3-D Global Induction in the Oceans and Solid Earth: Recent Progress in Modeling Magnetic and Electric Fields from Sources of Magnetospheric, Ionospheric and Oceanic Origin

Abstract

Electromagnetic induction in the Earth’s interior is an important contributor to the near-Earth magnetic and electric fields. The oceans play a special role in this induction due to their relatively high conductivity which leads to large lateral variability in surface conductance. Electric currents that generate secondary fields are induced in the oceans by two different processes: (a) by time varying external magnetic fields, and (b) by the motion of the conducting ocean water through the Earth’s main magnetic field. Significant progress in accurate and detailed predictions of the electric and magnetic fields induced by these sources has been achieved during the last few years, via realistic three-dimensional (3-D) conductivity models of the oceans, crust and mantle along with realistic source models. In this review a summary is given of the results of recent 3-D modeling studies in which estimates are obtained for the magnetic and electric signals at both the ground and satellite altitudes induced by a variety of natural current sources. 3-D induction effects due to magnetospheric currents (magnetic storms), ionospheric currents (Sq, polar and equatorial electrojets), ocean tides, global ocean circulation and tsunami are considered. These modeling studies demonstrate that the 3-D induction (ocean) effect and motionally-induced signals from the oceans contribute significantly (in the range from a few to tens nanotesla) to the near-Earth magnetic field. A 3-D numerical solution based on an integral equation approach is shown to predict these induction effects with the accuracy and spatial detail required to explain observations both on the ground and at satellite altitudes.

Appendices

Appendices

1.1 Appendix 1: Governing Equations of the 3-D Volume Integral Equation Approach

The 3-D Earth model considered consists of a number of 3-D inhomogeneities of conductivity σ_3D(r, ϑ, φ), embedded in 1-D host section of conductivity σ _b (r). Complex-valued conductivity σ(σ_3D and σ _b ) can incorporate the effects of displacement currents or induced polarization. Note, also, that σ_3D may account for anisotropy of the electric conductivity and thus it is represented in general by a 3 × 3 matrix. For this problem statement, the EM fields in the frequency domain obey Maxwell’s equations (1). Equations (1) are solved by a modern version of the volume integral equation approach. A brief review of the approach is given below.

First, some “reference” radially-symmetric (1-D) section of conductivity σ_o(r) is introduced. The electric, E°, and magnetic, H°, fields in the reference medium obey Maxwell’s equations

$$ \nabla \times {\mathbf{H}}^{\text{o}} = \sigma_{\text{o}} {\mathbf{E}}^{\text{o}} + {\mathbf{j}}^{ext} ,\quad \nabla \times {\mathbf{E}}^{\text{o}} = i\omega \mu_{\text{o}} {\mathbf{H}}^{\text{o}} , $$

(A.1.1)

and can be determined via fundamental solutions (dyadic Green’s functions) of Eq. A.1.1, G _o ^ej and G ^hj_o as

$$ {\mathbf{E}}^{\text{o}} = \int\limits_{{V^{ext} }} {G_{\text{o}}^{ej} ({\mathbf{r}},{\mathbf{r}}'){\mathbf{j}}^{ext} ({\mathbf{r}}')dv'} ,\quad {\mathbf{H}}^{\text{o}} = \int\limits_{{V^{ext} }} {G_{\text{o}}^{hj} ({\mathbf{r}},{\mathbf{r}} '){\mathbf{j}}^{ext} ({\mathbf{r}} ')dv'} . $$

(A.1.2)

Here V ^ext is the volume occupied by j ^ext, ${\mathbf{r}} = (r,\vartheta ,\varphi ) , {\mathbf{r}} '= (r',\vartheta ',\varphi ')$. Explicit forms for dyadic Green’s functions G ^ej_o and G ^hj_o are presented in Appendix 2. Introducing “scattered” fields, E ^s = E − E° and H ^s = H − H°, and subtracting (A.1.1) from (1), Maxwell’s equations for the scattered fields are obtained

$$ \nabla \times {\mathbf{H}}^{\text{s}} = \sigma_{\text{o}} {\mathbf{E}}^{\text{s}} + {\mathbf{j}}^{q} ,\quad \nabla \times {\mathbf{E}}^{\text{s}} = i\omega \mu_{\text{o}} {\mathbf{H}}^{\text{s}} , $$

(A.1.3)

where

$$ {\mathbf{j}}^{q} = (\sigma - \sigma_{\text{o}} I){\mathbf{E}}^{{\text{s}}} + {\mathbf{j}}^{{\text{s}}} ,\quad {\mathbf{j}}^{\text{s}} = (\sigma - \sigma_{\text{o}} I){\mathbf{E}}^{\text{o}} . $$

(A.1.4)

Here I is identity matrix. E ^s, in analogy with (A.1.2), can be written, as

$$ {\mathbf{E}}^{\text{s}} = \int\limits_{{V^{mod} }} {G_{\text{o}}^{ej} \left( {{\mathbf{r}},{\mathbf{r}}^{\prime}} \right){\mathbf{j}}^{q} ({\mathbf{r}}^{\prime})dv' = {\mathbf{E}}_{\text{o}} ({\mathbf{r}}) + \int\limits_{{V^{mod} }} {G_{\text{o}}^{ej} ({\mathbf{r'}}, {\mathbf{r}})(\sigma ({\mathbf{r}} ') - \sigma_{\text{o}} ({{r^{\prime}}})I){\mathbf{E}}^{\text{s}} ({\mathbf{r}} ')dv '} } , $$

(A.1.5)

which gives the conventional scattering equation with respect to the unknown field E ^s, with a free term, ${\mathbf{E}}_{\text{o}} ({\mathbf{r}}) = \int_{{V^{mod} }} {G_{\text{o}}^{ej} ({\mathbf{r}},{\mathbf{r}} ')(\sigma ({\mathbf{r}} ') - \sigma_{\text{o}} ({\mathbf{r}} ')I){\mathbf{j}}^{\text{s}} ({\mathbf{r}} ')dv'} .$Here V ^mod is a region, where σ − σ _o I differs from 0. After discretization, equation (A.1.5) can be solved by conjugate gradients (CG) iterations. However, for models with strong scatterers the resulting system of linear equations appears to be poorly conditioned. The remedy is to modify equation (A.1.5) to another integral equation (cf. Singer 1995; Pankratov et al. 1995, 1997)

$$ {\varvec{\chi}}({\mathbf{r}}) - \int\limits_{{V^{mod}}} K ({\mathbf{r}},{\mathbf{r}}')R({\mathbf{r}}'){\varvec{\chi}}({\mathbf{r}}')dv' = {\varvec{\chi}}_{\text{o}} ({\mathbf{r}}), $$

(A.1.6)

where

$$ R = (\sigma - \sigma_{\text{o}} I)(\sigma + \sigma_{\text{o}}^{*} I)^{ - 1} , $$

(A.1.7)

$$ K({\mathbf{r}},{\mathbf{r}}') = \delta ({\mathbf{r}} - {\mathbf{r}}'){{I}} + 2\sqrt {\text{Re} \sigma_{\text{o}} ({\text{r}})} G_{\text{o}}^{ej} ({\mathbf{r}},{\mathbf{r}}')\sqrt {\text{Re} \sigma_{\text{o}} (r')} , $$

(A.1.8)

$$ {\varvec{\chi}}_{\text{o}} = \int\limits_{{V^{mod}}} K ({\mathbf{r}},{\mathbf{r}}')\sqrt {\text{Re} \sigma_{\text{o}} } (\sigma + \sigma_{\text{o}}^{*} {{I}})^{ - 1} {\mathbf{j}}^{{\text{s}}} ({\mathbf{r}}')dv', $$

(A.1.9)

where $\delta ({\mathbf{r}} - {\mathbf{r}}')$ is Dirac’s delta function, $\sigma_{\text{o}}^{*}$ and Reσ _o are the respective complex conjugate and real part of σ _o, and where

$$ {\varvec{\chi}} = \frac{1}{{2\sqrt {\text{Re} \sigma_{\text{o}} } }}((\sigma + \sigma_{\text{o}}^{*} I){\mathbf{E}}^{{\text{s}}} + {\mathbf{j}}^{{\text{s}}} ). $$

(A.1.10)

Note that in order to derive (A.1.6) the term $(\sigma - \sigma_{\text{o}} ){\mathbf{E}}^{{\text{o}}} /2\text{Re} \sigma_{\text{o}}$ is added to both sides of (A.1.5) and a change of variables is performed. The specific form of equation (A.1.6) is motivated by the energy inequality for the scattered EM field, which expresses a fundamental physical fact that the energy flow of the scattered field outside the domain with inhomogeneities is always non-negative (cf. Singer 1995; Pankratov et al. 1995). The advantage of this form of integral equation is that operator A

$${A}{\varvec{\chi}} = (I - KR){\varvec{\chi}} = {\varvec{\chi}}_{\text{o}},$$

(A.1.11)

is well conditioned irrespective of the conductivity contrast in the model. Indeed it can be shown (cf. Avdeev et al. 2000) that a condition number, $k(A) = \left\| A \right\| \cdot \left\| {A^{ - 1} } \right\|,$ can be estimated as $k(A) \simeq \sqrt M ,$ where M is the maximum lateral contrast in the model, provided the reference medium is chosen in an “optimal” way (Singer 1995). Specifically, outside the depths occupied by inhomogeneities, the reference medium coincides with the conductivity of the host section, σ _b (r), but at depths with a laterally inhomogeneous distribution of conductivity it has a form $\sigma_{\text{o}} (r) = \sqrt {\mathop {\min }\limits_{\vartheta ,\varphi } \sigma (r,\vartheta ,\varphi )\mathop {\max }\limits_{\vartheta ,\varphi } \sigma (r,\vartheta ,\varphi )} .$ From the estimate for k(A) it follows that even for media with large lateral contrasts of conductivity (say, on land-ocean boundaries), the operator A still remains well conditioned.

Numerical solutions based on this volume integral approach can be represented schematically as a sequence of the following steps:

• (1) A discretized form of A cf. (A.1.11) is calculated on V ^mod (cf. (A.1.7) and (A.1.8));

• (2) j^s and χ_o are calculated on V^mod (cf. (A.1.4) and (A.1.9));

• (3) The scattering equation (A.1.6) is solved on V ^mod by the CG iterations;

• (4) E^s is calculated on V^mod from (A.1.10), and then j^q is calculated from (A.1.4);

• (5) E^s and H^s are calculated at ${\mathbf{r}} \in \;V^{obs}$ (region of interest) as

$$ {\mathbf{E}}^{{\text{s}}} = \int\limits_{{V^{mod}}} {G_{\text{o}}^{ej} ({\mathbf{r}},{\mathbf{r}}'){\mathbf{j}}^{q} ({\mathbf{r}}')dv'} ,\quad {\mathbf{H}}^{\text{s}} = \int\limits_{{V^{mod}}} {G_{\text{o}}^{hj} ({\mathbf{r}},{\mathbf{r}}'){\mathbf{j}}^{q} ({\mathbf{r}}')dv';} $$

(A.1.12)

• (6) E° and H° are calculated at ${\mathbf{r}} \in \;V^{obs}$ from (A.1.2) and finally the total fields are calculated as E = E^s + E° and H = H^s + H°;

The cornerstone of the integral equation solution is the derivation and calculation of dyadic Green’s functions, $G_{\text{o}}^{ej} ({\mathbf{r}},{\mathbf{r}}')$ and $G_{\text{o}}^{hj} ({\mathbf{r}},{\mathbf{r}}')$which are discussed below.

1.2 Appendix 2: Explicit Forms of 3 × 3 Dyadic Green’s Functions of Radially-Symmetric Section

In this Appendix the final expressions for the elements of 3 × 3 dyadic Green’s functions of radially-symmetric section, $G_{\text{o}}^{ej} ({\mathbf{r}},{\mathbf{r}}')$ and $G_{\text{o}}^{hj} ({\mathbf{r}},{\mathbf{r}}')$ are presented. The details of the derivation can be found in Kuvshinov (2008). These functions express electric and magnetic fields induced in a radially-symmetric section of conductivity $\sigma_{\text{o}} ({\text{r}})$ by an impressed current j

$$ {\mathbf{E}}(r,\vartheta ,\varphi ) = \int\limits_{V} {G_{\text{o}}^{ej} (r,r',\vartheta ,\vartheta ',\varphi - \varphi '){\mathbf{j}}(r',\vartheta ',\varphi ')dv'} , $$

(A.2.1)

$$ {\mathbf{H}}(r,\vartheta ,\varphi ) = \int\limits_{V} {G_{\text{o}}^{hj} (r,r',\vartheta ,\vartheta ',\varphi - \varphi '){\mathbf{j}}(r',\vartheta ',\varphi ')dv'} . $$

(A.2.2)

Here V is a 3-D volume occupied by a current j, $dv^{\prime} = {r^{\prime}}^{2} \sin \vartheta^{\prime} d\vartheta^{\prime} d\varphi^{\prime} dr^{\prime} ,$ and G ^ej(hj) are

$$ G_{\text{o}}^{ej(hj)} = {\mathbf{e}}_{\vartheta } g_{\vartheta \vartheta^{\prime} }^{ej(hj)} {\mathbf{e}}_{\vartheta^{\prime}} + {\mathbf{e}}_{\vartheta } g_{\vartheta \varphi^{\prime}}^{ej(hj)} {\mathbf{e}}_{\varphi^{\prime} } + \cdots + {\mathbf{e}}_{r} g_{rr^{\prime} }^{ej(hj)} {\mathbf{e}}_{r^{\prime} } , $$

(A.2.3)

where ${\mathbf{e}}_{\vartheta }$, ${\mathbf{e}}_{\varphi }$ and ${\mathbf{e}}_{r}$, ${\mathbf{e}}_{{\vartheta^{\prime}}}$, ${\mathbf{e}}_{{\varphi^{\prime}}}$ and ${\mathbf{e}}_{{r^{\prime}}}$ are the unit vectors of spherical system of coordinates at points $(r,\vartheta ,\varphi )$ and $(r',\vartheta ',\varphi ')$ respectively. For example if Eqs. A.1.12 are considered then ${\mathbf{E}} \equiv {\mathbf{E}}^{{\text{s}}} ,\;\;{\mathbf{H}} \equiv {\mathbf{H}}^{{\text{s}}} ,\;\;{\mathbf{j}} \equiv {\mathbf{j}}^{q} ,\;\;V \equiv V^{\bmod } .$ The expressions for elements $g_{{\vartheta \vartheta^{\prime}}}^{ej(hj)} ,\,g_{{\vartheta \varphi^{\prime}}}^{ej(hj)}$ … are

$$ g_{\vartheta \vartheta^{\prime} }^{ej} = \frac{1}{\sin \vartheta }\frac{1}{\sin \vartheta^{\prime} }\partial_{\varphi } \partial_{\varphi^{\prime} } P\left[ {\frac{1}{{r^{\prime}r}}\frac{{G^{t} }}{n(n + 1)}} \right] + \partial_{\vartheta } \partial_{\vartheta^{\prime} } P\left[ {\frac{1}{r^{\prime} r}\frac{{G^{p} }}{n(n + 1)}} \right], $$

(A.2.4)

$$ g_{{\vartheta \varphi^{\prime}}}^{ej} = - \frac{1}{\sin \vartheta }\partial_{\varphi } \partial_{\vartheta '} P\left[ {\frac{1}{{r^{\prime}r}}\frac{{G^{t} }}{n(n + 1)}} \right] + \frac{1}{\sin \vartheta '}\partial_{\vartheta } \partial_{\varphi '} P\left[ {\frac{1}{{r^{\prime}r}}\frac{{G^{p} }}{n(n + 1)}} \right], $$

(A.2.5)

$$ g_{\vartheta r^{\prime} }^{ej} = - \partial_{\vartheta } P\left[ {\frac{1}{{r^{\prime 2} r}}\frac{{\beta^{p} G^{p} }}{{\sigma_{\text{o}} (r^{\prime} )}}} \right], $$

(A.2.6)

$$ g_{{\varphi \vartheta^{\prime}}}^{ej} = - \frac{1}{\sin \vartheta^{\prime} }\partial_{\vartheta } \partial_{\varphi^{\prime} } P\left[ {\frac{1}{r^{\prime} r}\frac{{G^{t} }}{n(n + 1)}} \right] + \frac{1}{\sin \vartheta }\partial_{\varphi } \partial_{\vartheta^{\prime} } P\left[ {\frac{1}{r^{\prime} r}\frac{{G^{p} }}{n(n + 1)}} \right], $$

(A.2.7)

$$ g_{\varphi \varphi^{\prime} }^{ej} = \partial_{\vartheta } \partial_{\vartheta^{\prime} } P\left[ {\frac{1}{r^{\prime} r}\frac{{G^{t} }}{n(n + 1)}} \right] + \frac{1}{\sin \vartheta }\frac{1}{\sin \vartheta^{\prime} }\partial_{\varphi } \partial_{\varphi^{\prime} } P\left[ {\frac{1}{r^{\prime} r}\frac{{G^{p} }}{n(n + 1)}} \right], $$

(A.2.8)

$$ g_{\varphi r^{\prime} }^{ej} = - \frac{1}{\sin \vartheta }\partial_{\varphi } P\left[ {\frac{1}{{r^{\prime 2} r}}\frac{{\beta^{p} G^{p} }}{{\sigma_{\text{o}} (r^{\prime} )}}} \right], $$

(A.2.9)

$$ g_{r\vartheta '}^{ej} = \partial_{\vartheta '} P\left[ {\frac{1}{{r^{\prime} r^{2} }}\frac{{\alpha^{p} G^{p} }}{{\sigma_{\text{o}} (r)}}} \right], $$

(A.2.10)

$$g_{r\varphi^{\prime} }^{ej} = \frac{1}{\sin \vartheta^{\prime}}\partial_{\varphi^{\prime} } P\left[{\frac{1}{r^{\prime}r^{2}}}\frac{{\alpha^{p} G^{p}}}{{\sigma_{\text{o}} (r)}} \right], $$

(A.2.11)

$$ g_{rr^{\prime} }^{ej} = - \frac{\delta (r - r^{\prime} )\delta (\vartheta - \vartheta^{\prime} )\delta (\varphi - \varphi^{\prime} )}{{\sin \vartheta^{\prime} r^{\prime 2} \sigma_{\text{o}} (r)}} + P\left[ {\frac{1}{{r^{\prime 2} r^{2} }}\frac{{n(n + 1)\alpha^{p} \beta^{p} G^{p} }}{{\sigma_{\text{o}} (r)\sigma_{\text{o}} (r^{\prime} )}}} \right], $$

(A.2.12)

for electric field, and

$$ g_{\vartheta \vartheta '}^{hj} = - \frac{1}{\sin \vartheta '}\partial_{\vartheta } \partial_{\varphi '} P\left[ {\frac{1}{r'r}\frac{{\alpha^{t} G^{t} }}{n(n + 1)}} \right] - \frac{1}{\sin \vartheta }\partial_{\varphi } \partial_{\vartheta '} P\left[ {\frac{1}{r'r}\frac{{\alpha^{p} G^{p} }}{n(n + 1)}} \right], $$

(A.2.13)

$$ g_{{\vartheta \varphi^{\prime}}}^{hj} = - \frac{1}{\sin \vartheta }\frac{1}{\sin \vartheta '}\partial_{\varphi } \partial_{\varphi '} P\left[ {\frac{1}{{r^{\prime}r}}\frac{{\alpha^{t} G^{t} }}{n(n + 1)}} \right] + \partial_{\vartheta } \partial_{\vartheta '} P\left[ {\frac{1}{{r^{\prime}r}}\frac{{\alpha^{p} G^{p} }}{n(n + 1)}} \right], $$

(A.2.14)

$$ g_{\vartheta r^{\prime} }^{hj} = \frac{1}{\sin \vartheta }\partial_{\varphi } P\left[ {\frac{1}{{r^{\prime 2} r}}\frac{{\alpha^{p} \beta^{p} G^{p} }}{{\sigma_{\text{o}} (r^{\prime} )}}} \right], $$

(A.2.15)

$$ g_{\varphi \vartheta^{\prime} }^{hj} = \partial_{\vartheta } \partial_{\vartheta^{\prime} } P\left[ {\frac{1}{r^{\prime} r}\frac{{\alpha^{t} G^{t} }}{n(n + 1)}} \right] - \frac{1}{\sin \vartheta^{\prime} }\partial_{\vartheta } \partial_{\varphi^{\prime} } P\left[ {\frac{1}{r^{\prime} r}\frac{{\alpha^{p} G^{p} }}{n(n + 1)}} \right], $$

(A.2.16)

$$ g_{{\varphi \varphi^{\prime}}}^{hj} = \frac{1}{\sin \vartheta }\partial_{\varphi } \partial_{\vartheta '} P\left[ {\frac{1}{{r^{\prime}r}}\frac{{\alpha^{t} G^{t} }}{n(n + 1)}} \right] + \frac{1}{\sin \vartheta '}\partial_{\vartheta } \partial_{\varphi '} P\left[ {\frac{1}{{r^{\prime}r}}\frac{{\alpha^{p} G^{p} }}{n(n + 1)}} \right], $$

(A.2.17)

$$ g_{\varphi r^{\prime} }^{hj} = - \partial_{\vartheta } P\left[ {\frac{1}{{{r^{\prime}}^{2} r}}\frac{{\alpha^{p} \beta^{p} G^{p} }}{{\sigma_{\text{o}} (r^{\prime} )}}} \right], $$

(A.2.18)

$$ g_{r\vartheta \prime }^{hj} = \frac{1}{\sin \vartheta^\prime }\partial_{\varphi^\prime } P\left[ {\frac{1}{{r^{\prime}r^{2} }}\frac{{G^{t} }}{{i\omega \mu_{\text{o}} }}} \right], $$

(A.2.19)

$$ g_{r\varphi^{\prime} }^{hj} = - \partial_{\vartheta^{\prime} } P\left[ {\frac{1}{{r^{\prime} r^{2} }}\frac{{G^{t} }}{{i\omega \mu_{\text{o}} }}} \right], $$

(A.2.20)

$$ g_{rr^{\prime} }^{hj} = 0 $$

(A.2.21)

for magnetic field. Here P[f] stands for the summation of the series

$$ P[f](\cos \gamma ,r,r^{\prime} ) = \sum\limits_{n = 0}^{\infty } {\frac{2n + 1}{4\pi }f(n,r,r^{\prime} )P_{n} } (\cos \gamma ), $$

(A.2.22)

where $\cos \gamma = \cos \vartheta \cos \vartheta^{\prime} + \sin \vartheta \sin \vartheta^{\prime} \cos (\varphi - \varphi^{\prime} ),$ and P _n are Legendre polynomials. Scalar Green’s functions $G^{t(p)} (n,r,r^{\prime} )$ are given as continuous solutions of equation

$$ \partial_{r} \left( {\frac{1}{{p^{t(p)} (r)}}\partial_{r} G^{t(p)} (n,r,r^{\prime} )} \right) = q^{t(p)} (r)G^{t(p)} (n,r,r^{\prime} ) + \delta (r - r^{\prime} ), $$

(A.2.23)

with the boundary conditions $G^{t(p)} (n,r,r^{\prime} ) \to 0$ when r → 0 and r → ∞. The coefficients p ^t(p)(r) and q ^t(p)(r) are determined either as

$$ p^{t} (r) = - i\omega \mu_{\text{o}} ,\quad q^{t} (r) = - \frac{{\kappa^{2} }}{{i\omega \mu_{\text{o}} }}, $$

(A.2.24)

for the toroidal mode, or

$$ p^{p} (r) = \frac{{\kappa^{2} }}{{\sigma_{\text{o}} (r)}},\quad q^{p} (r) = \sigma_{\text{o}} (r), $$

(A.2.25)

for the poloidal mode. Here $\kappa^{2} = n(n + 1)/r^{2} - i\omega \mu_{\text{o}} \sigma_{\text{o}} (r).$ The final result for $G^{t(p)} (n,r,r')$ is given as

$$ G^{t(p)} (n,r,r') = - \;\frac{1}{{Y^{l,t(p)} (n,r') + Y^{u,t(p)} (n,r')}}{\text{e}}^{{\int\limits_{{r^{\prime}}}^{r} {p^{t(p)} (n,\xi )\alpha^{t(p)} (n,\xi ,r')d\xi } }} , $$

(A.2.26)

where the function $\alpha^{t(p)} (n,\xi ,r^{\prime})$ is

$$ \alpha^{t(p)} (n,\xi ,r^{\prime} ) = \left\{ \begin{array}{ll} - Y^{u,t(p)} (n,\xi ),& \xi > r^{\prime} \hfill \\ Y^{l,t(p)} (n,\xi ),& \xi < r^{\prime} \hfill \\ \end{array} \right.. $$

(A.2.27)

Note that the functions β^t(p) in Eqs. A.2.6, A.2.9, A.2.12, A.2.15 and A.2.18 are defined via α^t(p) as $\beta^{t(p)} (n,r,r^{\prime}) = \alpha^{t(p)} (n,r^{\prime} ,r)$.

It is seen from Eqs. A.2.26–A.2.27 that calculation of $G^{t(p)} (n,r,r^{\prime} )$ for arbitrary r and $r^{\prime}$ requires calculation of admittances $Y^{l,t(p)} (r^{\prime} )$ and $Y^{u,t(p)} (r^{\prime})$, and calculations of exponentials, which in its turn contain Y ^l,t(p) and Y ^u,t(p). In order to calculate these functions radially-symmetric sections are introduced that consist of N layers $\{ r_{k + 1} < r \le r_{k} \}_{k = 1,2, \ldots ,N} ,$ where within each layer the conductivity varies as $\sigma_{\text{o}} (r) = \sigma_{k} \left( {\frac{{r_{k} }}{r}} \right)^{2}$ (e.g. Fainberg et al. 1990). Then the recurrences for calculating $Y^{p,l} (r_{k} ),$ $Y^{p,u} (r_{k} ),$ $Y^{t,l} (r_{k} )$ and Y ^t,u(r _k ) for arbitrary r _k are

$$ Y_{k}^{l,p} \equiv Y^{l,p} (r_{k} ) = g_{k} \frac{{Y_{k + 1}^{l,p} (b_{k} - 0.5\tau_{k} ) - g_{k} \eta_{k} \tau_{k} }}{{g_{k} \eta_{k} (b_{k} + 0.5\tau_{k} ) - b_{k}^{ + } b_{k}^{ - } \tau_{k} Y_{k + 1}^{l,p} }},\quad k = N - 1,\; \ldots ,2,\;1,\,Y_{N}^{l,p} = \frac{{\sigma_{N} r_{N} }}{{b_{N}^{ - } }}, $$

(A.2.28)

$$ Y_{k + 1}^{u,p} = g_{k} \eta_{k} \frac{{Y_{k}^{u,p} (b_{k} + 0.5\tau_{k} ) - g_{k} \tau_{k} }}{{g_{k} (b_{k} - 0.5\tau_{k} ) - b_{k}^{ + } b_{k}^{ - } \tau_{k} Y_{k}^{u,p} }},\quad k = 1,\;2, \ldots ,N - 1,\quad Y_{1}^{u,p} = \frac{{\sigma_{1} r_{1} }}{{b_{1}^{ + } }}, $$

(A.2.29)

$$ Y_{k}^{l,t} = \frac{1}{{q_{k} }}\frac{{q_{k + 1} Y_{k + 1}^{l,t} (b_{k} - 0.5\tau_{k} ) + b_{k}^{ + } b_{k}^{ - } \tau_{k} }}{{(b_{k} + 0.5\tau_{k} ) + q_{k + 1} \tau_{k} Y_{k + 1}^{l,t} }},\;\;k = N - 1,\; \ldots ,2, 1,\quad Y_{N}^{l,t} = - \frac{{b_{N}^{ + } }}{{q_{N} }}, $$

(A.2.30)

$$ Y_{k + 1}^{u,t} = \frac{1}{{q_{k + 1} }}\frac{{q_{k} Y_{k}^{u,t} (b_{k} + 0.5\tau_{k} ) + b_{k}^{ + } b_{k}^{ - } \tau_{k} }}{{(b_{k} - 0.5\tau_{k} ) + q_{k} \tau_{k} Y_{k}^{u,t} }},\quad k = 1,\;2, \ldots ,N - 1,\quad Y_{1}^{u,t} = - \frac{{b_{1}^{ - } }}{{q_{1} }}, $$

(A.2.31)

where $q_{k} = i\omega \mu_{\text{o}} r_{k}$, $b_{k}^{ - } = b_{k} - \frac{1}{2},\quad b_{k}^{ + } = b_{k} + \frac{1}{2},\quad b_{k} = \{ (n + \frac{1}{2})^{2} - i\omega \mu_{0} \sigma_{k} r_{k}^{2} \}^{{\frac{1}{2}}}$, $\eta_{k} = \frac{{r_{k} }}{{r_{k + 1} }}$, $\tau_{k} = \frac{{1 - \varsigma_{k} }}{{1 + \varsigma_{k} }}$, $\varsigma_{k} = \eta_{k}^{{2b_{k} }}$ and $g_{k} = \sigma_{k} r_{k}$. Finally G ^t(p)(n,r _i, r _j ) for r _i  < r _j can be written as

$$ G^{t} (n,r_{i} ,r_{j} ) = - \frac{1}{{Y_{j}^{t,\;l} + Y_{j}^{t,\;u} }}\prod\limits_{k = j}^{i} {F_{k}^{t} } ,\quad r_{i} \le r_{j} ,\,F_{k}^{t} = \frac{1}{{1 + \zeta_{k} }}\frac{{2b_{k} \eta_{k}^{{b_{k}^{ - } }} }}{{(b_{k} + 0.5\tau_{k} ) + q_{k} \tau_{k} Y_{k + 1}^{l,t} }}, $$

(A.2.32)

$$ G^{p} (n,r_{i} ,r_{j} ) = - \frac{1}{{Y_{j}^{p,\;l} + Y_{j}^{p,\;u} }}\prod\limits_{k = j}^{i} {F_{k}^{p} } ,\;r_{i} \le r_{j} ,\,F_{k}^{p} = \frac{1}{{1 + \zeta_{k} }}\frac{{2g_{k} b_{k} \eta_{k}^{{b_{k}^{ - } }} }}{{g_{k} \eta_{k} (b_{k} + 0.5\tau_{k} ) - b_{k}^{ + } b_{k}^{ - } \tau_{k} Y_{k + 1}^{l,p} }}. $$

(A.2.33)

Note that, due to the symmetry of scalar Green’s functions, $G^{t(p)} (n,r,r^{\prime}) = G^{t(p)} (n,r^{\prime},r)$, one can readily obtain the results for r _i  > r _j . Finally, it is relevant to mention that C-responses, C _n , at the Earth’s surface are connected to $Y_{1}^{l,t}$ as

$$ C_{n} = - 1/i\omega \mu_{\text{o}} Y_{1}^{l,t} , $$

(A.2.34)

whereas Q _n are connected with $Y_{1}^{l,t}$ as

$$ Q_{n} = \frac{n}{n + 1}\frac{{i\omega \mu_{\text{o}} a\,Y_{1}^{l,t} + n + 1}}{{i\omega \mu_{\text{o}} a\,Y_{1}^{l,t} - n}}. $$

(A.2.35)