We present a new method for interpreting electromagnetic (EM) data using ray tomography. Direct application of ray tomography to low‐frequency EM data is difficult because of the diffusive nature of the field. Diffusive EM fields can, however, be mathematically transformed to wavefields defined in a time‐like variable. The transform uniquely relates a field satisfying a diffusion equation in time, or in frequency, to an integral of the corresponding wavefield. If the corresponding wavefields can be computed from low‐frequency EM data, one should be able to interpret these data using techniques developed for the wavefields. To test the idea, numerically calculated transient magnetic fields were first transformed to wavefields. The typical window of the time‐domain data required for the transform is 1.5 decades. Traveltimes from a source to the receivers were estimated from the reconstructed wavefields. Time‐domain data with a Gaussian noise of 3 percent gave a traveltime resolution of better than one percent. For the tomographic inversion, the cross‐section between the transmitter and receiver boreholes is divided into a number of rectangular elements, and a continuous slowness is assigned to each of these elements. A functional is formulated by invoking Fermat’s principle for the traveltime data. Imposing a stationary condition on the functional gives an iterative procedure for the slowness model. Rays are allowed to bend smoothly within each cell. Incorporating smoothly bending rays is extremely important when the velocity contrast is large. A model with a conductivity contrast of ten (10) has been successfully imaged in 120 iterations with 5 CPU hours on a SUN SPARCstation 2.

A ray series solution for Maxwell's equations provides an efficient numerical technique for calculating wavefronts and raypaths associated with electromagnetic waves in anisotropic media. Using this methodology and assuming weak anisotropy, we show that a perturbation of the anisotropic structure may be related linearly to a variation in the traveltime of an electromagnetic wave. Thus, it is possible to infer lateral variations in the dielectric permittivity and magnetic permeability matrices. The perturbation approach is used to analyze a series of crosswell ground-penetrating radar surveys conducted at the Idaho National Engineering Laboratory. Several important geological features are imaged, including a rubble zone at the interface between two basalt flows. Linear low-velocity anomalies are imaged clearly and are continuous across well pairs.

This paper presents a simple, generalized parameter constraint using a priori information to obtain a stable inverse of geophysical data. In the constraint the a priori information can be expressed by two limits: lower and upper bounds. This is a kind of inequality constraint, which is usually employed in linear programming. In this paper, we have derived this parameter constraint as a generalized version of positiveness constraint of parameter, which is routinely used in the inversion of electrical and EM data. However, the two bounds are not restricted to positive values. The width of two bounds reflects the reliability of ground information, which is obtained through well logging and surface geology survey. The effectiveness and convenience of this inequality constraint is demonstrated through the smoothness-constrained inversion of synthetic magnetotelluric data.

A rigorous full waveform inversion of seismic data has been a challenging subject partly because of the lack of precise knowledge of the source. Since currently available approaches involve some form of approximations to the source, inversion results are subject to the quality and the choice of the source information used. We propose a new full waveform inversion methodology that does not involve source spectrum information. Thus potential inversion errors due to source estimation can be eliminated. A gather of seismic traces is first Fourier-transformed into the frequency domain and a normalized wavefield is obtained for each trace in the frequency domain. Normalization is done with respect to the frequency response of a reference trace selected from the gather, so the complex-valued normalized wavefield is dimensionless. The source spectrum is eliminated during the normalization procedure. With its source spectrum eliminated, the normalized wavefield allows us construction of an inversion algorithm without the source information. The inversion algorithm minimizes misfits between measured normalized wavefield and numerically computed normalized wavefield. The proposed approach has been successfully demonstrated using a simple two-dimensional scalar problem.

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We present a new method, dubbed the modified extended Born approximation (MEBA), for efficient three‐dimensional (3D) simulation and inversion of geophysical frequency‐domain electromagnetic (EM) data for a targeted object lodged in a layered half‐space. Based on the integral equation method and modified from an extended Born approximation technique, the MEBA method calculates the total electric field in an electrical conductivity inhomogeneity without any need for solving a huge matrix equation. This is done by multiplying the background electric field by a depolarization tensor. The Fourier transform and the convolution theorem are used to dramatically increase the computational efficiency. Comparisons of MEBA‐generated numerical data for tabular targets with data generated by other means are used to verify the scheme and check its range of validity. The results indicate that the MEBA technique yields better accuracy when current channeling in the conductivity anomaly dominates over the induction process. The MEBA algorithm has been incorporated into a least‐squares inversion scheme which is used to interpret borehole‐to‐surface EM tomography field data. The survey served to monitor the subsurface conductivity change associated with the extraction of a volume of saltwater previously injected into a known aquifer.

Abstract:Since most oil wells are cased with steel pipes, electromagnetic (EM) signals undergo a severe attenuation as they diffuse across the casing. This paper presents the effect of non-uniform casing properties on EM fields measured in a steel-cased well embedded in a layered formation. We use a finite-element method to compute secondary azimuthal electric fields for a cylindrically symmetric conductivity model, while primary fields are analytically obtained for a homogeneous casing in a whole space. Although EM signals induced by a layered formation are greatly masked by the steel casing, phase responses are more pronounced than amplitude responses. The effect of casing non-uniformity is quite large but highly localized. When the conductivity rapidly varies in the casing wall, the resulting EM fields also fluctuate rapidly. These fluctuations are so similar that their cross correlation function is strongly peaked at two points, whose distance is equal to the separation between source and receiver. The high-frequency coherent noise caused by the non-uniform casing event can be largely suppressed by low-pass filtering to enhance EM signals from the formation conductivity.

Electrical properties in most geologic materials have been known to be frequency dependent, and resulting dispersion relationship can be a useful diagnostic tool for investigating the shallow subsurface. In this paper we investigate the determination of dispersive electrical properties of the shallow subsurface with inversion of high-frequency electromagnetic (EM) fields. We have limited the dispersive characteristics to the electrical permittivity and used the Cole-Cole model to describe the frequency dependence of the permittivity. For horizontally layered earth models high-frequency EM fields are successfully inverted via Marquardt-Levenberg least-squares method and simulated annealing method. Inversion experiments show that the simulated annealing yields slightly better parameter resolution than the leastsquares inversion.