To comprehend the bases and the interpretational techniques of electrical prospecting methods, requires first a knowledge of the tools of electromagnetic theory. The ability to solve a boundary-value problem in electromagnetic theory then becomes the objective. All electromagnetic phenomena are governed by the empirical Maxwell's equations; we must start with them. Maxwell's equations are uncoupled first-order linear differential equations but can be coupled by the empirical constitutive relations which reduce the number of basic vector field functions from five to two. Care must be taken in selecting the form of the constitutive relations pertinent to the earth. In particular, for most earth problems, we assume isotropy, homogeneity, linearity, and temperature-time-pressure independence of the electrical parameters of local regions of the earth. A more complicated earth model is formed by juxtaposition of several such regions.A concept fundamental in electricity and magnetism is that of magnetic and electric polarization. We introduce the polarization vector in a symmetrical sense which demands recognition of, at least, the theoretical existence of magnetic current density and magnetic monopoles, as well as electric current density and electric monopoles. Maxwell's equations are generalized and made more symmetric in the process with a conceptual advantage afforded when we come to deal with scalar and vector potentials. Total vector polarization functions, consisting of the algebraic sums of induced and source parts, frequently facilitate description of a physical problem.Many boundary-value problems can be solved in terms of the vector electric and magneticfield intensity functions. In an earth composed of several juxtaposed homogeneous isotropic linear regions, a solution of a wave equation is postulated for each region. These solutions must be matched at every boundary according to prescribed boundary conditions on two vector field functions or potentials. The wave equation used for each region is derived directly 132 Ward and Hohmann from Maxwell's equations and really represents a compact form of these equations. If the wave equation is homogeneous, no sources are present in that region. If the wave equation is inhomogeneous, sources exist in that region. Often a boundary-value problem is difficult to solve in terms of vector field functions andis easier to solve in terms of vector and/or scalar potential functions from which the vector field functions may be derived. Several different sets of potential functions appear in the literature; we use the Schelkunoff potentials because of their symmetry and because of the ease of relating them to the TE and TM modes of excitation. Any electromagnetic field, in a homogeneous, source-free region, may be decomposed into a part for which the electric field is transverse to some axis (TE mode) and a part for which the magnetic field is transverse to this same axis (TM mode). Mode decomposition simplifies the solution of boundary-value problems. The development of a se...

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response varies significantly with conductivity contrast; the peak response occurs at higher contrasts for two‐dimensional bodies than for bodies of limited length. Very conductive bodies can produce negative IP response due to EM induction. An electrically polarizable body produces a small magnetic field, so that it is possible to measure IP with a sensitive magnetometer. Calculations show that horizontal loop EM response is enhanced when the background resistivity in the earth is reduced, thus confirming scale model results.

An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homoge We have developed a finite-difference solution for neous Dirichlet condition is employed along the subsur three-dimensional (3-D) transient electromagnetic face boundaries. Numerical dispersion is alleviated by problems. The solution steps Maxwell's equations in using an adaptive algorithm that uses a fourth-order dif time using a staggered-grid technique. The time-step ference method at early times and a second-order meth ping uses a modified version of the Du Fort-Frankel od at other times. Numerical checks against analytical, method which is explicit and always stable. Both integral-equation, and spectral differential-difference so conductivity and magnetic permeability can be func lutions show that the solution provides accurate results. tions of space, and the model geometry can be arbi Execution time for a typical model is about 3.5 trarily complicated. The solution provides both elec hours on an IBM 3090/600S computer for computing tric and magnetic field responses throughout the earth. the field to 10 ms. That model contains 100 x 100 x 50 Because it solves the coupled, first-order Maxwell's grid points representing about three million unknowns equations, the solution avoids approximating spatial and possesses one vertical plane of symmetry, with derivatives of physical properties, and thus overcomes the smallest grid spacing at 10 m and the highest many related numerical difficulties. Moreover, since resistivity at 100 n . m. The execution time indicates the divergence-free condition for the magnetic field is that the solution is computer intensive, but it is valu incorporated explicitly, the solution provides accurate able in providing much-needed insight about TEM results for the magnetic field at late times. responses in complicated 3-D situations.

by a large rectangular loop is substantial when host currents are strong near the conductor. The more con ductive the host, the longer the galvanic responses will The three-dimensional (3-D) electromagnetic scatter persist. Large galvanic responses occur if a 3-D conduc ing problem is tirst formulated in the frequency domain tor is in contact with a conductive overburden. For a in terms of an electric field volume integral equation.thin vertical dike embedded within a conductive host, Three-dimensional responses are then Fourier trans the 3-D response is similar in form but differs in mag formed with sine and cosine digital filters or with the nitude and duration from the 2-D response generated decay spectrum. The digital filter technique is applied to by two infinite line sources positioned parallel to the a sparsely sampled frequency sounding, which is re strike direction of the 2-D structure.placed by a cubic spline interpolating function prior to We have used the 3-D solution to study the appli convolution with the digital filters. Typically, 20 to 40 cation of the central-loop method to structural interpre frequencies at five to eight points per decade are re tation. The results suggest variations of thickness of quired for an accurate solution. A calculated transient is conductive overburden and depth to sedimentary struc usually in error after it has decayed more than six ture beneath volcanics can be mapped with one orders in magnitude from early to late time. The decay dimensional inversion. Successful 1-D inversions of 3-D spectrum usually req uires ten frequencies for a satisfac transient soundings replace a 3-D conductor by a con tory solution. However, the solution using the decay ducting layer at a similar depth. However, other pos spectrum appears to be less accurate than the solution sibilities include reduced thickness and resistivity of the using the digital filters, particularly after early times.(-0 host containing the body. Many different l-D Checks on the 3-D solution include reciprocity and con models can be fit to a transient sounding over a 3-D vergence checks in the frequency domain, and a com structure. Near-surface, 3-D geologic noise will not per parison of Fourier-transformed responses with results manently contaminate a central-loop apparent resistivi from a direct time-domain integral equation solution. ty sounding. The noise is band-limited in time and even The galvanic response of a 3-D conductor energized tually vanishes at late times.

The electromagnetic fields scattered by a three‐dimensional (3-D) inhomogeneity in the earth are affected strongly by boundary charges. Boundary charges cause normalized electric field magnitudes, and thus tensor magnetotelluric (MT) apparent resistivities, to remain anomalous as frequency approaches zero. However, these E‐field distortions below certain frequencies are essentially in‐phase with the incident electric field. Moreover, normalized secondary magnetic field amplitudes over a body ultimately decline in proportion to the plane‐wave impedance of the layered host. It follows that tipper element magnitudes and all MT function phases become minimally affected at low frequencies by an inhomogeneity. Resistivity structure in nature is a collection of inhomogeneities of various scales, and the small structures in this collection can have MT responses as strong locally as those of the large structures. Hence, any telluric distortion in overlying small‐scale extraneous structure can be superimposed to arbitrarily low frequencies upon the apparent resistivities of buried targets. On the other hand, the MT responses of small and large bodies have frequency dependencies that are separated approximately as the square of the geometric scale factor distinguishing the different bodies. Therefore, tipper element magnitudes as well as the phases of all MT functions due to small‐scale extraneous structure will be limited to high frequencies, so that one may “see through” such structure with these functions to target responses occurring at lower frequencies. About a 3-D conductive body near the surface, interpretation using 1-D or 2-D TE modeling routines of the apparent resistivity and impedance phase identified as transverse electric (TE) can imply false low resistivities at depth. This is because these routines do not account for the effects of boundary charges. Furthermore, 3-D bodies in typical layered hosts, with layer resistivities that increase with depth in the upper several kilometers, are even less amenable to 2-D TE interpretation than are similar 3-D bodies in uniform half‐spaces. However, centrally located profiles across geometrically regular, elongate 3-D prisms may be modeled accurately with a 2-D transverse magnetic (TM) algorithm, which implicitly includes boundary charges in its formulation. In defining apparent resistivity and impedance phase for TM modeling of such bodies, we recommend a fixed coordinate system derived using tipper‐strike, calculated at the frequency for which tipper magnitude due to the inhomogeneity of interest is large relative to that due to any nearby extraneous structure.