B-polarization induction in two generalized thin sheets at the surface of a conducting half-space

Abstract

A new closed-form solution is obtained analytically for a Bpolarization induction problem of geophysical interest, in which a local region of the Earth is represented by a generalized thin sheet at the surface of and in electrical contact with a uniformly conducting half-space. The generalized sheet, first introduced by Ranganayaki & Madden, is a mathematical idealization of a double layer which consists, in this problem, of two adjacent half-planes with distinct conductances representing a surface conductivity discontinuity such as an ocean-coast boundary, underlain by a uniform sheet of finite integrated resistivity representing the lower crust. The resistive sheet exerts a considerable mathematical influence on the solution causing, under certain conditions, an additional pole to appear in one of the forms of contour integral by which the solution can be expressed; it also weakens or eliminates field singularities that would otherwise occur at the conductance discontinuity. A numerical calculation is made for model parameters typifying an ocean-coast boundary underlain by a highly resistive crust. It is found that the residue of the pole associated with the resistive sheet dominates the solution for this example, the main consequence of which is a huge increase in the horizontal range over which the induced currents adjust themselves between the different 'skin-effect' distributions at infinity on either side of the model. Moreover the solution shows that this 'adjustment distance' has a more complicated dependence on the conductance and integrated resistivity of the sheet' than that given simply by the square root of their product which was the length parameter proposed by Ranganayaki & Madden. astr. Soc., 56,419-438. Price's analysis" by R. P. Ranganayaki and T. R. Madden, Ceophys. J. R. astr, Soc., 65,237-238. edition, Academic Press, New York. Equations, Pergamon Press, London. 1999-2015. appl. Math., 2, 283-310.

The problem of current channelling: A critical review

“…However, Dawson et al (1982) obtained an exact analytical solution for this double layer problem in the B-polarisation mode, and showed that the exponential decay of the anomalous field on either side of a conductivity discontinuity in the top sheet (for this analysis the lower sheet must have the same characteristics on both sides of the discontinuity) is governed by the attenuation constants -Im (v~) and -Im (v~), where v? = ao~l,~z7 a'h,~z/a2 (16) This above defined 'adjustment distance', 6a, is obviously correct only for a 2D model.…”

Okak Bay AMT data‐set case study: Lessons in dimensionality and scale

“…However, as discussed by Ranganayaki and Madden (1980), Dawson et al (1982), and Jones (1983), the appropriate scale length is given by the adjustment distance, not the skin depth, and at low frequencies this distance can be many hundreds of kilometers. As formulated by Ranganayaki and Madden (1980) and Dawson et al (1982), this distance applies on the ocean side of the coastline, so it cannot be directly applied as there is no conducting upper thin sheet on the land side. Nevertheless, the induction arrows clearly indicate that there is an observable effect.…”

Modelling of electromagnetic fields in thin heterogeneous layers with application to field generation by volcanoes-theory and example

“…A number of 2‐D algorithms produce consistent results. What is also important is that the 2‐D algorithms can be tested using analytical solutions found by Weidelt (1971) for E ‐polarization, and by Dawson et al (1982) and Dawson (1996) for B ‐polarization. The 2‐D code used in our test was developed by Poll (1994) and based on the scheme by Brewitt‐Taylor & Weaver (1976) and Weaver & Brewitt‐Taylor (1978).…”

“…Generalized thin‐sheet models have successfully been used by Dawson et al (1982), Berdichevsky & Zhdanov (1984), Weaver & Dawson (1992), Weaver (1994), and other authors. We believe that the opportunities offered by this class of models are far from exhausted.…”