The Born and Rytov approximations, widely used for solving scattering problems, are of limited utility for low-frequency electromagnetic scattering in geophysical applications where conductivity can vary over many orders of magnitude. We present four new, relatively simple nonlinear estimators that can be used for rapid electromagnetic modeling. The first, termed the static localized nonlinear approximation, is designed specifically to correct the magnitude of the electric field internal to the scatterer. The second, termed the localized nonlinear approximation, improves the estimate of the phase of the scattered field and includes some of the cross-polarization effects due to full wave scattering. Two further new estimators, based on the Rytov transformation (the localized nonlinear Rytov and the static localized nonlinear Rytov approximations) are designed to further improve the estimation of the phase of the scattered field, especially at high frequency and for larger size scatterers. Although these approximations are nonlinear functions in conductivity, they are generally much faster to compute than the full forward problem, and are almost as efficient as the Born or Rytov approximations. Moreover, the enhanced accuracy of the new estimators has made us optimistic about their application to low-frequency three-dimensional inverse problems in electromagnetics. The approximations developed in this paper will also be applicable to fields such as quantum mechanics. ¿ American Geophysical Union 1993