A two-layer situation is encountered often in electrical prospecting. two-layer cases were interpreted with the aid of characteristic curves.
These theoretical curves, calculated for a particular four-electrode array, take into account the change in depth penetration when current lines cross the boundary to a layer with different resistivity.
The electrical boundary conditions require continuity of the component of current density J normal to the interface and of the component of electric field E tangential to the interface.
At a boundary the current lines behave like optical or seismic rays, and are guided by similar laws of reflection and refraction. For example, if u is the angle between a current line and the normal to the interface, the electrical “law of refraction” is
Characteristic curves can also be computed for the interpretation of structures with multiple horizontal layers, The apparent resistivity curve for a three-layer structure generally has one of four typical shapes, determined by the vertical sequence of resistivities in the layers.
The type K curve rises to a maximum then decreases, indicating that the intermediate layer has a higher resistivity than the top and bottom layers.
The type H curve shows the opposite effect; it falls to a minimum then increases again due to an intermediate layer that is a better conductor than the top and bottom layers.
The type A curve may show some changes in gradient but the apparent resistivity generally increases continuously with increasing electrode separation, indicating that the true resistivities increase with depth from layer to layer.
The type Q curve exhibits the opposite effect; it decreases continuously along with a progressive decrease of resistivity with depth.
Once the observed resistivity profile has been identified as of K, H, A, or Q type, the next step is equivalent to the one-dimensional inversion of the field data.
The method assumes the equations for the theoretical response of a multi-layered ground. Each layer is characterized by its thickness and resistivity, each of which must be determined.
A first estimate of these parameters is made for each layer and the predicted curve of apparent resistivity versus electrode spacing is computed.
The discrepancies between the observed and theoretical curves are then determined point by point.