Impulse radiating antennas, proposed by C. E. Baum in 1989 [1], allow to generate impulse-like, broadband, high-amplitude electromagnetic fields. Applications include electromagnetic compatibility testing and remote sensing [2,3]. Analytical and semi-analytical expressions for the electric and magnetic fields emitted by such antennas are challenging to obtain because of the broadband nature of the signals and the interactions between fields, reflectors, feeder plates, and matching resistors. Analytical formulas can be obtained by integration of the tangential electric field on the antenna aperture [1], later completed by the inclusion of the prepulse [4] (that is, the spurious radiation of the feeder plates) and the diffraction of the feeder plates [5]. An analytical formula for the near- and far-fields valid in the boresight cone was presented in [6]. Ad hoc formulas valid everywhere have been developed, notably for impulse-radiating antennas fed by transverse-electromagnetic coplanar plates [7]. However, analytical models that both incorporate the prepulse and are valid outside the aperture of the antenna have not yet been developed. In this chapter, we propose to generalize such semi-analytical expressions for the fields radiated by any impulse radiating antenna to the whole space thanks to a "gray-box" modeling approach, i.e., we fit experimental data to a model whose physical behavior is known, but specific parameters are not. We first introduce the method, based on the time-domain Cartesian multipole expansion, with some adaptations to inhomogeneous media. Then, we show how to implement the proposed method as an optimization problem. This approach can be seen as a data-driven model calibration. Next, we apply the technique to an impulse-radiating antenna modeled by a time-domain finite-element simulation and evaluate its performance. Finally, we conclude with some comments.