Introduction Electromagnetic methods have a special place in the arsenal of geophysical tools available for environmental investigations. They owe their status to five factors. First, many environmental investigations are concerned with detecting the location of contaminants in groundwater and identifying pathways for contaminant transport. These types of problems boil down to learning as much as possible about the fluids in the pore spaces. Contaminants can modify both the electrical conductivity and dielectric constant of the pore fluid, and electromagnetic and electrical methods are the only geophysical methods that are directly influenced by the electrical properties of pore fluids. Second, electromagnetic techniques are sensitive to changes in geology such as rock type, porosity, grain size, fractures, and clay content. This sensitivity allows electromagnetic methods to be used for geologic mapping, which is of great value in developing geological and hydrological models needed for environmental studies. Third, there is a wide range of electromagnetic equipment available for purchase or rental. Over the past 30 years companies such as Geonics, Zonge Engineering, GeoInstruments, Geophex, Iris Instruments, Geosystems, and others have developed modern, reliable instrumentation making electromagnetic surveys relatively easy to perform. Hopefully, this new found ease of data collection does not lull the naïve and untrained into thinking that these methods are without pitfalls. Fourth, electromagnetic techniques are noninvasive. While drilling is often necessary to unequivocally confirm an interpretation, drilling can worsen an environmental problem by puncturing an intact, buried container, or by producing a high permeability flow path for further contaminant migration. Finally, compared to other geophysical methods and to drilling, electromagnetic techniques are relatively inexpensive due to their ease of use and low labor requirements.

The magnetotelluric (MT) method, pioneered by Tikhonov (1950) and Cagniard (1953), has been applied with varying degrees of success to yield information about the Earth's electrical conductivity, temperature regimes, and geologic structure. The strengths and weaknesses of the MT method are now generally appreciated and the majority of research over the past few years has concentrated on data processing, forward modeling and inversion. Historically, MT interpretation has used one-dimensional (1-D) forward modeling algorithms, 1-D inversions, and 2-D forward modeling. Algorithms for 2-D inversion and 3-D forward modeling are just now reaching maturity. Although all researchers recognize that the Earth is geologically complicated,1-D inversion is still an important source of information. First, there are geologic regions where the lateral variation is small and therefore a 1-D interpretation is directly applicable. Second, 1-D inverse solutions provide good starting models for 2-D inversion algorithms. Third, a thorough understanding of the 1-D inverse problem, especially the nonuniqueness of the solution, provides a valuable foundation for solving inverse problems in higher dimensions. Many years ago, some believed that the 1-D inverse problem was adequately solved and that further research would yield few additional benefits. Higher dimensional solutions were thought to be the only remaining frontier. Fortunately, the 1-D problem was further explored, and valuable insight into the nature of the electromagnetic induction problem was developed. For example, the excellent paper by Parker (1980) showed that a conductivity structure producing the least-squares fit to the observations consisted of an infinitely resistive Earth interleaved with a set of infinitely conducting sheets having finite conductance.

An MT response function derived from vertically incident, plane electromagnetic waves diffusing into a 1-D conductivity σ ( z ) must satisfy certain mathematical constraints. In practice, these theoretical constraints may be violated because measurement and processing of field records yield responses contaminated with errors. The constraints may also be violated because of the discrepancy between the idealized mathematical representation and the true physical situation. For example, in most areas the real Earth conductivity is 3-D, and the source fields are a complex superposition of time-varying waves propagating in different directions. Therefore, when interpreting MT responses in terms of a 1-D σ ( z ) model, it is important to answer first the question of existence; that is, does there exist a σ ( z ) profile which reproduces the observations? The question of existence is equivalent to determining whether or not the observations satisfy the theoretical 1-D constraints dictated by the mathematical model. We discuss first the mathematical form of these theoretical constraints. The dispersion relations and inequalities are good theoretical tests for existence, but they are difficult to apply in practice because field data are discretely sampled within a finite bandwidth. The integrals and derivatives required to implement these existence tests cannot be calculated precisely. Weidelt (1986) overcame this difficulty by deriving a set of existence tests which do not involve derivatives. The conditions require the positivity of 2N determinants derived from the discrete data measured at N frequencies. Importantly, these constraints are necessary and sufficient conditions for the existence of a 1-D conductivity model. Parker (1983) showed that the dispersion relations and inequalities are necessary but not sufficient conditions. Parker (1980) also examined the conditions on the response required for the existence of a solution by using a linear programming formulation.

Tikhonov (1965) was the first to consider the uniqueness theorem for the 1-D MT inverse problem. He proved that different, piecewise analytic, conductivity profiles correspond to different response functions, and also indicated that the converse may be proved in a similar manner. He therefore established a one-to-one relationship between every complete data set and a distinct conductivity profile. Hence, surface measurements are, in principle, sufficient to determine the true Earth σ ( z ). In fact, responses known precisely on any open interval ω 1 < ω < ω 2 , or at an infinite number of equally spaced frequencies are sufficient to uniquely define σ ( z ) (Parker, 1983). Other uniqueness proofs, valid for different types of conductivity models, also exist. Bailey (1970) proved a uniqueness theorem for the class of nonzero, bounded, infinitely differentiable conductivity functions. He showed that complete knowledge of a single surface harmonic mode is sufficient to give the unique σ (T) of a radially symmetric Earth. MacBain and Bednar (1986) extended Bailey's work to prove that c(w) can be uniquely inverted when σ (z) is from the class of all homogeneous layered models, all piecewise polynomials, and all piecewise finite trigonometric-series functions. Loewenthal (1975) proved that models consisting of homogeneous, isotropic, parallel layers each with equal field attenuation are uniquely determined by the surface impedance. MT uniqueness theorems assume that an infinite number of responses are known precisely. In contrast, for all practical data sets consisting of a finite number of inaccurate data the inverse problem is always nonunique. The nonuniqueness arises for two reasons. First, there are an infinite number of ways to interpolate and extrapolate a finite number of accurate data, and each such realizable response corresponds to a different conductivity profile. Second, since any practical algorithm must permit misfits to the noisy observations, errors in the data make the space of acceptable interpolations even larger.

Asymptotic methods are approximate inversion schemes which consider various limiting or approximate forms of the governing MT equations. All methods exploit the fact that low frequencies penetrate more deeply into the Earth than high frequencies, and therefore, successively lower frequencies determine successively deeper conductivities. Unfortunately, some of the approaches do not utilize the phase information. Moreover, because many do not incorporate the data errors the constructed σ(z) may not adequately fit all of the observations. Despite their heuristic nature and innate deficiencies, these methods are easily programmable and can therefore be used in the field to obtain almost immediate information about the conductivity structure. Also these approaches are still used routinely in more sophisticated multidimensional interpretations and sometimes do generate conductivities which fit the data. Lastly, since one of our goals is to illustrate the numerous ways in which the MT inverse problem can be attacked, it is especially interesting to examine these methods in detail because many of them yield essentially the same end formula even though the derivation is considerably different. For instance, asymptotic analysis for several layers over a half-space, approximation of the induction equations, and low frequency correspondence with de resistivity all approach the MT problem from a different viewpoint. The fact that a similar algorithm is produced in each case is a strong testimonial to the overwhelming smoothing of the electromagnetic fields as they diffuse into the Earth. Physical understanding of this diffusion provides a firm foundation upon which to build a simple algorithm to yield information about the subsurface conductivity.

Two fundamental deficiencies of the asymptotic methods of Chapter 4 are that they do not incorporate observational errors directly in the inversion, and they cannot guarantee that the constructed model reproduces the observations. To correct these problems, numerous authors have linearized the MT equations and developed algorithms which iterate to an acceptable σ(z). Unfortunately, as shown in the excellent paper by Parker (1980), the conductivity with the least-squares misfit is the geophysically unreasonable D + delta function model. Although other models may exist which fit the data equally well, the fact that a D + model always achieves the minimum χ 2 misfit is of fundamental importance to iterative algorithms which attempt to find a model that reproduces the data. D + solutions imply that the iterative formulation is inherently unstable if considered in terms of misfit alone, and our conjecture is that linearized algorithms will naturally tend to an oscillatory model which in some sense resembles a D + solution. Such oscillatory models are generally undesirable, and consequently artificial stabilization or smoothing must be applied. This stabilization may hinder convergence. For linearized approaches, additional problems are the choice of starting model and the sensitivity of the final σ( z) to the starting model. Anderssen (1975) discussed some of the hazards of linearization such as oscillating iterations and convergence to an incorrect result. He concluded that linearization is only appropriate if the starting model is sufficiently close to a possible solution. Nevertheless, despite these difficulties, the lure of linear equations has resulted in numerous linearized and iterated MT inversion schemes. Many of these schemes work exceedingly well. The first linearized approach discussed in Section 5.1 is the Born approximation of Coen et al. (1983). These authors solved a linearized integral equation written in terms of the normalized electric field. Section 5.2 shows that the inversion method of Schmucker (1972) described by Larsen (1975) is quite similar to a Born inverse. This method is derived from the layer to layer recursion relation of a response approximately equal to r(w). Larsen (1981) extended this method and made it more precise. The parametric inversion discussed in Section 5.3 is used extensively to interpret MT data in terms of a layered σ (z). Finally, Section 5.4 presents the Frechet derivative approach of Oldenburg (1979). This linearization is a continuous version of the parametric method, and it constructs smooth σ (z) models.

The linearized approach discussed in Section 5.4 constructs models built up by adding perturbations to the current best estimate of σ(z). The method minimizes a penalty functional of the perturbation. For example, the "small" penalty functional is the integrated square of the perturbation values, and the "fiat" functional is the integrated square of the first derivatives of the perturbations with respect to depth; however, these do not require the final σ(z) to be smallest or flattest in any sense. Clearly, it is advantageous to reformulate the inverse problem so that the Fredholm equation of the first kind to be solved has the model in the integrand rather than a perturbation of the model. We call penalty functionals of the entire model "global" to contrast them with the "local" functionals which involve a model perturbation. In the global case, the arsenal of linear inverse theory techniques are applied to the equation to compute weighted smallest or flattest models, as well as models deviating least from a given σ ( z ) profile. These minimum structure solutions are valuable since they are less likely to have spurious features to mislead an interpreter. Interesting features of such solutions are hopefully essential characteristics required by the data, and not just artifacts of noise or the inversion method. An important benefit of reformulating the equations to optimize directly a global functional of the total model is that additional constraints on the conductivity are straightforward to incorporate. For instance, the conductivity may be known to be bounded between two functions, or the average value of conductivity over a depth range may be known to within a given uncertainty. Conductivity models which optimize a global penalty functional, satisfy the observations, and satisfy additional imposed constraints are more likely to be representative of the true Earth. The negative aspect of the optimization is the iterative solution required because the kernels are model dependent. Consequently, there is concern about convergence of the algorithm and the possibility of being trapped in a local minimum. There is no way of determining analytically or numerically whether the true global minimum has been achieved. If there is doubt, the pragmatic approach is to repeat the inversion beginning with different starting models and different partitions of the depth axis. In most cases explored by us, the algorithms converged to the same conductivity model and thus provided optimism that the global solution had been found.

The random search, or Monte-Carlo, method has been used by many interpreters to construct conductivity models satisfying MT data. The method guesses a σ ( z ) profile and uses forward modeling to test its fit to the observed data. If the fit is acceptable, the proposed model is saved, otherwise the model is rejected and a new σ ( z ) is randomly selected. Monte-Carlo methods require no approximations because forward modeling is a simple, direct calculation. Moreover, by finding a large collection of acceptable yet diverse σ ( z ), the method explores to some extent the nonuniqueness of the problem. The disadvantage of Monte-Carlo methods is that the time and expense required to propose and test thousands of σ ( z) models can be prohibitive. A finite search can never completely explore the infinite-dimensional space of acceptable models. For L layers and M possible choices of σ for each layer, the total number of possible models is M L . To reduce the possibilities, the number of layers is usually limited to less than ten, and bounds are placed on the range of permissible layer conductivities and thicknesses. If these restrictions are well-founded then the method has a better chance of finding a model that is close to reality. Otherwise, arbitrarily limiting the search of model space to a smaller region decreases the chances of a correct interpretation. Thus, the method is heavily dependent on the model parameterization.

Despite the nonlinearity of the MT equations, several exact methods exist which calculate a σ ( z ) consistent with the data. For these approaches there is no need for any approximations other than those required for computer implementation. The solution is not iterative so no starting model is required and there are no convergence problems; however, these algorithms are not entirely without problems. Some do not incorporate the data errors, others have numerical solutions which are unstable, and still others may not guarantee a positive σ ( z ) model. All exact methods subdivide into at least two stages. The first major stage is to complete the measured data somehow to obtain realizable responses at all frequencies. The second stage maps these completed responses to a unique conductivity profile. The different approaches to the completion and mapping problems account for the different inversion methods. Some techniques are more sensitive to noise than others. We have divided the exact methods into rough categories based on their completion and mapping schemes; however, there are overlaps between categories in many cases. Bailey (1970,1973) derived an exact inversion scheme for finding a radially symmetric conductivity σ (r). As a response, the method uses the frequency-domain ratio of the induced to the inducing magnetic field of any spherical harmonic mode. This response satisfies a Riccati equation as well as dispersion relations which guarantee causality. Integrating the Riccati equation over frequency and using the causality condition gives the conductivity in a shell adjacent to the level at which the responses are known. The Riccati equation can then be used to downward continue the responses through this layer. These new responses define the next deeper conductivity value, and the process is repeated. Importantly, Bailey showed that this inversion technique is unique for the class of all nonzero, bounded, infinitely differentiable σ ( r). That is, there is only one conductivity profile which keeps the responses causal at all radii.

In the preceding sections, we discussed many techniques which construct σ ( z ) profiles using a limited number of inaccurate MT data. For these practical data, the inverse problem is nonunique. Even for the wide-band test data used here, the range of models constructed is large (see Figures 17 and 18 for example). The vast space of acceptable solutions contains delta function, piecewise constant, and infinitely differentiable σ ( z ). The interpreter must grapple with this nonuniqueness in order not to be misled by a model with features not required by the data. As Jackson (1973) stated, it is essential not only to find a solution, but to represent in a meaningful way the degree of nonuniqueness permitted by the data. In the preceding sections, we discussed many techniques which construct σ ( z ) profiles using a limited number of inaccurate MT data. For these practical data, the inverse problem is nonunique. Even for the wide-band test data used here, the range of models constructed is large (see Figures 17 and 18 for example). The vast space of acceptable solutions contains delta function, piecewise constant, and infinitely differentiable σ ( z ). The interpreter must grapple with this nonuniqueness in order not to be misled by a model with features not required by the data. As Jackson (1973) stated, it is essential not only to find a solution, but to represent in a meaningful way the degree of nonuniqueness permitted by the data. For the MT problem, we cannot put bounds on the conductivity value at a particular depth within a constructed profile. At any depth it is possible to include arbitrarily thin layers of large or small conductivity without significantly affecting the misfit of the responses. Thus, for any practical data set, the lower bound on the conductivity is zero and the upper bound is unlimited. This leads naturally to the concept that only spatial averages of conductivity can be quantitatively assessed. An average of the conductivity may be written as

This monograph presents a review of various approaches which invert MT data under the assumptions that the conductivity is 1-D and isotropic. Obviously, the Earth conductivity is always in disagreement with these assumptions, but this should not prevent a thorough examination of 1-D techniques. There are geologic environments (for example, under the ocean floors and in sedimentary basins) where the 1-D assumption is sometimes an adequate representation. If the conductivity is 2-D, then a 1-D inversion of a particular polarization mode gives useful information in itself. In more complicated environments, 1-D inversions of impedance tensor averages such as the square root of the determinant, or the average of its off-diagonal elements (Berdichevsky and Dmitriev, 1976) may provide a reasonable conductivity estimate to initiate more realistic 2-D or 3-D inversions. Therefore, irrespective of geology, 1-D inversions provide useful information about the conductivity. There is another, and perhaps equally important reason for thoroughly investigating 1-D MT inversion techniques. MT represents one extreme of electromagnetic sounding methods; the other extreme is dc resistivity. An MT experiment has induction effects but no source geometry. The inducing field is assumed to be a vertically incident harmonic wave, and the field data are processed to be compatible with this assumption. On the other hand, dc resistivity experiments involve source geometry, but no induction. All other active sounding experiments have both source geometry and electromagnetic induction. Therefore, it follows that techniques developed for inverting MT data, and insight acquired regarding what information about the conductivity can and cannot be obtained, are of value when developing inversion algorithms for other types of sounding experiments.

Abstract In this short monograph, 1D inversion methods are examined collectively using a uniform notation. One-dimensional inversion methods are still important because there are geologic regions where lateral variation is small and 1D interpretation is directly applicable; 1D inverse solutions provide good starting models for 2D inversion; and understanding the 1D inverse problem provides a foundation for solving inverse problems in higher dimensions. The 10 chapters are “Introduction”, “Existence”, “Uniqueness,” “Asymptotic Methods,” “Linearized/Iterated Methods,” “Global Penalty Functional Methods,” “Monte Carlo Methods,” “Exact Methods,” “Appraisal Methods,” and “Conclusion.”