0 Introduction

The fact that the rocks in the crust and the upper mantle of the Earth can display electrical anisotropy to some degree has been proven by a large number of field observations and laboratory studies^[1-19]. With the enormously important development of geophysical observation techniques and the increasing understanding of the EM wave propagation in the Earth, the study of the electrical anisotropy has gained increasing attention^[2]. Now it has been realized that the electrical anisotropy has significant implications in many fields related to the electromagnetic induction, such as economic resource evaluation, interpretation of hydrological flows, and models of material composition and transport in the deep crust and upper mantle^[7].

The magnetotelluric (MT) method, which was developed by Tikhonov^[20-21] and Cagniard^[22], is a passive electromagnetic (EM) technique to make inferences about conductivity distribution of subsurface by utilizing the natural, time varying electric and magnetic field components recorded at right angles at the surface of the Earth. Many algorithms^[23-29] for solving the two-dimensional forward problems related to the MT technique in the electrical anisotropy media have been well developed. However, sometimes one may encounter the situation where the results given by different algorithms differ from each other, and then it is essential to validate the numerical procedures by the analytic solution of some simple models with special conductivity structures^[30-33]. With regards to the analytic solution of the MT fields for the models with electrical anisotropy, many studies have been published. For example, the propagation of the EM wave in a horizontally N-layered model was considered by O’Brien and Morrison^[34]. Subsequently, a similar model but with more general anisotropy structures was investigated by Loewenthal and Landisman^[35], in which the theory for MT observations at the surface of a layered anisotropic half space was presented. The MT fields for a half-space with dipping anisotropy was studied firstly by Chetayev^[36] and subsequently by Reddy and Rankin^[37]. The EM fields for a stratified model with uniaxial anisotropic conductivity, where in each layer one of the components of the conductivity tensor differs from the other two, was reported by Kong^[38]. With the aid of the Rayleigh expansions technique, Osella and Martinelli^[28] obtained the MT fields for a 2-D earth consisting of anisotropic layers with smooth irregular boundaries. The quasi-analytic solutions of the MT responses on the surface of an infinite fault with diagonal anisotropy was derived by Qin, Yang and Chen^[33].

Most recently, the authors have developed the analytic MT responses to a two-segment model with axially anisotropic conductivity structures overlying a perfect conductor^[39]. However, the electrical conductivity of the Earth materials may vary, ranging from 10⁶ to 10^-14Ω·m, as was pointed out by Negi and Saraf^[2]. Hence the basement in different regions may comprised by electrically insulating or conducting materials, which makes the basement may be considered as either an electrical insulator or conductor. In our previous paper^[39], only the case involving a conductor basement is treated. In this contribution, the contrary case where the basement is an insulator is considered. For these two cases, the boundary conditions to be satisfied will be diverse for the different basement. Therefore, there exist some differences in the solution of the EM fields.

The rest of this study is structured as follows: Firstly, the model considered and governing equations are described in section 1. Secondly, the technique to solve the analytic solution of the induction problem is outlined in section 2. Next, in section 3, the analytic solution is subsequently validated by two special models. And then we calculate the analytic solutions of the MT responses for a particular model in section 4. Finally the study is closed with conclusions in section 5.

1 Model and governing equations

The problem of EM induction considered in this contribution is approached based on the Cartesian coordinate system where the x-axis and y-axis are assumed to be parallel and perpendicular to the structure strike of the geo-electric model, respectively, while the z-axis extends vertically downwards and thus perpendicular to the xy plane. The anisotropic model to be treated is shown in Figure 1. The model is composed of two segments (medium①and②) with axially anisotropic conductivity structures underlain by a perfect insulator basement. The region 0 < z < d is occupied by the segments with thickness of d. The segments are underlain by an isotopic perfect insulator which is located at the interface z=d and extends vertically downwards to infinity. The three principal values of the anisotropic conductivity of the two segments are represented by σ₁^j, σ₂^j, and σ₃^j(j=1, 2), where the superscripts 1 and 2 denote the medium①and②, respectively.

Considering the 2-D symmetry of the conductivity distribution in the media and also the 1-D symmetry of the MT source field, the governing equations for the EM fields in cases of the two polarization modes (transverse magnetic and transverse electric, i.e. TM and TE), in international system of units (SI), to be solved in each region are exactly the same with the different conductivity inserted and have been given by Qin, Yang and Chen^[33] and Qin and Yang^[39] as

Where H_x and E_x are the strike-parallel electric and magnetic components for the two polarization modes, respectively; i=, which is the imaginary unit; ω is the angular frequency; μ is the magnetic permeability which is usually supposed to be the value of free space (μ₀=4π×10^-7N/A²); σ_i^j(i=1, 2, 3;j=1, 2) is the conductivity value along the direction of the ith principal axes in the medium j; ω=2π/t, which is the angular frequency, and t is period considered.

These two equations must be treated in each region to seek the solutions of the induction problem subject to appropriate boundary conditions on the inner and outer boundaries of the model.

2 Analytic solution of induction problems

Based on the fact that no difference exists between the anisotropy and isotropic case in the TE mode^[33], and the isotropic case had been investigated by Rankin^[40] and also by d’Erceville and Kunetz^[41], the case of TE mode will not be considered hereinafter, and only the TM mode will be analyzed.

2.1 The solution of MT fields in the TM mode

In order to seek the analytic solution of the EM fields in this case, the equation (1) must be solved subject to some boundary conditions, which are similar to that of the isotropic model^{[30, 32, 42]}. The conditions to be satisfied could be stated as follows:

(i)H_x¹=H_x²=H₀ (constant) on z=0, where the superscript 1, 2 indicate the medium①and②, respectively (similarly hereinafter);

(ii)H_x¹=H_x²=0 on z=d;

(iii);

(iv)H_x¹=H_x² on y=0;

(v)The model tends to be 1-D as |y|→∞.

According to the approach stated in Qin and Yang^[39], the total horizontal magnetic fields could be easily obtained. Here only the final results are stated. The reader is referred to Qin and Yang^[39] for detailed information.

The horizontal magnetic fields H_xand the horizontal electric fields E_y could be written as

and

Where

2.2 MT responses functions

After the horizontal electric and magnetic fields are solved, the impedance Z_TM in the case of the TM mode could be determined from the ratio of the electric and magnetic fields, and then the apparent resistivity and impedance phase can be obtained^[43].

Where ρ_{a, TM} and φ_TM are the apparent resistivity and impedance phase in the TM mode; |Z_TM| is the module of the impedance; Im{Z_TM} and Re{Z_TM} are the imaginary and real parts of the impedance, respectively.

3 Validation of the analytic solution

In this section, we will validate the analytic solution obtained in the previous section with the responses of a vertical fault with axially anisotropic conductivity which was investigated by Qin, Yang and Chen^[33].

The model considered in this work will reduce to an infinite fault with diagonal anisotropy as d→∞. The depth of the basement in the calculation is assigned as d=200 km. The resulting comparison is illustrated in Figure 2. It could be seen a very good consistence between the two different results. Note that the difference in the abscissa (i.e., y) between this work and that of Qin, Yang and Chen^[33].

4 Analytic responses of a particular model

In this section, we will consider the analytic solution of the MT fields for the two-segment model with specified parameters. The analytic results could be used as an initial standard solution to validate the numerical modeling results given by various computer programs. Here we give the apparent resistivity and impedance phase at some selected observation points at the surface of the particular model. The parameters used in the calculation are listed as follows:t=1 s, d=20 km, and ρ₁¹=10Ω·m, ρ₂¹=100Ω·m, ρ₃¹=2 000Ω·m, and ρ₁²=1 000Ω·m, ρ₂²=200Ω·m, ρ₃²=40Ω·m.

The apparent resistivity and impedance phase at different observation points are shown in Table 1. And then, the MT responses at the surface of the particular model in the previous paragraph for different periods (or frequencies) are also investigated. The typical frequencies encountered in the MT sounding ranges from 10^-4 to 10³ Hz, and then the corresponding period is 10⁴ to 10^-3 s, here the MT responses will be calculated for the period at every half-order of magnitude on a logarithmic scale in the computation. We select two observation points (say, site-A and site-B) which are located at either side of the fault, respectively. The distances from the fault to these two points are both assumed to be 50 m, that is, y=-50 m for site-A and for site-B. The calculation results for various values of period t are given in Table 2. It can be seen that the apparent resistivity calculated at the short period is close to the true resistivity of the segment, while the apparent resistivity at the long period is close to the value of the resistivity of the basement.We hope that these values shown above could serve as at least an initial standard solution against which the results given by various numerical codes developed for modeling the MT responses of a 2-D medium with more general anisotropy.

5 Conclusions

This study presents the analytic solution of the MT responses to a two-segment model with axially anisotropic conductivity structures overlying a perfect insulator. The analytic solution of the MT responses at some selected observation points along two different levels for a particular model are calculated and then compared with the numerical results. The MT responses at the very short and the very long periods tend to be the responses of a half-space model in which the resistivity is the same as the value in the segments or the insulator.

The analytic solution presented is beneficial to further understand the propagation of the EM wave in the Earth within which the crust and the upper mantle are believed to be anisotropy to some extent. Additionally, the analytic solution could be used to examine the reliability and the accuracy of the numerical programs developed for modeling the MT responses to the 2-D anisotropic medium.