Abstract
The magnetotelluric (MT) responses are explicitly solved for a stratiform earth containing multiple transitional layers in which the conductivity varies linearly with depth. In the model under consideration, any one homogeneous layer with constant conductivity or transitional one may be absent in the geometry. The traditional onedimensional (1D) models with sharp boundaries will be obtained if all the transitional layers are absent in the geometry, while a special 1D model consisting of a sequence of contiguous transitional layers may be obtained if all the homogeneous layers (except the basement layer) are removed from the geometry. The tangential electric and magnetic fields as well as the surface impedance are analytically expressed by Airy functions. The analytical formula is validated in three theoretical examples by comparing with the results from available codes. The apparent resistivity and impedance phase on the surface of three different transitional models are illustrated to analysis the influence of the transitional layers on MT responses. The new formula provides an alternative way to obtain the analytic MT responses for the special layered earth.
1 Introduction
The electromagnetic (EM) geophysical techniques have been widely used in probing the pipelines [1,2], exploring mineral resources [3,4], estimating soil electrical conductivity [5,6], monitoring contamination [7,8], etc. The magnetotelluric (MT) technique is one of EM geophysical methods to measure the electrical conductivity distribution inside the earth through the use of lowfrequency EM induction [9,10,11]. With the rapid development of the electronic computer, the MT data are mainly inverted and interpreted assuming that the conductivity inside earth is twodimensional (2D) [12–26] or threedimensional (3D) [27–42]. Although 2D or 3D inversion schemes have been widely adopted, algorithms for onedimensional (1D) still have a very important role in MT experiments, especially when the data are acquired in the sediment basin or marine environment.
With regard to the 1D algorithms, the model is usually assumed to be composed of a sequence of homogeneous layers with constant conductivity [43–46] or a series of layers containing at least one transitional layer in the geometry [47–51]. The transitional layer refers to certain layer in which the conductivity is dependent on depth, or in other words, conductivity varies with depth according to some mathematic rule (for example, linear or exponential variation). It seems that the transitional layer may better represent the conductivity variation inside earth in some geologic formation. For example, as stated by Patella [52], the conductivity in a fissured geological formation (e.g., limestone) may vary with depth. Therefore, it is still necessary to study the theoretical MT responses of the 1D earth with transitional layers [7].
The analytical theoretical MT soundings of 1D model with transitional layer had been investigated in detail by some authors. For example, the MT responses of 1D model with transitional layer was first investigated by Mallick [47]. The inductive sounding of a stratified earth with transition layer resting on dipping anisotropic beds was presented by Negi and Saraf [53]. Berdichevsky et al. investigated the deep EM sounding of the gradient media in which resistivity exponentially decreases with depth [54]. Kao and Rankin reported the MT response for a threelayer earth, formed by constant conductivity in the first and the third layer and conductivity varying linearly with depth in the second layer [49]. MT response in an inhomogeneous earth having resistivity varying linearly with depth was investigated by Kao [50]. MT response in an inhomogeneous earth having conductivity varying linearly with depth was also investigated by Kao [51]. Pal presented the MT response on a layered earth with nonmonotonic resistivity distribution [55]. Fang and Luo solved the MT response on vertically inhomogeneous earth having conductivity varying linearly with depth by layers [56]. Chumchob published the mathematical modeling of EM response on a multilayered earth with a layer having exponentially varying conductivity [57]. Qin et al. investigated the MT responses of a vertical inhomogeneous and anisotropic resistivity structure with a transitional layer [58]. Qin et al. also studied the MT responses of an anisotropic 1D earth with a layer of exponentially varying conductivity [59].
However, in those publications mentioned previously, the model considered contains only one transitional layer. Obviously, it is not enough to approximate the conductivity distribution inside the earth with only one transitional layer. Recently, Qin and Yang [60] examined the MT soundings on a stratified earth with two transitional layers. However, as described by the well logging results [61], the subsurface of earth should contain many layers. Hence, it is necessary to study the MT responses of the model with many transitional layers. In this work, following the algorithm given by Qin and Yan [60], theoretical MT response for 1D layered earth with multiple transitional layers is investigated. The new formula provides an alternative way to obtain the analytic MT responses for the special layered earth.
2 Model description
In this work, we consider a planepolarized EM wave with the time factor
where
For the problem considered in this study, the Maxwell’s equations need to be solved in different layers (regions) to obtain the MT field components.
3 Solution of the induction problem
For the model under consideration in this work, it consists of two different types of media: one medium with constant conductivity in odd layers, and one medium with conductivity varying with the depth in even layers. In the following contexts, the Maxwell’s equations will be solved in two different media, respectively.
3.1 MT fields in odd layers
For the theory of MT probing, it is convenient to ignore the effect of the displacement current from the start [63]. Therefore, the Helmholtz’s equation derived from Maxwell’s equations should be satisfied in the isotropic homogeneous media [54,64,65], that is, the following equation holds in the odd layers,
where
A general expression for electric field may be obtained by solving equation (2) [49,66].
where
Then, corresponding horizontal magnetic field in the layer
3.2 MT fields in even layers
For the transitional layers, the Helmholtz’s equation [54,64,65] to be satisfied may be written as follows:
where
Now substituting equation (1) in equation (5) immediately leads to
Following the method presented in Qin and Yang [60], after some algebra, the general solution of equation (6) may be obtained and written as follows:
where
Then, the corresponding horizontal magnetic field may be obtained from the derivative of the electric field and written as follows:
where
3.3 Determination of coefficients
In this section, the coefficients in the expressions of the EM fields are determined by applying appropriate boundary conditions.
In the MT theory, the electric field
Considering the boundary conditions above and the EM fields vanish at infinite depth, we have the following system of equations:
where
The system of equations is solved recursively from
After some algebraic operations, we can obtain the recursive formulas as follows:
The ratio of the coefficients at each interface boundary may be easily obtained through equation (11) once the model parameters are known.
3.4 MT apparent resistivity and impedance phase
In order to calculate the MT apparent resistivity and impedance phase at the surface of the model, the surface impedance needs to be determined first. According to the definition of the impedance, we can obtain
where
Then, the apparent resistivity and impedance phase may be easily obtained from the following equation [9]:
where
4 Ratio of the coefficients in special circumstances
In this section, we present the ratio of the coefficients in special circumstances in which any one transitional or homogeneous layer in the model is absent.
4.1 Absence of any one transitional layer
Suppose that the transitional layer having the conductivity
The above procedure may be repeated for any transitional layer which is supposed to be absent. If all the transitional layers in Figure 1 (even layers) are removed from the model, then a traditional 1D model is obtained.
4.2 Absence of any one homogeneous layer
Suppose that the homogeneous layer having the conductivity
Then, the relationship of coefficients related to the upper layer above this homogeneous layer and the lower one may be easily given from equation (15) as follows:
The procedure mentioned above may be repeated for any homogeneous layer which is supposed to be absent. When the procedure is used for all homogeneous layers except the basement, a 1D earth model consisting of contiguous transitional layers in which conductivity varies linearly with depth (Figure 2) is obtained. In other words, the model is obtained if all the layers with constant conductivity in Figure 1 (odd layers except the basement) are removed from the model.
5 Synthetic examples
In this section, three synthetic models are considered for investigating the theoretical MT responses.
First, we consider a 1D layered earth model characterized by an alternate sequence of homogeneous and transitional layers, as displayed in Figure 3. In this layered geometry, the model is composed of four homogeneous layers and three transitional layers, and the detailed model parameters are listed at the bottom of Figure 3. The apparent resistivity and impedance phase are calculated by the algorithm in this study. In addition, the MT transfer functions of this model are also computed using the opensource code MT1D in OccamCSEM1D publicized by Key [67] (referred to as Key1dCode hereinafter). The variations in the MT responses at the surface of the model shown in Figure 3 with the periods are displayed in Figure 4. The results calculated using the present algorithn are denoted by black solid curves, while the ones from available codes are represented by the magenta dashed curve. Figure 4 shows pretty good consistency between two results. In other words, the algorithm presented in this work is validated
Subsequently, let us now examine a second theoretical example. The layered geometry in this example is a 1D layered earth model characterized by a sequence of contiguous transitional layers, as shown in Figure 5. In other words, all the homogeneous layers with constant conductivity (except the basement layer) are removed from the 1D model, based on the procedure described in Section 5. Then, a sequence of contiguous transitional layers is obtained, and it may be supposed to approximate a continuous variation in the conductivity with depth, from the model surface down to the basement. The detailed model parameters are given at the bottom of Figure 5. The MT responses at the surface of the model as a function of period are shown in Figure 6. The results calculated using the present algorithn are denoted by black solid curves, while the ones from available codes are represented by the magenta dashed curve. The comparisons between the apparent resitivity and impedance phases given by different algorithms show a good consistency, and the algorithm in this study is further validated.
Finally, in the third theoretical example, the model is a 1D layered earth model characterized by a random sequence of homogeneous and transitional layers (Figure 7). The 1D model is supposed to be composed of a random combination of homogeneous and transitional layers, in the sense that they do not have an alternate disposition in the layered geometry. The MT responses (apparent resistivity and impedance phase) to the model shown in Figure 7 are computed using the algorithm in this work and the opensource code Key1dCode. The MT responses at the surface of the model shown in Figure 7 as a function of period are displayed in Figure 8. The resulting MT responses given by the present algorithm are displayed by the black solid curves and the ones from available code are denoted by magenta dashed curves. It is clear from Figure 8 that the results given by the algorithm in this study are consistent with that given by the opensource code. Therefore, this is another proof to validate the correctness of the algorithm in this study.
6 Conclusion
The theoretical MT responses ofo a 1D layered geometry composed of homogeneous and multiple transitional layers are investigated. The homogeneous layers have constant conductivity values while the transitional ones have conductivity varying linearly with depth. The new formula is derived to calculate the MT responses for the special 1D layered model. The new formula provides an alternative way to obtain the analytic MT responses for the special layered earth. For the model under consideration, the MT fields in the transitional layers can be explicitly expressed in term of the Airy functions.
Any of the homogeneous or transitional layers may be absent in the layered geometry in the calculation, which may be more suitable to approximate the conductivity distribution in the earth compared with traditional 1D model only having layers with constant conductivity. The conductivity in each transitional layers may be increased or decreased with the depth according to the values of the conductivity in adjacent layers. Three typical theoretical examples are presented to validate the algorithm in this study.
The algorithm and corresponding results of theoretical examples may be used to examine the propagation of the EM waves in 1D earth. However, there are still some limitations in future application since the algorithm is only related to the forward computation. The inversion technique needs to be developed for the future studies.
Acknowledgements
The authors gratefully acknowledge anonymous reviewers for their thorough reading of this manuscript and for their insightful questions and constructive suggestions, which significantly improved the quality of this article.

Funding information: This study is partly supported by National Key Research and Development Program of China (No. 2019YFB1600700).

Conflict of interest: The authors declare that they have no competing interests.

Data availability and statement: The data involved during the present study are available from the corresponding author upon reasonable request.
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