# Electromagnetic Field Variation of ELF Near-Region Excited by HED in a Homogeneous Half-Space Model
## Appendix A  
The strict solution of \(F_{1}\) (\(k_{1}\), \(d_{0}\))-\(F_{5}\) (\(k_{1}\), \(d_{0}\)) and \(G_{1}\) (\(k_{1}\), \(d_{1}\))-\(G_{5}\) (\(k_{1}\), \(d_{1}\)) in the Sommerfeld integral is as follows:  
\[F_{1}(k_{1},d_{0})=\int_{0}^{\infty}\frac{e^{j\gamma_{1}d_{0}}}{\gamma_{1}}J_{0}( \lambda\rho)\lambda d\lambda=-\frac{je^{ik_{1}r_{0}}}{r_{0}} \tag{10}\]  
\[F_{2}(k_{1},d_{0})=\int_{0}^{\infty}\frac{e^{j\gamma_{1}d_{0}}}{\gamma_{1}}[J_{ 0}(\lambda\rho)-I_{2}(\lambda\rho)]\lambda^{3}d\lambda=-2k_{1}\Bigg{[}\frac{k _{1}}{r_{0}}-\frac{2}{r_{0}^{2}}-\frac{2j}{k_{1}r_{0}^{3}}-\frac{d_{0}^{2}}{r_{ 0}^{2}}(\frac{ik_{1}}{r_{0}}-\frac{3j}{r_{0}^{2}})\Bigg{]}e^{ik_{1}r_{0}} \tag{11}\]  
\[F_{3}(k_{1},d_{0})=\int_{0}^{\infty}e^{j\gamma_{1}d_{0}}I_{1}(\lambda\rho) \lambda^{2}d\lambda=-pk_{1}\frac{d_{0}}{r_{0}}\Bigg{(}\frac{k_{1}}{r_{0}^{2}} +\frac{3i}{r_{0}^{3}}-\frac{3}{k_{1}r_{0}^{4}}\Bigg{)}e^{ik_{1}r_{0}} \tag{12}\]  
\[F_{4}(k_{1},d_{0})=\int_{0}^{\infty}e^{j\gamma_{1}d_{0}}J_{0}(\lambda\rho) \lambda d\lambda=-\frac{d_{0}}{r_{0}}(\frac{k_{1}}{r_{0}^{2}}-\frac{1}{r_{0}^ {2}})e^{ik_{1}r_{0}} \tag{13}\]  
\[F_{5}(k_{1},d_{0})=\int_{0}^{\infty}\frac{e^{j\gamma_{1}d_{0}}}{\gamma_{1}}I_{ 1}(\lambda\rho)\lambda^{2}d\lambda=-pk_{1}(\frac{1}{r_{0}^{2}}+\frac{1}{k_{0}^ {3}})e^{ik_{1}r_{0}} \tag{14}\]  
\[G_{1}(k_{1},d_{1})=\int_{0}^{\infty}e^{j\gamma_{1}d_{1}}J_{1}(\lambda\rho) \lambda^{2}d\lambda=-pk_{1}\frac{d_{1}}{r_{1}}\Bigg{(}\frac{k_{1}}{r_{1}^{2} }+\frac{3j}{r_{1}^{3}}-\frac{3j}{k_{1}r_{1}^{4}}\Bigg{)}e^{ik_{1}r_{1}} \tag{15}\]  
\[G_{2}(k_{1},d_{1})=\int_{0}^{\infty}e^{j\gamma_{1}d_{1}}[J_{0}(\lambda\rho)+J _{2}(\lambda\rho)]\lambda d\lambda=2\Bigg{(}\frac{e^{ik_{1}d_{1}}}{\rho^{2}} -\frac{d_{1}}{r_{1}}\frac{e^{ik_{1}r_{1}}}{\rho^{2}}\Bigg{)} \tag{16}\]  
\[G_{3}(k_{1},d_{1})=\int_{0}^{\infty}e^{j\gamma_{1}d_{1}}[J_{0}(\lambda\rho)-J _{2}(\lambda\rho)]\lambda d\lambda=-2(\frac{1}{\rho^{2}}+\frac{d_{1}}{r_{1}} \Bigg{(}\frac{ik_{1}}{r_{1}}-\frac{1}{r_{1}^{2}}-\frac{1}{\rho^{2}}\Bigg{)})e^ {ik_{1}r_{1}} \tag{17}\]  
\[G_{4}(k_{1},d_{1})=\int_{0}^{\infty}\gamma_{1}e^{j\gamma_{1}d_{1}}[J_{0}( \lambda\rho)-J_{2}(\lambda\rho)]\lambda d\lambda=-2k_{1}\Bigg{\{}\frac{e^{ik_ {1}d_{1}}}{\rho^{2}}+\frac{d_{1}^{2}}{k_{1}r_{1}^{3}}+\frac{d_{1}^{2}}{r_{1}^{ 2}}\Bigg{(}\frac{k_{1}}{r_{1}}-\frac{3}{r_{1}^{2}}-\frac{3j}{k_{1}r_{1}^{3}}- \frac{1}{\rho^{2}}\Bigg{)}\Bigg{]}e^{ik_{1}r_{1}}\Bigg{\}} \tag{18}\]  
\[G_{5}(k_{1},d_{1})=\int_{0}^{\infty}\gamma_{1}e^{j\gamma_{1}d_{1}}[J_{0}( \lambda\rho)+J_{2}(\lambda\rho)]\lambda d\lambda=-2k_{1}\Bigg{\{}\frac{e^{ik_ {1}d_{1}}}{\rho^{2}}+\frac{je^{ik_{1}r_{1}}}{k_{1}r_{1}^{3}}\Bigg{(}1+\frac{k_ {1}r_{1}d_{1}^{2}}{\rho^{2}}\Bigg{)}\Bigg{\}} \tag{19}\]  
where \(J_{n}\) (\(\lambda\rho\)), \(n=0,1,2\), is the expression of \(n\)-order Bessel function, \(\lambda\) is the wavelength, and \(\rho\) is propagation distance. \(\gamma_{1}\) is the propagation parameter in the seawater medium of region 1, and \(\gamma_{2}\) is the propagation parameter in the air medium of region 2. \(r_{0}\) is the geometric distance from the source point to the receiving point of the seawater medium, \(r_{1}\) is the geometric distance from the image point of the field source to the receiving point of the air medium, \(d_{0}\) is the vertical height between the source and receiving points of the electric dipole in the seawater medium, and \(d_{1}\) is the vertical height between the image point of the field source and receiving point of the air medium. \(k_{1}\) is the wave number in the seawater medium of region 1. \(j\) is an imaginary unit.