Michael S. Zhdanov

Chapter 6 – Electromagnetic Fields in Inhomogeneous Media

There are several techniques available for electromagnetic (EM) forward modeling in inhomogeneous media. They are based on numerical implementation of the differential equation (DE) approach (finite difference, FD, or finite element, FE, methods) or the integral equation (IE) approach. In this chapter, I discuss the principles of all these methods. The integral equation (IE) method is a powerful tool in electromagnetic (EM) modeling for geophysical applications. We derive the fundamental equations of the IE method in two and three dimensions and consider the methods of their solution in isotropic and anisotropic media. An effective approach to solve the system of integral equations is based on application of the contraction operator, which can be treated as an effective preconditioner of the original system of equations. We also consider a family of linear and nonlinear integral approximations of the EM fields, based on the IE formulation of Maxwells equations. Another powerful group of methods of numerical modeling of EM fields uses differential equation methods. We discuss in detail the most important of these methods, the finite difference (FD) and finite element (FE) methods. The finite difference method provides a simple but effective tool for numerically solving the electromagnetic boundary-value problem. We present a discretization of Maxwells equations using a staggered grid, and introduce a contraction preconditioner for a system of FD equations. Finally, this chapter concludes with the exposition of the most powerful technique for numerical modeling – the method of finite elements.

Airborne Electromagnetic Methods

Airborne electromagnetic methods provide the means for carrying geophysical surveying rapidly over a large area to be explored. The first practical application of the airborne electromagnetic (AEM) method was made as early as in 1951. Since then, AEM can be credited with directly aiding in the discovery of hundreds mineral deposits. Many different frequency- and time-domain AEM systems have been developed over the last 75 years. The theoretical foundation of these methods is the same as has already been discussed for surface-based applications. In Chapter 16, we discuss AEM methods mainly in terms of the operational features peculiar to the use of aircraft. We consider the principles of a typical frequency domain airborne EM survey. We also describe a typical time domain airborne EM system which uses an alternating polarity half-sine or square-wave pulses to energize a large, vertical-axis transmitter loop, which is stretched over the wing tip to tail around the aircraft. The airborne platform enables collecting a huge amount of data about the electrical properties of the earth. However, interpreting the massive amounts of data gathered poses a significant challenge. To overcome these difficulties, we present the concept of 3-D AEM inversion with a moving sensitivity domain (MSD), which makes it possible to invert entire AEM surveys with no approximations into high resolution 3-D earth geoelectrical models. Chapter 16 is concluded with a description of the AEM methods based on measuring the field generated by a ground-based transmitter or natural magneto fields. An example of the later method is the z-axis tipper electromagnetic method or ZTEM.

Magnetotelluric and Magnetovariational Methods

In this Chapter, we discuss a group of methods for determining the electrical structure of the earth using naturally existing electromagnetic fields. Two principal forms of such methods are the MagnetoTelluric (MT) method and the MagnetoVariational (MV) method. The foundations of the magnetotelluric method are based on the Tikhonov-Cagniard model of a plane wave vertically propagating in the layered earth. We outline the theory of the MT and MV methods and introduce the concept of the transfer functions and magnetotelluric operators: impedance and admittance, telluric and magnetic. The other important transfer functions considered in Chapter 13 are induction vectors and magnetic and electric tippers. The properties of the magnetotelluric fields in inhomogeneous media are discussed as well. The Chapter is concluded with the sections dedicated to qualitative and quantitative interpretation of the MT and MV data including rigorous MT impedance and phase tensors inversion.

Marine Electromagnetic Methods

Geophysical electromagnetic methods were originally introduced for land observations only. However, it was discovered in the mid-twentieth century that various electromagnetic methods can be used quite effectively on and in the World Ocean. Marine EM methods are used both in academic applications to study the deep structure of the earth crust and upper mantle, and in exploration for mineral resources. In this Chapter we discuss marine magnetotelluric and controlled source methods. The methodic of the marine magnetotelluric (MT) method is similar to that used on the land. The geoelectrical study at the sea bottom, however, is not the same as geoelectrical studies conducted on land. These differences result in different techniques for measuring the EM field in the seawater and in different behaviors of the observed electromagnetic fields in comparison with land observations, which are examined in this Chapter. We also discuss marine controlled source electromagnetic (MCSEM) methods. There are two major survey configurations of the MCSEM methods. One is based on the fixed sea-bottom receivers and a towed electric field transmitter. Another widely used marine EM method is based on the data acquisition system consisting of an electric bipole transmitter and an array of electric field receivers towed by a vessel. Both methods are described and analyzed in this Chapter.

Electromagnetic Methods in the Frequency and Time Domains

A major category of geoelectrical methods comprises the controlled-source electromagnetic (CSEM) methods. These differ from the DC methods described in Chapter 13 in that the parameters of the source of the electromagnetic field can be specified (in other words, we use a controlled source). These two differences make the discussion of the behavior of the fields more difficult. In this Chapter, we examine the principles of the CSEM methods both in the frequency and time domains. The concepts of apparent resistivity and electromagnetic sounding is introduced. A fast method of interpretation of the controlled-source time domain data using the thin-sheet approach is discussed as well. The Chapter is concluded with a summary of the principle survey configurations of the CSEM methods used in geophysical exploration.

Foundations of Field Theory

This chapter discusses the general principles of geophysical field theory and of field generations. It begins with the introduction of harmonic functions and Liouvilles theorem of the uniqueness of the harmonic function. We show that there can exist only two types of field excitation – sources and vortices. Source is a type of field generator that produces nonzero divergence of the vector field. Vortex is a type of field generator that produces a nonzero curl. The general physical meaning of a source or a vortex of a field is analyzed. The theory of field generation provides a basis for a classification of the vector fields. The concepts of the point source and Dirac singular functions are introduced as well. We also discuss the fundamental Greens function and its application for the solution of the Laplace equation. The last section of this chapter presents an introduction to the theory of differential forms and their application to the analysis of basic equations of nonstationary (time-dependent) vector fields.

Direct Current and Induced Polarization Methods

This Chapter examines in detail the first class of electrical geophysical methods – the DC methods. While both electromagnetic and DC methods first came into use in the early part of the 20th century, the DC methods gained early acceptance because of less demanding theoretical and instrumentation considerations. DC methods have become the most widely used geoelectric method. We discuss three important groups of DC electric techniques including those known as vertical electric sounding (VES), horizontal profiling, and the electrical mapping methods. A highly important time-varying electrical method called Induced Polarization (IP) method is presented as well. In this method, it is recognized that the current flow in the earth represents a very complex EM phenomenon characterized by charge polarization and accumulation in the rocks. Mathematically, the IP phenomena can be analyzed based on models with frequency-dependent complex conductivity distribution, e.g., a Cole-Cole model. We also consider a generalized effective-medium theory of induced polarization (GEMTIP) which was introduced recently in order to provide a link between the petrophysical properties of the rocks and their complex conductivity spectra

Electromagnetic Properties of Rocks and Minerals

The physical properties of rocks, which affect the propagation of the electromagnetic field, are electrical conductivity, dielectric permittivity, and magnetic permeability. In this chapter, we discuss these physical properties from three viewpoints: 1) the physical phenomena which cause the properties to behave as they do, 2) the values of these physical properties for rocks as they exist in the earth, and 3) the way in which these properties respond to environmental conditions and how they correlate with other physical and geological properties of rocks. We note also that, the electrical properties depend on many factors, some environmental, like temperature and pressure, and others which reflect the character of the rock, such as composition. We also introduce a phenomenon called induced polarization, associated with the current flow in the rocks. It is important to have a quantitative method, which would allow us to calculate the bulk conductivity of a complex rock formed by a mixture of conductive minerals and a host rock. This problem is solved using the methods of effective-medium theory. The properties of large-scale geoelectrical structures of the Earth, including the oceans and the atmosphere, are summarized as well.

Principles of Ill-Posed Inverse Problem Solution

The forward problem of electromagnetic modeling is almost always well posed. Given an earth model, we are certain that there is a single, unique solution of Maxwells equations which describes the EM field behavior. In contrast, in the inverse problem, while we may be almost certain that there is a unique physical inverse solution, widely differing numerical geoelectric models may be found, which may cause almost exactly the same electromagnetic field behavior. Moreover, the accuracy with which we observe the electromagnetic field behavior is contaminated to some extent by noise and measurement error. However, even in the case of very accurate observations, the inversion results may have arbitrarily large differences from one another. This lack of uniqueness on the practical level makes inverse solutions difficult if not impossible in some cases. For this reason, the inverse problem as usually encountered in electromagnetic methods is said to be ill posed. In this chapter, we discuss the principles of the regularization theory, which provides the foundations for the solution of the ill-posed inverse problems. We begin our discussion with the mathematical formulations of the well-posed and ill-posed problems. We then introduce stabilizing functionals and regularizing operators and formulate the principles of Tikhonov regularization for the construction of a solution of an ill-posed inverse problem.

Chapter 9 – Electromagnetic Migration

The conventional approach to interpretation of the observed electromagnetic data based on standard 3-D forward modeling and inversion meets significant difficulties because of the enormous amount of computations required in the case of the multitransmitter and multireceiver data acquisition systems typical for modern EM geophysical surveys. There exists, however, an alternative approach to the solution of this problem, called EM migration, based on the principles of EM holography, which extends to EM cases the methods of optical and radio holography. Electromagnetic migration is based on a special form of downward continuation of the observed field or one of its components. This downward continuation is obtained as the solution of the boundary value problem in the lower half-space for the adjoint Maxwells equations, in which the boundary values of the migration field on the earths surface are determined by the observed electromagnetic field data. Chapter 9 presents an overview of the principles of EM migration in the time and frequency domains. We also discuss the numerical methods of migration including the methods based on integral transformation, digital filters, and finite difference approach. The final section of this chapter presents an iterative migration, which is similar to the conventional iterative inversion, discussed in Chapter 8.