Spectral method for the inverse vertical electrical sounding problem in two-dimensional quasi-layered media (Q1841130)

Vertical electrical sounding involves determing the distribution of electrical conductivity in the Earth's interior from measurements of the electrical field excited on the Earth's surface by a current source at the point \(M_s=(0,y_s,0).\) They consider sounding of two-dimensional quasi-layered media in which the conductivity distribution is defined in the following form: \[ \sigma(y,z) = \begin{cases} \sigma_0=0 & \text{ for } 0>z , -\infty <y < \infty, \\ \sigma(z) &\text{ for } 0<z<H(y),\;-\infty <y < \infty, \\ \sigma_0=0 &\text{ for } H(y)<z<\infty,\;-\infty <y < \infty,\end{cases} \] \[ H(y) = \begin{cases} H(y) & \text{ for } 0<y<l \\ H_0=\text{const} & \text{ for } y<0 \text{ and }y>l,\end{cases} \] where \(\sigma(y,z)\) and \(H(y)\) are unknown functions which must be found. The measured electrical field \(E(M)\) is expressed in terms of the scalar potential \(U(M)\) in the form \(\bar E(M)=-\operatorname {grad} U(M).\) The scalar potential is the solution of the boundary-value problem \[ \begin{cases} \text{div}(\sigma(z)\operatorname {grad} U(M))=-I\delta(x)\delta(y-y_s)\delta(z), \cr \frac{\partial U}{\partial z}|_{z=0}=0; \quad \frac{\partial U}{\partial z}|_{z=H(y)}=0; \end{cases} \tag{1} \] where \(I\) is the source current. Introducing the potential spectrum \[ u_v(y,z)=\int\limits^{\infty}_{-\infty} U(x,y,z)e^{-ivx}dx, \] they transform (1) to the boundary-value problem \[ \begin{cases} \frac{\partial}{\partial y} \left(\sigma(z)\frac{\partial u_v}{\partial y}\right)+ \frac{\partial}{\partial z}\left(\sigma(z)\frac{\partial u_v}{\partial z}\right)- v^2\sigma(z)u_v=-I\delta(y-y_s)\delta(z), \cr \frac{\partial U_v}{\partial z}|_{z=0}=0; \quad \frac{\partial U_v}{\partial n}|_{z=H(y)}=0, \end{cases} \tag{2} \] where \(\bar n\) is the normal to the surface \(z=H(y).\) The problem (2) reduced to an integral equation, can be solved numerically.