We introduce a novel approximation to numerically simulate the electromagnetic response of point or line sources in the presence of arbitrarily heterogeneous conductive media. The approximation is nonlinear with respect to the spatial variations of electrical conductivity and is implemented with a source-independent scattering tensor. By projecting the background electric field(i.e., the electric field excited in the absence of conductivity variations) onto the scattering tensor, we obtain an approximation to the electric field internal to the region of anomalous conductivity. It is shown that the scattering tensor adjusts the background electric field by way of amplitude, phase, and cross-polarization corrections that result from frequency-dependent mutual coupling effects among scatterers. In general, these three corrections are not possible with the more popular first-order Born approximation. Numerical simulations and comparisons with a 2.5-dimensional finite difference code show that the new approximation accurately estimates the scattered fields over a wide range of conductivity contrasts and scatterer sizes and within the frequency band of a subsurface electromagnetic experiment. Furthermore, the approximation has the efficiency of a linear scheme such as the Born approximation. For inversion, we employ a Gauss-Newton search technique to minimize a quadratic cost function with penalty on a spatial functional of the sought conductivity model. We derive an approximate form of the Jacobian matrix directly from the nonlinear scattering approximation. A conductivity model is rendered by repeated linear inversion steps within range constraints that help reduce nonuniqueness in the minimization of the cost function. Synthetic examples of inversion demonstrate that the nonlinear approximation reduces considerably the execution time required to invert a large number of unknowns using a large number of electromagnetic data.