SH-TM mathematical analogy for the two-layer case . A magnetotellurics application

The same mathematical formalism of the wave equation can be used to describe anelastic and electromagnetic wave propagation. In this work, we obtain the mathematical analogy for the reflection/refraction (transmission) problem of two layers, considering the presence of anisotropy and attenuation – viscosity in the viscoelastic case and resistivity in the electromagnetic case. The analogy is illustrated for SH (shear-horizontally polarised) and TM (transversemagnetic) waves. In particular, we illustrate examples related to the magnetotelluric method applied to geothermal systems and consider the effects of anisotropy. The solution is tested with the classical solution for stratified isotropic media.


Introduction
The role of mathematical analogies has been well illustrated and explained by Tonti [1].Quoting Tonti: "Many physical theories show formal similarities due to the existence of a common mathematical structure.This structure is independent of the physical contents of the theory and can be found in classical, relativistic and quantum theories; for discrete and continuous systems".Carcione and Cavallini [2] found analogies between anelastic and electromagnetic vector wave fields, while Carcione et al. [3] relate the medium properties.Carcione and Cavallini [2] show that the 2-D Maxwell equations describing propagation of the transverse-magnetic mode in anisotropic media is mathematically equivalent to the SH wave equation in an anisotropic-viscoelastic solid where attenuation is described with the Maxwell mechanical model.Later, Carcione and Robinson [4] establish the analogy for the reflection-transmission problem at a single interface, showing that contrasts in compressibility yield the reflection coefficient for light polarized perpendicular to the plane of incidence (Fresnel's sine law -the electric vector perpendicular to the plane of incidence), and density contrasts yields the reflection coefficient for light polarized in the plane of incidence (Fresnel's tangent law).Carcione et al. [5] considered the reflection/transmission problem through an anisotropic and lossy layer.In particular, they obtained the analogy among P and SH elastic waves, TE and TM electromagnetic waves and wave mechanics in quantum theory.
Osella and Martinelli [6] have studied the effect of anisotropic layers on the apparent resistivity curves, concluding that anisotropy cannot be detected with the TE mode and the TM response should be used.An extension of the technique used in that paper to 3D space can be found in Martinelli and Osella [7].In this work, we solve the problem of horizontally polarized shear (SH) waves in a twolayer system and apply the analogy to obtain the TM solution.A geophysical application considering magnetotellurics in an anisotropic geothermal reservoir illustrates the use of the analogy.We analyze the apparent resistivity and phase angle for different orientations of the principal axis of anisotropy and angle of incidence of the plane wave.

Viscoelasticity. Propagation of SH waves
We start from the viscoelastic equations and then apply the analogy to obtain the equivalent electromagnetic expressions.Figure 1 shows two layers embedded between two isotropic half spaces with different properties.
In the following, we denote particle velocity by v, stress by σ, magnetic field by H, electric field by E, density by ρ, elasticity constant by c, viscosity by η, magnetic permeability by µ, dielectric permittivity by ǫ and electrical conductivity by σ (see below).Moreover, (x, y, z) indicates the spatial variables, ∂ x a partial derivative with respect to x and a dot above a variable denotes time differentiation.To distinguish between the stress and conductivity components, we use letters and numbers as subindices, respectively, e.g., σ xy is a stress component and σ 11 is a conduc-tivity component.
The viscoelastic medium is characterized by the mass density ρ and elasticity, compliance and viscosity matrices The displacement associated to a homogeneous viscoelastic SH plane wave has the form where x = (x, z) is the position vector, ω is the angular frequency, t is the time variable, i = √ −1 and defines the complex wavevector, with κ = (l 1 , l 3 ) ⊤ , defining the propagation direction through the direction cosines l 1 and l 3 .Replacing the plane wave (5) into equation ( 2) yields the dispersion relation where the p IJ are the components of P obtained as The relation ( 7) defines the complex velocity,

Reflection and transmission coefficients
The boundary conditions at the interfaces require continuity of [8] σ yz and v y .
In the electromagnetic case, continuity of the tangential components of the electric and magnetic fields is required [9] (see below).Let us assume that the incident, reflected and refracted waves are identified by the subscripts and superscripts I, R and T .For a single interface, say that at z = z 1 , the particle velocities of the incident, reflected and refracted waves are given by respectively, where (s x , s z ) ⊤ is the slowness vector, and R and T are the reflection and refraction (transmission) coefficients.The equations obtained below hold for incident inhomogeneous plane waves (non-uniform waves in electromagnetism), i.e., waves for which the wavenumber and attenuation vectors do not point in the same direction.In the special case where these two vector coincide, the wave is termed homogeneous (uniform in electromagnetism), and we have where θ is the incidence angle.
In the general case, the reflection and transmission coefficients (TM case in electromagnetism) are given by where with and with "pv" denoting the principal value [4,8].
The coefficients for the interfaces at z = 0 and z = z 2 have similar forms but assuming isotropy for the upper and lower media, respectively, with p 44 = p 66 and p 46 = 0.
To obtain the reflection and transmission coefficients of the two layers, we follow the procedure indicated in Section 6.4 of Carcione [8].At depth z in the second layer, the particle-velocity field is a superposition of upgoing and downgoing waves of the form (17) where V − and V + are upgoing-and downgoing-wave amplitudes.
From equation ( 2), the normal stress component is where r ′ IJ = s ′ IJ − iτ ′ IJ /ω are the components of matrix R ′ defined in equation ( 8).Using equation ( 17), we obtain Omitting the phase exp[iω(t − s x x)], the particlevelocity/stress vector can be written as where Then, the fields at z = z 1 and z = z 2 are related by the following equation: where where Note that when z 2 = z 1 , B ′ is the identity matrix.Similarly, we have where where Combining equations ( 22) and (25), we finally obtain On the other hand, using equations ( 11) and ( 14), the particle-velocity/stress field at z = 0 and z = z 2 can be expressed as where R and T are here the reflection and transmission coefficients of the two-layer system and for an incident homogeneous plane wave.Substituting equation ( 29) into (28), we have where a ij are the components of matrix A.
If the two layers have the same properties and the "46" stiffness components are zero, we obtain the equations given in Carcione et al. [5], where ϕ = −ωs T z z 2 and Equation ( 32) is similar to equation (5.22) of Born and Wolf [9] if the layers are isotropic and lossless.In the case in which the media above and below the layer have the same properties, i.e., when r 23 = −r 12 , equation (33) becomes On the other hand, the transmission coefficient is

Surface impedance and apparent viscosity
We define the impedance in the upper half-space as where where s z = cos θ/v and we have used equations ( 11), ( 12) and ( 19).Substituting equation (37) into (36) at z = 0, we obtain the surface impedance where we have used equations ( 30) and (31).
On the other hand, the surface impedance can be obtained from the fields of the first layer at z = 0. From equation (29) we have v y (0) = (a 11 −a 12 Z b )T, and σ yz (0) = (a 21 −a 22 Z b )T, (39) which, using (36), yields equation (38).
We define the apparent surface viscosity as where we have used equation (38).A phase angle can be defined as Note that Z s , p a and φ do not depend on Z 0 .Let us assume z 1 = z 2 = 0.Then, B, B ′ and A are identity matrices and α = −Z 0 /Z b .We obtain where we used (30).At normal incidence s x = 0, and Since, from equation (8), it is , where c b and η b are the elastic constant and viscosity of the lower half space, respectively, we obtain

SH-TM analogy
Let us consider Maxwell equations and assume that the propagation is in the (x, z)-plane, and that the material properties are invariant in the y-direction.Then, E x , E z and H y are decoupled from E y , H x and H z .In the absence of electric-source currents, the first three fields obey the TM (transverse-magnetic) differential equations: where E i and H i denote the electric and magnetic field components, µ is the magnetic permeability, and ǫ ij and σ ij are the permittivity and electrical conductivity components, respectively [8].Equations ( 2) and ( 45) are mathematically equivalent if From equation ( 8), in virtue of the acoustic-electromagnetic equivalence ( 46)-( 49), it follows that P corresponds to the inverse of the complex dielectric-permittivity matrix ǭ, namely: Therefore, all the equations obtained in the previous section can be used to obtain the electromagnetic properties using the mathematical analogies ( 46)-(50).

Surface impedance and apparent resistivity
In particular, the surface impedance (36) is and is given by equation (38).
The equivalent of the surface apparent viscosity (42) is the apparent resistivity In the case z 1 = z 2 = 0, we have from equation (44), where µ b , ǫ b and σ b are the magnetic permeability, dielectric permittivity and electric conductivity of the lower half-space.In magnetotellurics, the upper space is air and the magnetic permeability is assumed to the constant, i.e., µ b = µ 0 .Moreover, displacement currents are neglected (ǫ b ≪ σ b /ω).Hence, we obtain where ρb is the resistivity of the lower half-space, as expected.

Example: Magnetotellurics
Electrical anisotropy in the Earth can be due to preferred orientation of fracture porosity, fluidised, melt-bearing or graphitised shear zones, lithologic layering, oriented heterogeneity, or hydrous defects within shear aligned olivine crystals [10].Magnetotellurics is a technique to measure the apparent resistivity of the subsoil to interpret the nature of the geological formations.The method operates at low frequencies, where the displacement term is neglected in Maxwell equations [11], such that the apparent resistivity is given by equation ( 52), with in order to apply the analogy.Specifically, .
(57) Moreover, magnetotellurics assumes that plane waves are normally incident on the surface of the earth (zdirection).In this case, equation (45) simplify to Eliminating the magnetic field, we obtain The quantity between round parentheses can be seen as an effective conductivity.The standard magnetotelluric method cannot distinguish changes in the conductivity components if that quantity is kept constant [12,10].On the other hand, for a plane wave travelling along the x-direction and based on the vertical component E z , the effective conductivity σ 33 − σ 2 13 /σ 11 can be obtained.Another measurement is required with a plane wave incident at an intermediate angle to obtain the three conductivity components.Moreover, note that if σ 13 = 0, variations in σ 33 have no effects on the result when the plane wave is normally incident.
Let us consider the model shown in Figure 2 and analyze the apparent resistivity as the properties of the formation vary for plane waves incident at different angles.
The conductivity components of the geothermal zone are obtained from a clockwise rotation by an angle β about the y-axis of the conductivity matrix from the principal sys- A rotation of the tensor by an angle β gives the components σ ij in the system (x, z).
tem with components σ 1 and σ 3 , Here we study the apparent resistivity as a function of σ 3 and β and keep constant the other properties.In particular σ 1 = 0.2 S/m.The different cases are shown in Table 1, with Case 1 the standard magnetotelluric technique.As indicated above, the conductivity component σ 33 has no effect on the results if σ 13 = 0. Hence, in this case anisotropy cannot be detected.Figure 3 shows the apparent resistivity (a) and phase (b) for σ 3 = σ 1 (Case 1 in Table 1), where T = 1/(2πω) is the period.
It can be shown that the results coincide with the classical magnetotelluric solution (65) given in Appendix A. Figures 4 and 5 compare the isotropic and anisotropic cases for incidence angles of π/4 and π/2, respectively.Cases 2 and 4 have more apparent resistivity, since the σ 3 component has a lower value.The differences due to anisotropy, and the fact that wave is not normally incident, are significant.

Conclusions
Theories describing wave propagation and field diffusion in different fields of physics consist in partial differential equations, which have identical or similar mathematical expressions.Here, we have considered the reflection/transmission problem of SH through a two-layer anisotropic and lossy system.We have shown that the same mathematical equations can be used in electromagnetism to describe the propagation and diffusion of TM waves.An example shows how the SH-wave equation reduce to the differential equation describing the magnetotelluric technique.
The analogy can be useful in the space-time domain using numerical simulations.In this case, the same computer code, with appropriate input variables can be used to solve the different physical problems.[8] J.M. Carcione, Wave Fields in Real Media.Theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media, 3rd edition, Elsevier, 2014.

Figure 1 :
Figure 1: Plane wave propagating through two layers.The viscoelastic properties are indicated.

Figure 2 :
Figure2: Model of a geothermal reservoir.The conductivity tensor in its principal system has components σ 1 and σ 3 .A rotation of the tensor by an angle β gives the components σ ij in the system (x, z).

Table 1 :
Values of the rotation angle and conductivity component (in S/m).