The scaling properties of Maxwell's equations allow the existence of simple yet general nonlinear integral equations for electrical conductivity. These equations were developed in an attempt to reduce the generality of linearization to the exclusive scope of electromagnetic problems.The reduction is achieved when the principle of similitude for quasi-static fields is imposed on linearized forms of the field equations. The combination leads to exact integral relations which represent a unifying framework for the general electromagnetic inverse problem.
The equations are of the same form in both time and frequency domains and hold for all observations that scale as electric and magnetic fields do; direct current resistivity and magnetometric resistivity methods are considered as special cases. The kernel functions of the integral equations are closely related, through a normalization factor, to the Frechèt kernels of the conventional equations obtained by linearization. Accordingly, the sensitivity functions play the role of weighting functions for electrical conductivity despite the nonlinear dependence of the model and the data. In terms of the integral equations, the inverse problem consists of extracting information about a distribution of conductivity from a given set of its spatial averages.
The form of the new equations leads to the consideration of their numerical solution through an approximate knowledge of their kernel functions. The integral equation for magnetotelluric soundings illustrates this approach in a simple fashion.