The time-domain electromagnetic (TEM) modeling method of Oristaglio and Hohmann is reformulated here in terms of the secondary field. This finite-difference method gives a direct, explicit time-domain solution for a two-dimensional body in a conductive earth by advancing the field in time with DuFort-Frankel time-differencing. As a result, solving for the secondary field, defined as the difference between the total field and field of a half-space, is not only more efficient but is also simpler and eliminates several problems inherent in the solution for the total field. For example, because the secondary field varies slowly both in space and time, it can be modeled on a coarse grid with large time steps. In addition, for a simple body the field is local; therefore, because the field can be assumed to satisfy a simple boundary condition in the earth computation is greatly simplified. Our tests show that for the same accuracy, the secondary-field solution is roughly five times faster than the total-field solution.We compute and analyze the magnetic field impulse response for a suite of models, most of which consist of a thin body embedded in a conductive half-space--with or without overburden. The results indicate the conductive half-space will both delay and attenuate the response of the body and even obscure it if the conductivity contrast is small. The results also suggest that the conductive host can alter the decay rate of the response of the body from its free-space counterpart. Our results for multiple bodies illustrate the importance of early-time measurements to obtain resolution, particularly for measurements of the horizontal magnetic field. The vertical magnetic field, however, can be used to infer the dip direction of a dipping body by studying the migration of the crossover.The results for models which include overburden show that the effect of a conductive overburden, in addition to the half-space effect, is to delay the response of the body, because the primary current initially tends to concentrate and slowly diffuse through the overburden, and does not reach the body until later time. This effect also complicates the early-times profiles, becoming more severe as the conductivity of the overburden is increased.