A vertical contact separating two resistive domains, \(\rho_1\) and \(\rho_2\) (with \(\rho_2 > \rho_1\)), is shown in the figure below, with the strike direction along X, the profile direction (perpendicular to strike) along Y - where the electric field component \(E_y\) is discontinuous across the contact - and depth along Z. The magnitude of the discontinuity in the apparent resistivity, when the electric field, \(E\), is perpendicular to the strike, is
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E perpendicular to strike is the H-polarization (TM) mode - use continuity of tangential H and of normal current density across the contact.
When the electric field is perpendicular to strike, current flows across the vertical contact - this is the H-polarization (TM) mode of a 2-D resistivity structure, exactly the mode the figure highlights by marking \(E_y\) as discontinuous.
Step 1: Boundary conditions across the vertical contact. The magnetic field component along strike, \(H_x\), is tangential to the contact and must be continuous across it. The current density perpendicular to the contact, \(J_y\), must also be continuous (charge conservation - no sources or sinks of current at the boundary itself).
Step 2: Relate the electric fields on the two sides. Since \(J_y = E_y/\rho\) and \(J_y\) is continuous, \(E_{y1}/\rho_1 = E_{y2}/\rho_2 = J_y\), so \(E_{y1} = J_y\rho_1\) and \(E_{y2}=J_y\rho_2\), giving \(E_{y1}/E_{y2} = \rho_1/\rho_2\).
Step 3: Apparent resistivity on each side. Apparent resistivity from the impedance \(Z_{yx}=E_y/H_x\) goes as \(\rho_{a}\propto |E_y/H_x|^2\). Since \(H_x\) is the same (continuous) on both sides of the contact, the ratio of the apparent resistivities computed just on the \(\rho_1\) side to that on the \(\rho_2\) side is \(\rho_{a1}/\rho_{a2} = (E_{y1}/E_{y2})^2 = (\rho_1/\rho_2)^2\).
So the magnitude of the discontinuity in apparent resistivity across the vertical contact, for E perpendicular to strike, is \(\boxed{(\rho_1/\rho_2)^2}\) - option (A).
