Appendix A: Semi-analytical solutions expressed as wavenumber integrations
According to Eqs. (30) and (31) of Sun et al. (2021) and utilizing recurrence formulas of Bessel function, once the expansion coefficients of seismic displacement uT,m(z, k), uS,m(z, k), and uR,m(z, k) are obtained, the radial, azimuthal, and vertical components of seismic displacement ur, uθ, and uz (i.e., seismic solutions) can be written as
$$\begin{gathered} u_{r} (r,\theta ,z) = \frac{1}{{4\pi }}\sum\limits_{{m = - \infty }}^{{ + \infty }} {e^{{im\theta }} \left\{ {\int\limits_{0}^{{ + \infty }} {\left[ {iu_{{T,m}} (z,k) + u_{{S,m}} (z,k)} \right]J_{{m - 1}} (kr)k{\text{d}}k} } \right.} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad + \left. {\int\limits_{0}^{{ + \infty }} {\left[ {iu_{{T,m}} (z,k) - u_{{S,m}} (z,k)} \right]J_{{m + 1}} (kr)k{\text{d}}k} } \right\}, \hfill \\ \end{gathered}$$
(A1)
$$\begin{gathered} u_{\theta } (r,\theta ,z) = \frac{1}{{4\pi }}\sum\limits_{{m = - \infty }}^{{ + \infty }} {e^{{im\theta }} \left\{ {\int\limits_{0}^{{ + \infty }} {\left[ {iu_{{S,m}} (z,k) - u_{{T,m}} (z,k)} \right]J_{{m - 1}} (kr)k{\text{d}}k} } \right.} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad + \left. {\int\limits_{0}^{{ + \infty }} {\left[ {iu_{{S,m}} (z,k) + u_{{T,m}} (z,k)} \right]J_{{m + 1}} (kr)k{\text{d}}k} } \right\}, \hfill \\ \end{gathered}$$
(A2)
$$u_{z} (r,\theta ,z) = \frac{1}{{2\pi }}\sum\limits_{{m = - \infty }}^{{ + \infty }} {e^{{im\theta }} } \int\limits_{0}^{{ + \infty }} {\left[ { - u_{{R,m}} (z,k)} \right]J_{m} (kr)k{\text{d}}k} .$$
(A3)
Similarly, according to Eqs. (31), (66), and (67) of Sun et al. (2021), EM solutions can be written as
$$\begin{gathered} \left[ {\begin{array}{*{20}c} {E_{r} (r,\theta ,z)} \\ {H_{r} (r,\theta ,z)} \\ \end{array} } \right] = \frac{1}{{4\pi }}\sum\limits_{{\zeta = SH,PSV}} {\;\sum\limits_{{p = 0, + 1, - 1}} {\;\;\sum\limits_{{m = - \infty }}^{{ + \infty }} {e^{{im\theta }} \times } } } \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \left\{ {\int\limits_{0}^{{ + \infty }} {\left[ {\begin{array}{*{20}c} {iE_{{T,m + p}}^{{p,\zeta }} (z,k) + E_{{S,m + p}}^{{p,\zeta }} (z,k)} \\ {iH_{{T,m + p}}^{{p,\zeta }} (z,k) + H_{{S,m + p}}^{{p,\zeta }} (z,k)} \\ \end{array} } \right]J_{{m + p - 1}} (kr)k{\text{d}}k} } \right. \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad + \left. {\int\limits_{0}^{{ + \infty }} {\left[ {\begin{array}{*{20}c} {iE_{{T,m + p}}^{{p,\zeta }} (z,k) - E_{{S,m + p}}^{{p,\zeta }} (z,k)} \\ {iE_{{T,m + p}}^{{p,\zeta }} (z,k) - E_{{S,m + p}}^{{p,\zeta }} (z,k)} \\ \end{array} } \right]J_{{m + p + 1}} (kr)k{\text{d}}k} } \right\}, \hfill \\ \end{gathered}$$
(A4)
$$\begin{gathered} \left[ {\begin{array}{*{20}c} {E_{\theta } (r,\theta ,z)} \\ {H_{\theta } (r,\theta ,z)} \\ \end{array} } \right] = \frac{1}{{4\pi }}\sum\limits_{{\zeta = SH,PSV}} {\;\sum\limits_{{p = 0, + 1, - 1}} {\;\;\sum\limits_{{m = - \infty }}^{{ + \infty }} {e^{{im\theta }} \times } } } \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \left\{ {\int\limits_{0}^{{ + \infty }} {\left[ {\begin{array}{*{20}c} {iE_{{S,m + p}}^{{p,\zeta }} (z,k) - E_{{T,m + p}}^{{p,\zeta }} (z,k)} \\ {iH_{{S,m + p}}^{{p,\zeta }} (z,k) - H_{{T,m + p}}^{{p,\zeta }} (z,k)} \\ \end{array} } \right]J_{{m + p - 1}} (kr)k{\text{d}}k} } \right. \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad + \left. {\int\limits_{0}^{{ + \infty }} {\left[ {\begin{array}{*{20}c} {iE_{{S,m + p}}^{{p,\zeta }} (z,k) + E_{{T,m + p}}^{{p,\zeta }} (z,k)} \\ {iE_{{S,m + p}}^{{p,\zeta }} (z,k) + E_{{T,m + p}}^{{p,\zeta }} (z,k)} \\ \end{array} } \right]J_{{m + p + 1}} (kr)k{\text{d}}k} } \right\}, \hfill \\ \end{gathered}$$
(A5)
$$\left[ {\begin{array}{*{20}c} {E_{z} (r,\theta ,z)} \\ {H_{z} (r,\theta ,z)} \\ \end{array} } \right] = \frac{1}{{2\pi }}\sum\limits_{{\zeta = SH,PSV}} {\;\sum\limits_{{p = 0, + 1, - 1}} {\;\;\sum\limits_{{m = - \infty }}^{{ + \infty }} {e^{{im\theta }} } } } \int\limits_{0}^{{ + \infty }} {\left[ {\begin{array}{*{20}c} { - E_{{R,m + p}}^{{p,\zeta }} (z,k)} \\ { - H_{{R,m + p}}^{{p,\zeta }} (z,k)} \\ \end{array} } \right]J_{{m + p}} (kr)k{\text{d}}k} .$$
(A6)
Obviously, both the seismic and EM solutions, i.e., Eqs. (A1, A2, A3, A4, A5, and A6), are integrations with respect to wavenumber k. The order ‘m’ in above equations will be limited as |m|≤ 2 if a ‘source-center’ cylindrical coordinate system is chosen to let the point source located in z-axis (Chen 1993, 1999). The above wavenumber integrations can be regarded as summation of several inverse Hankel Transforms, because Bessel function of first kind is contained in every integrand.
Appendix B: Amplitude decay factor of the evanescent EM waves
Evanescent EM waves are generated at an interface when θ > θc (or k > ω/Vem) which causes \(\gamma _{\text{em}}\) (\(= \sqrt {k^{2} - \omega ^{2} /V_{{{\text{em}}}}^{2} }\)) to have a non-zero real part. Following Sun et al. (2021), we can assume the general solutions of the non-localized EM waves (which have EM phase velocity) contain factors like \(\exp (\gamma _{{{\text{em}}}} z_{{{\text{rcv}}}} - i\omega t)\), where \(\text{Re} \{ \gamma _{{{\text{em}}}} \} > 0\) and zrcv represents the depth of an in-air receiver. In this work, we consider the z axis downward positive. Thus, zrcv < 0 and − zrcv is the normal distance from the in-air receiver to the ground surface (z = 0 m). Then, the amplitude decay factor of the evanescent EM waves fdecay is determined by
$$f_{{{\text{decay}}}} (\omega ,z) = \exp (\gamma _{{{\text{em}}}} z_{{{\text{rcv}}}} ).$$
(B1)
Considering the relation \(k = \sin \theta ({\omega \mathord{\left/ {\vphantom {\omega {V_{{{\text{sei}}}} }}} \right. \kern-\nulldelimiterspace} {V_{{{\text{sei}}}} }})\), we can obtain
$$\gamma _{{{\text{em}}}} = k\sqrt {1 - (V_{{{\text{sei}}}} \csc\theta /V_{{{\text{em}}}} )^{2} } ,$$
(B2)
where csc is the cosecant, i.e., cscθ = 1/sinθ. Substituting the above equation into equation (B1) gives
$$f_{{{\text{decay}}}} (\omega ,z){\text{ = }}\exp \left[ {k\sqrt {1 - (V_{{{\text{sei}}}} \csc\theta /V_{{{\text{em}}}} )^{2} } z_{{{\text{rcv}}}} } \right].$$
(B3)
The EM wave velocity Vem is usually much greater than the seismic wave velocity Vsei; therefore, the inequation Vseicscθ/Vem ≪ 1 is usually satisfied unless the seismic incident angle θ is a small value close to zero. Thus, the amplitude decay factor will be determined by
$$f_{{{\text{decay}}}} (\omega ,z) \approx \exp \left( {\omega \frac{{\sin \theta }}{{V_{{{\text{sei}}}} }}z_{{{\text{rcv}}}} } \right).$$
(B4)
The above equation suggests that the amplitude decay speed of the evanescent EM waves greatly depends on the frequency ω, seismic incident angle θ, and seismic wave velocity Vsei.