Theodore R. Madden

Induced polarization, a study of its causes

The causes of induced electrical polarization include not only the polarization of metal‐solution interfaces, but also effects associated with the coupling of different flows. Electro‐osmotic, thermal electric, and ion diffusion effects are among such examples. A study of the physical properties of geologic materials indicates that only electrode interface and diffusion flow phenomena are important sources of induced polarization effects. It was attempted to find characteristic differences between these two phenomena. Theoretical and experimental considerations show that the kinetic processes involved are quite similar in the two cases. This leads to difficulties in identifying the polarizing agent from electrical measurements, although the effects of well mineralized zones are easily recognized.

Three‐dimensional electromagnetic modeling using finite difference equations: The magnetotelluric example

We have developed a robust and efficient finite difference algorithm for computing the magnetotelluric response of general three-dimensional (3-D) models using the minimum residual relaxation method. The difference equations that we solve are second order in H and are derived from the integral forms of Maxwells equations on a staggered grid. The boundary H field values are obtained from two-dimensional transverse magnetic mode calculations for the vertical planes in the 3-D model. An incomplete Cholesky decomposition of the diagonal subblocks of the coefficient matrix is used as a preconditioner, and corrections are made to the H fields every few iterations to ensure there are no H divergences in the solution. For a plane wave source field, this algorithm reduces the errors in the H field for simple 3-D models to around the 0.01% level compared to their fully converged values in a modest number of iterations, taking only a few minutes of computation time on our desktop workstation. The E fields can then be determined from discretized versions of the curl of H equations.

Three-dimensional magnetotelluric modeling using difference equations­ Theory and comparisons to integral equation solutions

We have developed an algorithm for computing the magnetotelluric response of three‐dimensional (3-D) earth models. It is a difference equation algorithm that is based on the integral forms of Maxwell’s equations rather than the differential forms. This formulation does not require approximating derivatives of earth properties or electromagnetic fields, as happens when using the second‐order vector diffusion equation. Rather, one must determine how averages are to be computed. Side boundary values for the H fields are obtained from putting two‐dimensional (2-D) slices of the model into a larger‐scale 2-D model and solving for the fields at the 3-D boundary positions. To solve the 3-D system of equations, we propagate an impedance matrix, which relates all the horizontal E fields in a layer to all the horizontal H fields in that same layer, up through the earth model. Applying a plane‐wave source condition and the side boundary H field values allows us to solve for the unknown fields within the model. The r...

Electromagnetic precursors to earthquakes in the Ulf band: A review of observations and mechanisms

Despite over 2 decades of international and national monitoring of electrical signals with the hope of detecting precursors to earthquakes, the scientific community is no closer to understanding why precursors are observed only in some cases. Laboratory measurements have demonstrated conclusively that self potentials develop owing to fluid flow and that both resistivity and magnetization change when rocks are stressed. However, field experiments have had much less success. Many purported observations of low-frequency electrical precursors are much larger than expectations based on laboratory results. In some cases, no precursors occurred prior to earthquakes, or precursory signals were reported with no corresponding coseismic signals. Nonetheless, the field experiments are in approximate agreement with laboratory measurements. Maximum resistivity changes of a few percent have been observed prior to some earthquakes in China, but the mechanism causing those changes is still unknown. Anomalous electric and magnetic fields associated with fluid flow prior to earthquakes may have been observed. Finally, piezomagnetic signals associated with stress release in earthquakes have been documented in measurements of magnetic fields.

3-D resistivity forward modeling and inversion using conjugate gradients

We have developed rapid 3-D dc resistivity forward modeling and inversion algorithms that use conjugate gradient relaxation techniques. In the forward network modeling calculation, an incomplete Cholesky decomposition for preconditioning and sparse matrix routines combine to produce a fast and efficient algorithm (approximately 2 minutes CPU time on a Sun SPARC‐station 2 for 50 × 50 × 20 blocks). The side and bottom boundary conditions are scaled impedance conditions that take into account the local current flow at the boundaries as a result of any configuration of current sources. For the inversion, conjugate gradient relaxation is used to solve the maximum likelihood inverse equations. Since conjugate gradient techniques only require the results of the sensitivity matrix A˜ or its transpose A˜T multiplying a vector, we are able to bypass the actual computation of the sensitivity matrix and the inversion of A˜TA˜, thus greatly decreasing the time needed to do 3-D inversions. We demonstrate 3-D resistivit...

RANDOM NETWORKS AND MIXING LAWS

Random networks are investigated as models of heterogeneous media. A general approximate structure is used where the networks are described as a system of embedded networks, and the critical behavior and averaging behavior of such networks are developed. These results are applied to a study of the electrical conductivity of porous media, with special attention to an Archie9s law behavior. It appears that the wide range of crack and pore widths in rocks makes the resulting conductivity relatively insensitive to the topology of their interconnections and allows one to make reasonable predictions of rock conductivities, given the distribution of crack and pore widths. It also appears that with low porosity rocks the conductivity is controlled by the microcrack population which only accounts for a fraction of the total porosity. It would seem, therefore, that Archie9s law is a feature of some general trend between porosity and crack and pore width distributions rather than a fundamental property of porous media. The law of the geometric mean is an accurate predictor of the physical properties of a mixture of different materials. This mixing law can result from an equal balance of series and parallel arrangements which can be produced by an appropriate distribution of shapes. A brief look is given to problems of anisotropic distributions for the conductivity problem and it is shown how the averaging process greatly dilutes the microscopic anisotropy in producing the macroscopic properties.

An analysis of the magnetotelluric impedance for three‐dimensional conductivity structures

The eigenstate analysis of Lanczos, also known as singular value decomposition (SVD), is used to define eight parameters which uniquely describe the magnetotelluric impedance Z. These parameters are independent of a priori assumptions about Z and can be interpreted in terms of three‐dimensional conductivity structures. Through SVD, the impedance is represented by two characteristic states. These states consist of two pairs (E and H) of complex vectors and two corresponding, real, singular values which together describe the extremal properties of Z. The singular values are the maximum and minimum |E|/|H| ratios possible at the observation site and therefore yield the true maximum and minimum apparent resistivities. We use a variation of SVD analysis by incorporating phases in the singular values, which are then called characteristic values. These phases reflect the delay (caused by the earth’s conductivity) of the electric fields relative to their associated magnetic fields. In this analysis of Z, the char...

Effects of three‐dimensional structure on magnetotelluric sounding curves

Significant errors may result when applying one‐dimensional (1-D) or two‐dimensional (2-D) interpretation methods to magnetotelluric (MT) data collected in three‐dimensional (3-D) environments. Both depths and resistivities can be grossly incorrect if 1-D or 2-D methods are applied in a 3-D setting. We present examples of MT sounding curves generated using a 3-D modeling program (Madden and Park, 1982) and illustrate some interpretation pitfalls if 3-D effects are not considered. The 3-D effects discussed herein are attributed to a surface layer heterogeneity, and can be readily identified in MT data from a well‐designed MT survey. The MT survey must include stations chosen to yield information about regional structure. Alternatively, carefull examination of a geologic map will help the intepreter estimate the regional effects present in the data.

Magnetotelluric evidence for crustal suture zones bounding the Southern Great Valley, California

A geoelectric section inferred from a regional magnetotelluric study across the Coast Ranges, the Great Valley, and the Sierra Nevada reveals significant variations in electrical resistivity. Zones of lower resistivity interpreted at depths from 10 km to at least 30 km lie near mapped geologic boundaries between the Coast Ranges and the Great Valley and beneath the eastern side of the Great Valley. The former boundary is inferred by others to separate the subduction complex of the Coast Ranges from the mafic basement of the Great Valley. The lower resistivities are most likely associated with metasediments trapped between the Coast Ranges ophiolite and the former oceanic crust beneath the Great Valley. The latter boundary is problematic, but may be evidence for a deep metasedimentary section trapped between the ophiolites beneath the Great Valley and the granitic rocks of the Sierra Nevada. The lack of change in the magnetotelluric phase across the Great Valley indicates that a suture zone marked by lower resistivities is unlikely to be present beneath the valley. However, this does not preclude the existence of a resistive suture zone.

A magnetotelluric investigation of the San Andreas Fault at Carrizo Plain, California

High quality, wide-band magnetotelluric data were collected along two profiles crossing the San Andreas fault at Carrizo Plain, California for crustal imaging as part of the San Andreas Deep Drilling Project. Two-dimensional inversions of the data indicate that the upper crustal part of the San Andreas fault does not appear here as an anomalously conductive zone. Additionally, a broad resistive zone under the Temblor Mountains (east of the fault) suggests resistive crystalline or metamorphic rocks may be present here as opposed to the more conducting Franciscan assemblage, contradicting the generally-accepted geologic model.