ABSTRACT

Geoelectrical fields in a layered earth with arbitrary anisotropy

Changchun Yin* and Peter Weidelt^*

In many cases in geoelectrical exploration, the standard model of a layered isotropic half-space is a good approximation to geophysical reality. But sometimes it is useful to extend this model to uniform anisotropic layers. For example, in regions with distinct dipping stratification, one will try to represent this conductivity variation in the form of dipping anisotropic layers. In this paper, the layered isotropic half-space is extended to a halfspace with general anisotropy such that to each layer is assigned a symmetrical (3x3) resistivity tensor. After formulating the general layered problem, the numerical implementation is treated. A very fast algorithm exists only if the resistivity tensor is rotationally symmetric around a vertical axis (transverse isotropy). In all other cases, one has to calculate a double integral with respect to the horizontal components of the wavenumber vector. Furthermore, we have also calculated the magnetic field. Whereas in the isotropic case this field does not contain any information about the resitivity distribution, in the case of arbitrary anisotropy it carries some information about the distribution of resistivity inside the earth. No correct results are obtained if one interprets data resulting from an anisotropic structure by an isotropic model.

Manuscript received by the Editor September 8, 1997; revised manuscript received July 16, 1998.
¤Technical Univ. of Braunschweig, Inst. of Geophysics and Meterology, Mendelssohn Strasse 3, D-38106 Braunschweig, Germany. E-mail: yin@geophys.nat.tu-bs.de, weidelt@geophys.nat.tu-bs.de.
C 1999 Society of Exploration Geophysicists. All rights reserved.

INTRODUCTION

In the last few years, a great number of interpretation methods with variable isotropic electrical conductivity have been developed, but not enough attention has been paid to the effects of anisotropy, especially the arbitrary anisotropy of the earth. The effect of anisotropy on apparent resistivity was mentioned earlier in the geophysical literature by Slichter (1933) and Pirson (1935), but their research concentrated on the uniform transverse isotropic half-space only. In that case, the solution for the potential is usually obtained by replacing the anisotropic resistivity by an isotropic medium with the resistivity being the geometric mean of the vertical and radial resistivity. Also for the layered transverse isotropic earth, one can use the equivalent isotropic models to replace the anisotropic resistivity (Niwas and Upadhyaya, 1974; Pal and Das Gupta,1984). The model of a dipping anisotropic half-space is proposed for the first time by Bhattacharya and Patra (1968). The most advanced model considered so far is that of Gurevich (1975). It consists of a layered half-space with dipping anisotropy, the only restriction being that the horizontal strike direction is the same in all layers. Pal and Mukherjee (1986) calculated the electrical potential due to a point current source over a two-layer earth with a dipping anisotropic top layer. Negi and Saraf (1989) review the research on electrical anisotropy, which has been the subject of much attention in electrical prospecting (e.g.,Kunz and Moran, 1958; Asten, 1974;Moranand Gianzero, 1979, 1982). In this paper, the current density and magnetic field as source-free vectors are represented by poloidal and toroidal scalars. This representation has been applied by Maurer (1993) to model the problem of electric dipole induction in arbitrary anisotropic half-space. In the present direct current (dc) case, the system of differential equations leads to a simple stable recursive algorithm for both the electric and magnetic fields. For the latter, a Green’s function approach is used.