# Three-Dimensional Wide-Band Electromagnetic Forward Modelling Using Potential Technique
## 2 Theory and Methodology  
### Formulation of [PERSON]'s Equations in the Secondary Field  
Considering [PERSON]'s equations for geophysical application in the quasi-static frequency domain, we have  
\[\
abla\times\mathbf{E}=i\mu_{o}\omega\mathbf{H} \tag{1}\]  
\[\
abla\times\mathbf{H}=\mathbf{J} \tag{2}\]  
where the time dependence is \(e^{-i\omega t}\), \(\mathbf{E}\) is the electric field, \(\mu_{o}\) indicates the magnetic permeability in the free space, \(\omega\) denotes the angular frequency, \(\mathbf{H}\) represents the magnetic field, \(\mathbf{J}=\sigma\mathbf{E}\) which signifies the current density, and \(\sigma\) implies the conductivity [35]. By applying the curl to Equation (1) and substituting Equation (2), one can obtain the second derivative equation for the total \(\mathbf{E}\) field. In addition, we can partition the total field into the primary and secondary fields [36]. The secondary fields satisfy  
\[\
abla\times\
abla\times\mathbf{E}_{\mathrm{s}}-i\mu_{o}\omega\sigma\mathbf{E}_{ \mathrm{s}}=i\mu_{o}\omega\Delta\sigma\mathbf{E}_{\mathrm{p}}\text{ in }\Omega \tag{3}\]  
Here, \(\mathbf{E}_{p}\) and \(\mathbf{E}_{\mathrm{s}}\) are the primary and secondary electric fields respectively. the (complex valued) conductivity can be expressed as \(\sigma=\Delta\sigma+\sigma_{b}\) with \(\Delta\sigma=\sigma-\sigma_{b}\), \(\sigma_{b}\) is the known background conductivity and \(\Omega\) indicates the computed domain. For the Equation (3), the primary field are computed for a layered background conductivity model. We can decompose the magnetic induction (\(\mathbf{B}\)) field and the electric field (\(\mathbf{E}\)) as  
\[\mathbf{B}=\
abla\times\mathbf{A} \tag{4}\]\[\mathbf{E}=i\omega(\mathbf{A}+\
abla\varphi) \tag{5}\]  
where, \(\mathbf{A}\) is the magnetic vector potential and \(\varphi\) denotes the electric scalar potential. By implementing the formulation in terms of secondary potentials, the curl-curl Equation (3) can be expressed in terms of the primary potentials \((\mathbf{A}_{p},\varphi_{p})\) and secondary potentials \((\mathbf{A}_{s},\varphi_{s})\) as  
\[\
abla\times\
abla\times\mathbf{A}_{s}-i\mu_{o}\omega\sigma_{b}(\mathbf{A}_{s}+\
abla \varphi_{s})=i\mu_{o}\omega\Delta\sigma(\mathbf{A}_{p}+\
abla\varphi_{p}) \tag{6}\]  
Using the vector identity \(\
abla\times\
abla\times\mathbf{A}=-\
abla^{2}\mathbf{A}+\
abla(\
abla.\mathbf{A})\), and enforcing the Lorenz gauge condition \(\
abla.\mathbf{A}=i\mu_{o}\omega\sigma_{b}\varphi\) (it allows the uniqueness of the magnetic vector \((\mathbf{A})\) potential), we can obtain  
\[\
abla^{2}\mathbf{A}_{s}-\
abla(i\mu_{o}\omega\sigma_{b}\varphi_{s})+i\mu_{o} \omega\sigma_{b}(\mathbf{A}_{s}+\
abla\varphi_{s})=-i\mu_{o}\omega\Delta\sigma( \mathbf{A}_{p}+\
abla\varphi_{p}) \tag{7}\]  
We need an additional equation to solve the Equation (7). Taking the divergence of Equation (6), we get  
\[\
abla.\{i\mu_{o}\omega\sigma_{b}(\mathbf{A}_{s}+\
abla\varphi_{s})\}=\
abla.\{i \mu_{o}\omega\Delta\sigma(\mathbf{A}_{p}+\
abla\varphi_{p})\} \tag{8}\]  
which is equivalent to enforcing the continuity of current density at boundaries. Equations (6) and (8) with homogenous (Dirichlet) boundary condition \((\mathbf{A}_{s},\varphi_{s})\equiv(0,0)\) are used to obtain the unknowns.