RESISTIVITY SOUNDING ON A MULTI-LAYERED EARTH WITH TRANSITIONAL LAYERS. PART I: THEORY*

The electrical potential generated by a point source of current on the ground surface is studied for a multi-layered earth formed by layers alternatively characterized by a constant conductivity value and by conductivity varying linearly with depth. The problem is accounted for by solving a Laplace's differential equation for the uniform layers and a Poisson's differential equation for the transitional layers. Then, by a simple algorithm and by the introduction of a suitable kernel function, the general expression of the apparent resistivity for a Schlumberger array placed on the surface is obtained. Moreover some details are given for the solution of particular cases as 1) the presence of a infinitely resistive basement, 2) the absence of any one or more uniform layers, and 3) the absence of any one or more transitional layers. The new theory proves to be rather general, as it includes that for uniform layers with sharp boundaries as a particular case. Some mathematical properties of the kernel function are studied in view of the application of a direct system of quantitative interpretation. Two steps are considered for the solution of the direct problem: (i) The determination of the kernel function from the field measurements of the apparent resistivity. Owing to the identical mathematical formalism of the old with this new resistivity theory, the procedures there developed for the execution of the first step are here as well applicable without any change. Thus, some graphical and numerical procedures, already published, are recalled. (ii) The determination of the layer distribution from the kernel function. A recurrent procedure is proposed and studied in detail. This recurrent procedure follows the principle of the reduction to a lower boundary plane, as originally suggested by Koefoed for the old geoelectrical theory. Here the method differs mainly for the presence of reduction coefficients, which must be calculated each time when passing to a reduced earth section.