## 2 Materials and Methods
Thus, (9) and (15) can be rewritten as [8]:  
\[\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{z}}{\partial z}=i\omega \upmu_{0}\varepsilon_{0}\varepsilon_{2}^{\prime}U. \tag{3}\]8  
\[\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{z}}{\partial z}=i\omega \upmu_{0}\varepsilon_{0}\varepsilon_{2}^{\prime}U. \tag{3}\]9  
We determine the secondary potential of the spheroid, taking into account the water-air boundary, by substituting (18) and (19) in (3):  
\[ \begin{array}{l}U^{s}=\frac{1}{\omega\
u\mu_{0}\varphi_{2}\sigma}div\overset{ \rightarrow}{A}=-\frac{1}{\mu_{0}\
u_{2}}(\frac{\partial A_{x}}{\partial x}+ \frac{\partial A_{z}}{\partial z})=\\ =\frac{P}{4\pi\
u_{2}}\frac{x}{r}\left[\int\limits_{0}^{\infty}{l_{1}(\gamma r )e^{-q_{2}}\frac{\gamma d^{2}\zeta}{q_{2}}}-\int\limits_{0}^{\infty}{l_{1}( \gamma r)e^{-(2H-2h)q_{2}}\frac{\gamma^{2}d\zeta}{q_{2}}}\right]}.\end{array}  \tag{20}\]  
The function (20) is called the flow characteristic of a spheroid when it moves in a conducting layer. When calculating (20), it is taken into account that [9]  
\[{J_{0}}^{\prime}(\gamma r)=-\frac{x}{r}{J_{1}(\gamma r)},\frac{dA_{z}}{dz}=0.\]  
From the integral tables, we have [10]:  
\[U^{s}=\frac{P}{4\pi\sigma}[\frac{x}{R_{1}^{3}}(1+ik_{2}R_{1})\exp(-ik_{2}R_{1 })-\frac{x}{R_{1}^{(1)3}}(1+ik_{2}R_{1}^{(1)})\exp(-ik_{2}R_{1}^{(1)})]. \tag{21}\]  
where \(R_{1}\) is the distance from the center of the spheroid to the first receiving electrode; \(R_{1}^{(1)}=\sqrt{r^{2}+l^{2}-2 rl\sin\varphi+(2H-2h)^{2}}\) is the distance from the \"mirror\" source of the secondary field of the spheroid to the first receiving electrode (Figure 1). The equivalent dipole moment of the spheroid \(P\), taking into account the work of [7,11], has the form:  
\[P=\frac{32 Ul_{0}bl\sigma}{9}\frac{AC^{2}r(\alpha_{1}-i\beta_{1})}{1-\frac{AC^ {2}}{A^{2}-C^{2}}(\text{Art}\eta_{0}-\mu_{0})}, \tag{22}\]  
where \(U_{0}\) is the voltage of the primary harmonic field; \(A\) and \(C\) are the major and minor semi-axes of the elongated spheroid; \(\eta_{0}=\sqrt{1-\frac{C^{2}}{A^{2}}}\) is the eccentricity of the spheroid; \(\alpha_{1}\) and \(\beta_{1}\) are auxiliary functions; and:  
\[ \begin{array}{l}\alpha_{1}=\frac{e^{-k_{0}\sigma_{0}}}{r_{0}^{4}}\{(2+k_{0} r_{0})\cos k_{0}r_{0}+k_{0}r_{0}\sin k_{0}r_{0}\},\\ \beta_{1}=-\frac{e^{-k_{0}r}}{r_{0}^{4}}\{(2+k_{0}r_{0})\sin k_{0}r_{0}-k_{0}r_ {0}\cos k_{0}r_{0}\},\end{array} \]  
\(r_{0}=\sqrt{r^{2}+h^{2}}\)--the distance from the center of the spheroid to the center of the antenna system; \(k_{0}=\sqrt{\frac{\omega\mu_{0}\rho_{2}}{2}}\)--the modulus of the field propagation coefficient in seawater. Thus, to calculate the difference signal of the secondary field on the electrodes of the receiving EA, two factors must be taken into account:  
(1) The secondary field of the spheroid is excited by the sum of the fields of the grounded cable and its \"mirror\" image in the upper boundary of the layer;(2) The secondary field of the spheroid is equal to the difference between the secondary field of the spheroid excited by the transmitting EA and its \"mirror\" image at the upper boundary of the layer. The secondary field of the spheroid (21), taking into account the value of the dipole moment (22) on the receiving electrodes, has the form:  
\[ \begin{split}& U_{p}^{s}={P_{x}}^{\prime}{x_{1}}^{\prime}[(1+ik_{2}R_{ 1})\frac{e^{-ik_{2}R_{1}}}{R_{1}^{3}}-(1+ik_{2}R_{1}^{(1)})\frac{e^{-ik_{2}R_{1 }^{(1)}}}{R_{1}^{(1)}}]-{P_{x}}^{\prime}{x_{2}}^{\prime}[(1+ik_{2}R_{2})\frac{e ^{-ik_{2}R_{2}}}{R_{2}^{2}}\\ &-(1+ik_{2}R_{2}^{(2)})\frac{e^{-ik_{2}R_{2}^{(2)}}}{R_{2}^{(23)} }]+{P_{x}}^{\prime}h\left[(1+ik_{2}R_{1})\frac{e^{-ik_{2}R_{1}^{(1)}}}{R_{1}^{ 3}}-(1+ik_{2}R_{1}^{(1)})\frac{e^{-ik_{2}R_{1}^{(1)}}}{R_{1}^{(1)}}\right]-\\ &-{P_{z}}^{\prime}h\left[(1+ik_{2}R_{2})\frac{e^{-ik_{2}R_{2}^{(2 )}}}{R_{2}^{3}}-(1+ik_{2}R_{2}^{(2)})\frac{e^{-ik_{2}R_{2}^{(2)}}}{R_{2}^{(23)} }\right],\end{split}