of z at the wire.

Let j x,y represent the displacement of the mem- brane in the z direction. In the absence of external forces, the displace- ment of the membrane satisfies Laplace's equation in two dimensions, dx2 dy2 This condition would not be satisfied at any point of the membrane containing a peak or a trough.

Hence, Laplace's equation requires that maximum and minimum displacements can occur only on the frame, that is, on the boundary of the membrane. We might expect from the previous examples and soon will prove that a function that is harmonic throughout a region R must have all maxima and minima on the boundary of R and none within R itself.

The converse is not necessarily true, of course; a function with all maxima and minima on its boundary is not necessarily harmonic because it may not satisfy the three criteria listed. The definition of the second derivative of a one-dimensional function demonstrates another important property of a harmonic function. This is simply another way of stating the now familiar property of a potential: A function can have no maxima or minima within a region in which it is harmonic.

We will discuss a more rigorous proof of this statement in Chapter 2. All heat sources and sinks are restricted from the region. Heat flow J through a region R containing no heat sources or sinks. Region R is bounded by surface S, and n is the unit vector normal to S.

Consider the free flow of heat in and out of a region R bounded by surface 5, as shown by Figure 1. The total heat in region R is given by Tdv, 1. If the integrand, which we assume is continuous, is not zero throughout i?

If all heat sources and sinks lie outside of region R and do not change with time, then steady-state conditions eventually will be obtained and equation 1. Hence, temperature under steady-state conditions satis- fies Laplace's equation and is harmonic. The temperature distribution accompanying steady-state transfer of heat is an easily visualized example of a harmonic function, one which clarifies some of the theoretical results discussed earlier.

For example, imagine a volume of rock with no internal heat sources or heat sinks Figure 1. On the basis of previous dis- cussions, we can state a number of characteristics of the temperature distribution within the volume of rock. After all, some point of the boundary must be closer to any external heat sources or sinks than all interior points; likewise, some point of the boundary will be farther from any external heat source or sink than all interior points.

Suddenly the two ends are switched so that the hot end is in ice water and the cold end is in boiling water. Describe how the temperature of the rod changes with time. Is the temperature harmonic? Finding a solution to Laplace's equation, if indeed one exists, is a boundary-value problem of, in this case, the Dirichlet type; that is, find a representation for j throughout a region i? For example, the steady-state temperature can be calculated, in principle, throughout a spherically shaped region of homogeneous matter by solv- ing equation 1.

We will have considerably more to say about this subject in later chapters. The real and imaginary parts of a complex function are harmonic in regions where the complex function is analytic. For additional information about complex functions, the inter- ested reader is referred to the textbook by Churchill [59]. First we need some definitions. In the following, x and y are real variables describing a two-dimensional cartesian coordinate system.

An interior point of a set of points has some neighborhood containing only points of the set. Sets that contain only interior points are called open regions.

Open, connected regions of the complex plane are called domains.

The derivative of a complex function requires special consideration. In the complex plane, however, there are different paths along which Az can approach zero, and the value of the ratio r z may depend on that path.

In this latter case, the ratio has no limit and the derivative does not exist. The Cauchy-Riemann conditions provide an easy way to determine whether such conditions are met. The derivative of the complex function is given by dw du.

Consequently, the real part of a complex function satisfies the two- dimensional case of Laplace's equation in domains in which the func- tion is analytic, and since the necessary derivatives exist, the real part of w z must be harmonic.

Hence, if a complex function is analytic in domain T', it has a real part that is harmonic in T. Likewise, it can be shown that the imaginary part of an analytic complex function also is harmonic in domains of analyticity.

Prove that the intensity of a conservative force field is inversely proportional to the distance between its equipotential surfaces. If all mass lies interior to a closed equipotential surface S on which the potential takes the value C, prove that in all space outside of S the value of the potential is between C and 0.

If the lines of force traversing a certain region are parallel, what may be inferred about the intensity of the force within the region?

Two distributions of matter lie entirely within a common closed equipotential surface C. Show that all equipotential surfaces outside of C also are common. You are monitoring the magnetometer aboard an interstellar space- craft and discover that the ship is approaching a magnetic source described by a Remembering Maxwell's equation for B, will you report to Mis- sion Control that the magnetometer is malfunctioning, or is this a possible source?

The physical properties of a spherical body are homogeneous. De- scribe the temperature at all points of the sphere if the temperature is harmonic throughout the sphere and depends only on the distance from its center. As a crude approximation, the temperature of the interior of the earth depends only on distance from the center of the earth.

Explain your answer? Assume a spherical coordinate system and let r be a vector directed from the origin to a point P with magnitude equal to the distance from the origin to P.

Prove the following relationships: Paul Davies Only mathematics and mathematical logic can say as little as the physicist means to say. It was asserted that such po- tentials satisfy Laplace's equation at places free of all sources of F and are said to be harmonic. This led to several important characteristics of the potential. In the same spirit, this chapter investigates a number of additional consequences that follow from Laplace's equation. They are referred to as Green's identities.

He is perhaps best known for his paper, Essay on the Application of Mathematical Analysis to the Theory of Electricity and Magnetism, and was ap- parently the first to use the term "potential.

Let U and V be continuous functions with continuous partial deriva- tives of first order throughout a closed, regular region R, and let U have continuous partial derivatives of second order in R. The boundary of R is surface , and h is the outward normal to S. Several very interesting theorems result from Green's first identity if U and V are restricted a bit further.

It also can be shown Kellogg [, p. Region i? Surface S bounds region R. Unit vector n is outward normal at any point on S. Hence, equation 2. Equation 2. Suppose that vector field F has a potential U which is harmonic throughout some region. Hence, the flux of F into the region exactly equals the flux leaving the region, implying that no sources of F exist in the region.

If region R is in thermal equilibrium and contains no heat sources or sinks, the heat entering R must equal the heat leaving R. Therefore, U must be a constant. Hence, if U is harmonic and continuously differentiate in R and if U vanishes at all points of S, U also must vanish at all points of R. This result is intuitive from steady-state heat flow.

If temperature is zero at all points of a region's boundary and no sources or sinks are situated within the region, then clearly the temperature must vanish throughout the region once equilibrium is achieved. Green's first identity leads to a statement about uniqueness, some- times referred to as Stokes's theorem. The function U1 — U2 also must be harmonic in R. Consequently, a function that is harmonic and continuously differentiate in R is uniquely deter- mined by its values on S, and the solution to the Dirichlet boundary- value problem is unique.

Stokes's theorem makes intuitive sense when applied to steady-state heat flow. A region will eventually reach thermal equilibrium if heat is allowed to flow in and out of the region.

Potential theory in gravity and magnetic

It seems reasonable that, for any prescribed set of boundary temperatures, the region will always attain the same equilibrium temperature distribution throughout the region regardless of the initial temperature distribution. In other words, the steady-state temperature of the region is uniquely determined by the boundary temperatures. Again, steady-state heat flow provides some insight.

If the boundary of R is thermally insulated, equilibrium tem- peratures inside R must be uniform. Moreover, a single-valued harmonic function is determined throughout R, except for an additive constant, by the values of its normal derivatives on the boundary. Exercise 2. These last theorems relate to the Neumann boundary-value problem and show that such solutions are unique to within an additive constant. The uniqueness of harmonic functions also extends to mixed boundary- value problems.

We have shown that under many conditions Laplace's equation has only one solution in a region, thus describing the uniqueness of harmonic functions.

Potential Theory in Applied Geophysics

But can we say that even that one solution always exists? The answer to this interesting question requires a set of "existence theorems" for harmonic functions that are beyond the scope of this chapter.

Inter- ested readers are referred to Chapter XI of Kellogg [, p. This relationship will prove useful later in this chapter in discussing certain kinds of boundary-value problems. Derivation of Green's third identity. Point P is inside surface S but is excluded from region R. Angle dQ, is the solid angle subtended by dS at point P. First consider the integral over a Figure 2. The last integral of equation 2. As the sphere becomes arbitrarily small, the right-hand side of the pre- vious expression approaches 4TTU P 1 and equation 2.

In Chapter 3 equation 3. We will show in Chapter 5 equation 5. But remember that no physical meanings were attached to U in deriving Green's third identity; that is, U was only required to have a sufficient degree of continuity. Green's third identity shows, therefore, that any function with sufficient differentiability can be expressed as the sum of three potentials: Hence, we have the surprising result that any function with sufficient differentiability is a potential.

An important consequence follows from Green's third identity when U is harmonic. Then equation 2. This equation is called the representation formula Strauss [] , and we will return to it later in this chapter and again in Chapter It was shown earlier that a harmonic function satisfying a given set of Dirichlet boundary conditions is unique, but the converse is not true.

If U is harmonic in a region R, it also must be harmonic in each subregion of R. Likewise equation 2. It follows that the potential within any subregion of R can be related to an infinite variety of surface distributions.

Hence, no unique boundary conditions exist for a given harmonic function. This property of nonuniqueness will be a common theme in following chapters. If a is the radius of the sphere, then equation 2.

This relationship is called Gauss's theorem of the arithmetic mean. We discussed the maximum principle by example in Section 1. IfU is harmonic in region R, a closed and bounded region of space, then U attains its maximum and minimum values on the boundary of R, except in the trivial case where U is constant. Now we are in a position to prove it. The proof is by contradiction. E cannot equal the total of R because we have stated that U is not constant, and E must be closed because of the continuity of U.

Suppose that E contains at least one interior point of R. It can be shown that if E has one point interior to R it also has a frontier point interior to R, which we call PQ.

Because Po is interior to R, a sphere can be constructed centered about PQ that lies entirely within R. Region R includes a set of points E at which U attains a maximum.

Point PQ is a frontier point of E. A contradiction arises, and our original suppositions, that Po a n d at least one point of E lie interior to R, must be in error. Hence, no maxima of U can exist interior to R. Conversely, if F has a scalar potential, then F is conservative. Each of the three cartesian components of W has a form like 2J4 l At this point, we borrow a result from Chapter 3: With W defined as in equation 2.

Hence, the Helmholtz theorem is proven: If F is continuous and vanishes at infinity, it can be represented as the gradient of a scalar potential plus the curl of a vector potential. The Helmholtz theorem is useful, however, only if the scalar and vec- tor potentials can be derived directly from F. This should be possible because of the way ft and A were defined, and the relationships can be seen by taking the divergence and curl of both sides of equation 2.

Comparing this result with equations 2. Consequently, the scalar potential ft and vector potential A can be de- rived from integral equations taken over all space and involving the di- vergence and curl, respectively, of F itself. If both the divergence and curl vanish at all points, then the field itself must vanish or be constant everywhere.

In addition to this statement, the following important observations follow directly from the Helmholtz theorem and from the integral repre- sentation for scalar and vector potentials. Such fields have no vorticity or "eddies.

Examples of irrotational fields are common and include gravitational attraction, of considerable importance to future chapters. Hence, if the divergence of F vanishes in a region, the normal component of the field vanishes when integrated over any closed surface within the region. Or put another way, the "number" of field lines entering a region equals the number that exit the region, and sources or sinks of F do not exist in the region.

For example, gravitational attraction is solenoidal in regions not occupied by mass. It was stated in Section 2. Hence, if the divergence of a conservative field vanishes in a region, the potential of the field is harmonic in the region. Furthermore, the converse can be shown to be true by taking the curl of both sides of equation 2.

To see this, we use the magnetic field as an example and anticipate the results of future chapters. One of Maxwell's equations relates magnetic induction B and magnetization M in the absence of macroscopic currents: Moreover, equation 2.

This fact plus equation 2. This is called the forward prob- lem when applied to the geophysical interpretation of measured magnetic fields. We will return to this equation in Chapter 5 and subsequent chapters. A heuristic approach will be used, first considering a mechanical system and then extending this result to Laplace's equation.

One conceptual way to solve equation 2.

To find this velocity, we integrate both sides of equation 2. T T The first integral can be ignored if Ar is small and the particle has some mass. Combining this result with equation 2. It pre- sumes that the response of the particle at each instant of impact is independent of all other times. Given this property, the response of the particle to f t is simply the sum of all the instantaneous forces, and the particle is said to be a linear system.

Many mechanical and electrical systems and, as it turns out, many potential-field problems have this property. The function ip t,r is the response of the particle at time t due to an impulse at time r; it is called the impulse response or Green's function of the linear system. The Green's function, therefore, satisfies the initial conditions and is the solution to the differential equation 2.

In the limit as AT approaches zero, the impulse of equation 2. It should be considered rather as a "generalized function" characterized by the foregoing properties. Green's functions are very useful tools; equation 2. In Chapter 3, we will derive Poisson's equation 2 2. We seek a solution for U that satisfies the differential equation and the boundary condition that U is zero at infinity.

The density distribution in Poisson's equation is obviously the source of U and in this sense is analogous to the forcing function f t of the previous section. This is a very interesting result. This fundamen- tal equation relating gravitational potential to causative density distri- butions will be derived in a different way in Chapter 3.

It satisfies the required boundary condition, that ipi is zero at infinity, and is the solution to Poisson's differential equation when the density is an "impulse.

We should expect that such a simplification is possible because earlier results have shown that the potential is uniquely determined by its boundary conditions.

Let both U and V be harmonic in equation 2. In principle, equation 2. Unfortunately, the func- tion V is very difficult to derive analytically except for the simplest sorts of geometrical situations, such as half-spaces and spheres. We construct a point P' below the z — 0 plane that is the image of point P. Hence, V defined in this way satisfies the necessary requirements to be used in equation 2.

Such calculations are called upward continuation, a subject that will be revisited at some length in Chapter This simple integral expression for the potential in terms of density and the Green's function will prove useful in following chapters.

In the following, T is temperature in region R bounded by surface S, and h is the unit vector normal to S. Show, starting with Green's first identity, that if U is harmonic throughout all space, it must be zero everywhere.

Show that the three cartesian components of B are each harmonic in such situations. Function U satisfies the two-dimensional Laplace's equation at every point of a circle.

Function U is harmonic everywhere inside a sphere of radius a. Lawrence Morley and Andre Larochelle [] recognized that these lineations reflect a recording of the reversing geomagnetic field by the geologic process of seafloor spreading. A second major advance in magnetometer design was the development of the proton-precession magnetometer by Varian Associates in The French believed. By the mid s. Airy proposed. Elevated regions. Various models of the LaCoste and Romberg gravity meter are commonly used in land-based and shipborne studies and.

Many gravity meters of various types were developed and patented dur- ing to as U.

Sterneck of Austria reported the first pendulum instrument and used it to measure gravity in Europe. Both Airy and Pratt argued that in the absence of forces other than gravity. Near the end of the same century. Other types of pendulum in- struments followed. The application of gravity measurements to geological problems can be traced back to the rival hypotheses of John Pratt and George Airy published between and concerning the isostatic support of topography.

Pratt explained this observation in terms of lateral variations in density. The Hungarian geode- sist. The Ecuador expedition was led by several prominent French scientists. They noted that plumb lines near the Himalayas were de- flected from the vertical by amounts less than predicted by the topo- graphic mass of the mountain range. Kater in Roland von Eotvos. Most modern instruments suitable for field studies. Vening Meinesz of The Netherlands in Brozena and Peters [43].

The reversible pendulum was constructed by H.. Indeed the first oil discovered in the United States by geophysical methods was located in using gravity measurements Jakosky []. About This Book Considering this long and august history of the gravity and magnetic methods. About This Book xvii so mountain ranges rise above the surrounding landscape by virtue of underlying crustal roots. I quickly found my- self involved in a thesis topic that required a firm foundation in potential theory.

These books cover the wide range of geophysical methodologies. The classic texts on potential theory published during the middle of this century are still available today.

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Hugo V. These books deal thoroughly with the fundamentals of po- tential theory. The gravity method also has played a key role in exploration geo- physics. As a graduate student at Stanford University.. Chapters 1 and 2 define the meaning. It seemed to me then. The first six chapters build the founda- tions of potential theory. I believe.. On the other hand. Ramsey []. I have attempted to do this by structuring the book into essentially two parts. Boeckh used an Eotvos balance to measure gravity over anticlines and domes and explained his observations in terms of the densities of rocks that form the structures.

Stacey []. He thus was apparently the first to recognize the application of the gravity method in the ex- ploration for petroleum Jakosky []. This book attempts to fill the gap by first exploring the principles of potential theory and then applying the theory to problems of crustal and lithospheric geophysics. These schemes are divided into the forward method Chapter 9. Those readers wishing to make use of these subroutines should remember that the programming is designed to instruct rather than to be particularly efficient or "elegant.

Chapters 7 and 8 examine the gravity and magnetic fields of the earth on a global and regional scale and describe the calculations and underlying theory by which measurements are transformed into "anomalies. The last six chapters apply the foregoing principles of potential theory to gravity and magnetic studies of the crust and lithosphere. They include some of the "classic" techniques. Geophysics the technical journal of the U.

Here I have concentrated on the mathematical rather than the technical side of the methodology. Some of the methods discussed in Chapters 9 through 12 are accom- panied by computer subroutines in Appendix B. Chapters 4 and 5 expand these discussions to magnetic fields caused by distributions of magnetic media. During alone. Mul- tiply that number by the several dozen international journals of similar stature and then times the 50 some-odd years that the modern method- ology has been actively discussed in the literature.

Chapter 3 focuses these theoretical principles on Newtonian potential. Special at- tention is given therein to the all-important Green's identities. Green's functions.

Chapter 6 then formulates the theory on a spherical surface. I am responsible for the programming therein user beware. Geologi- cal Survey. Tiki Ravat. Tammy and Jason. Acknowledgments xix that each technique could not be given its due. I am especially grateful to Lauren Cowles. I am grateful to Richard Saltus and Gregory Schreiber for carefully checking and critiquing all chapters. The final scope of the book. Andrew Griscom. My colleagues at the U. Robert Simpson. Foremost are my former professors at Stanford University dur- ing my graduate studies.

Acknowledgments The seeds of this book began in graduate-level classes that I prepared and taught at Oregon State University and Stanford University be- tween and Robert Langel. George Thompson. I thank my wife. Gordon Ness. Richard J. Oregon State University. This book is dedicated to Diane. Thomas Hildenbrand. Richard Saltus. Stanford University. Gerald Connard. No other single equation has so many deep and diverse mathematical relationships and physical applications.

The first few chapters of this book describe some general aspects of potential theory of most interest to practical geophysics. This chapter defines the meaning of a potential field and how it relates to Laplace's equation. Henry Wadsworth Longfellow Two events in the history of science were of particular significance to the discussions throughout this book.

Isaac Newton put forth the Universal Law of Gravitation: Each particle of matter in the uni- verse attracts all others with a force directly proportional to its mass and inversely proportional to the square of its distance of separation.

Chapter 2 will delve into some of the consequences of this relationship. Readers finding. Naylor Every arrow that flies feels the attraction of the earth. Duff and D. Pierre Simon. These two hallmarks have subsequently devel- oped into a body of mathematics called potential theory that describes not only gravitational attraction but also a large class of phenomena. Nearly a century later. A vector field. Fields also can be classed as either scalar or vector.

Materialfieldsdescribe some physical property of a material at each point of the material and at a given time. In later discussions. The cartesian coordinate system will be used in the following devel- opment. A scalar field is a single function of space and time. We will be concerned primarily with two kinds of fields.

Appendix A describes the vector notation employed throughout this text. A force field describes the forces that act at each point of space at a given time. The gravitational attraction of the earth and the magnetic field induced by electrical currents are examples of force fields. Both are vector fields.

For example. A vector field can be characterized by its field lines also known as lines offlowor lines of force. Gravitational and magnetic attraction will be the principal focus of later chapters. Kellogg []. We begin by building an understanding of the general term field and. Small displacements along a field line must have x. Let Q be at the origin and use equation 1. The distinction between boundary and frontier is a fine one but will be an issue in one derivation in Chapter 2.

A set of points is bounded if all points of the set fit within a sphere of finite radius. A limit point does not necessarily belong to the set. A set of points refers to a group of points in space satisfying some condition. A set of points is closed if it contains all of its limit points and open if it contains only interior points. A domain is an open set of points such that any two points of the set can be connected by a finite set of connected line segments composed entirely of interior points.

Such physical associations are not considered until later chapters. While under the influence of forcefieldF. The test particle could be a small mass m acted upon by the gravitational field of some larger body or an electric charge moving under the influence of an electric field.

Newton's second law of motion requires that the momentum of the particle at any instant must change at a rate proportional to the magnitude of the force field and in a direction parallel to the direction taken by the force field at the location of the particle.

A region is a domain with or without some part of its boundary. The kinetic energy expended by the force field in moving the particle from one point to another is defined as the work done by the force field. If the particle moves from point Po to P during time interval to to t Figure 1. Equation 1. The quantity W P. In general.

Then W P. Po is the work required to move the particle from point P o to P. We assume now that the field is conservative and move the particle an additional small distance Ax parallel to the x axis. A vector field is said to be conservative in the special case that work is independent of the path of the particle. With equations 1. The integral can be solved by dividing both sides of the equation by Ax and applying the law of the mean.

The vector force field F is completely specified by the scalar field W. We have shown. As Ax becomes arbitrarily small. If the work function W has continuous.

A particle of mass travels through a conservative field. A corollary to equation 1. In other words.. Kellogg [] summarizes these conventions as follows: If particles of like sign attract each other e. If particles of like sign repel each other e.

In the latter case.. Then P equals Po. In the following we discuss another property of potential fields: If s is a unit vector lying tangent to an equipotential surface of F. Only one equipotential surface can exist at any point in space. It follows that field lines at any point are always perpendicular to their equipotential surfaces and. We start by discussing the physical meaning of Laplace's equation. The distance between equipotential surfaces is a measure of the density of field lines.

Several surprising and illustrative results follow from this statement. Exercise 1. Displacement in the y direction of a stretched rubber band due to an applied force F x. Laplace's equation is not satisfied along any part of the band containing a local minimum or maximum. Now consider a membrane stretched over an uneven frame. This is simply the one-dimensional case of Laplace's equation. The stretched rubber band has no curvature in the absence of external forces.

Note that the membrane reaches maximum and minimum values of z at the wire. Let j x. Laplace's equation requires that maximum and minimum displacements can occur only on the frame. As a three-dimensional example. It is useful. In the absence of external forces. If the salt is concentrated at some point within the fluid. Stretched membrane attached to an uneven loop of wire.

All heat sources and sinks are restricted from the region. We will discuss a more rigorous proof of this statement in Chapter 2. According to Fourier's law. We might expect from the previous examples and soon will prove that a function that is harmonic throughout a region R must have all maxima and minima on the boundary of R and none within R itself.

This is simply another way of stating the now familiar property of a potential: A function can have no maxima or minima within a region in which it is harmonic.

The definition of the second derivative of a one-dimensional function demonstrates another important property of a harmonic function. Field lines for J describe the pattern and direction of heat transfer. The converse is not necessarily true. The total heat in region R is given by Tdv. The divergence theorem Ap- pendix A can be used to convert the surface integral into a volume. Heat flow J through a region R containing no heat sources or sinks.

The change in total heat within R must equal the net flow of heat across boundary 5. Consider the free flow of heat in and out of a region R bounded by surface 5.

Region R is bounded by surface S. If the integrand. On the basis of previous dis- cussions. This is a reasonable result. If all heat sources and sinks lie outside of region R and do not change with time. The temperature distribution accompanying steady-state transfer of heat is an easily visualized example of a harmonic function. Describe how the temperature of the rod changes with time. Sets that contain only interior points are called open regions.

First we need some definitions. In the following. We will have considerably more to say about this subject in later chapters. The coordinate system represents the complex plane. After all. The real and imaginary parts of a complex function are harmonic in regions where the complex function is analytic.

Is the temperature harmonic? Finding a solution to Laplace's equation. Suddenly the two ends are switched so that the hot end is in ice water and the cold end is in boiling water.

An interior point of a set of points has some neighborhood containing only points of the set. For additional information about complex functions.

The derivative of a complex function requires special consideration. In the complex plane. Complex functions can be written in terms of their real and imaginary parts.

The Cauchy-Riemann conditions provide an easy way to determine whether such conditions are met. In this latter case. In order for a real function f x to have a derivative. The derivative of the complex function is given by dw du. The two-dimensional Laplacian of its real part u x. If all mass lies interior to a closed equipotential surface S on which the potential takes the value C.

De- scribe the temperature at all points of the sphere if the temperature is harmonic throughout the sphere and depends only on the distance from its center.

As a crude approximation. The physical properties of a spherical body are homogeneous. You are monitoring the magnetometer aboard an interstellar space- craft and discover that the ship is approaching a magnetic source described by a Remembering Maxwell's equation for B. Two distributions of matter lie entirely within a common closed equipotential surface C. Show that all equipotential surfaces outside of C also are common.

Prove that the intensity of a conservative force field is inversely proportional to the distance between its equipotential surfaces. If the lines of force traversing a certain region are parallel. Prove the following relationships: Explain your answer? Assume a spherical coordinate system and let r be a vector directed from the origin to a point P with magnitude equal to the distance from the origin to P.

It was asserted that such po- tentials satisfy Laplace's equation at places free of all sources of F and are said to be harmonic. They are referred to as Green's identities. Paul Davies Only mathematics and mathematical logic can say as little as the physicist means to say. Bertrand Russell In Chapter 1.

In the same spirit. This led to several important characteristics of the potential. He is perhaps best known for his paper. The boundary of R is surface. Let U and V be continuous functions with continuous partial deriva- tives of first order throughout a closed. Several very interesting theorems result from Green's first identity if U and V are restricted a bit further.

It also can be shown Kellogg [ Equation 2. Surface S bounds region R. Unit vector n is outward normal at any point on S. Suppose that vector field F has a potential U which is harmonic throughout some region. Region i? In other words. The right-hand side vanishes and. A region will eventually reach thermal equilibrium if heat is allowed to flow in and out of the region.

It seems reasonable that. Stokes's theorem makes intuitive sense when applied to steady-state heat flow. U also must vanish at all points of R. The function U1 — U2 also must be harmonic in R. Green's first identity leads to a statement about uniqueness. If region R is in thermal equilibrium and contains no heat sources or sinks. This result is intuitive from steady-state heat flow.

U must be a constant. If temperature is zero at all points of a region's boundary and no sources or sinks are situated within the region. The uniqueness of harmonic functions also extends to mixed boundary- value problems. These last theorems relate to the Neumann boundary-value problem and show that such solutions are unique to within an additive constant. Exercise 2. If the boundary of R is thermally insulated. Inter- ested readers are referred to Chapter XI of Kellogg [ We have shown that under many conditions Laplace's equation has only one solution in a region.

A similar proof could be developed to show that if U is single-valued. But can we say that even that one solution always exists? The answer to this interesting question requires a set of "existence theorems" for harmonic functions that are beyond the scope of this chapter.

This relationship will prove useful later in this chapter in discussing certain kinds of boundary-value problems.

Derivation of Green's third identity. We use the relationships. The last integral of equation 2. As the sphere becomes arbitrarily small. Point P is inside surface S but is excluded from region R. Angle dQ. First consider the integral over a Figure 2.

This equation is called the representation formula Strauss []. In Chapter 3 equation 3. Then equation 2. An important consequence follows from Green's third identity when U is harmonic. But remember that no physical meanings were attached to U in deriving Green's third identity. U was only required to have a sufficient degree of continuity.

We will show in Chapter 5 equation 5. Green's third identity demonstrates an important limitation that faces any interpretation of a measured potential field in terms of its causative. Green's third identity shows. If a is the radius of the sphere. Likewise equation 2.

It was shown earlier that a harmonic function satisfying a given set of Dirichlet boundary conditions is unique. Suppose that E contains at least one interior point of R. This relationship is called Gauss's theorem of the arithmetic mean. E cannot equal the total of R because we have stated that U is not constant. IfU is harmonic in region R.

Because Po is interior to R. If U is harmonic in a region R. This property of nonuniqueness will be a common theme in following chapters. By the definition of a.

The proof is by contradiction. It can be shown that if E has one point interior to R it also has a frontier point interior to R. Now we are in a position to prove it. We discussed the maximum principle by example in Section 1. It follows that the potential within any subregion of R can be related to an infinite variety of surface distributions. But PQ is also a member of E. Region R includes a set of points E at which U attains a maximum. Point PQ is a frontier point of E. A contradiction arises.

These concepts are a subset of the Helmholtz theorem Duff and Nay lor [81] which states that any vector field F that is continuous. With W defined as in equation 2. Each of the three cartesian components of W has a form like 2J4 l At this point. Poisson's equation: A vector identity Appendix A shows that V 2 W can be represented by a gradient plus a curl.

Comparing this result with equations 2. If F is continuous and vanishes at infinity. For convenience. This should be possible because of the way ft and A were defined. The Helmholtz theorem is useful. Because the integral holds for any closed surface within the region. Such fields have no vorticity or "eddies. Irrotational Fields A vector field is irrotational in a region if its curl vanishes at each point of the region.

Examples of irrotational fields are common and include gravitational attraction. In addition to this statement. If both the divergence and curl vanish at all points. The last term of this equation vanishes identically Appendix A. From the Helmholtz theorem. A physical meaning for solenoidal fields can be had by integrating the divergence of F over any volume V within the region. Or put another way.

It was stated in Section 2. To see this. We will return to this equation in Chapter 5 and subsequent chapters. This is called the forward prob- lem when applied to the geophysical interpretation of measured magnetic fields. This fact plus equation 2. Another of Maxwell's equations states that magnetic induction has no divergence. One of Maxwell's equations relates magnetic induction B and magnetization M in the absence of macroscopic currents: As soon as the force returns to zero.

A heuristic approach will be used. To find this velocity. T T The first integral can be ignored if Ar is small and the particle has some mass. One conceptual way to solve equation 2. The response of the particle to each blow should be independent of all other blows. Combining this result with equation 2.

Given this property. In the limit as AT approaches zero. Many mechanical and electrical systems and.Two loops of electric current Ia and h separated by a distance r. Note that gauss and oersted units have equivalent magnitude and dimensions in the emu system; the first quantity is used for magnetic induction, the latter for magnetic field intensity.

The test particle could be a small mass m acted upon by the gravitational field of some larger body or an electric charge moving under the influence of an electric field.

Point P is inside surface S but is excluded from region R. An important consequence follows from Green's third identity when U is harmonic. Distance between maximum and minimum values of Bx.