Computation of the attenuation function of the lightning return-stroke current from electromagnetic fields measured in a short-duration time-window, Part I: Theoretical investigation

In a recent paper, a procedure to reconstruct the attenuation function of a return-stroke current from the simultaneous measurements of the channel-base current and the radiated electromagnetic fields was presented. One of the assumptions of the whole framework was that the sensors can record the overall time-evolution of the lightning return-stroke, namely from its start to its end. However, this condition is too restrictive for most of the real data collected in the past decades, since they were generally acquired for other goals than the reconstruction of the attenuation function. Accordingly, in the present paper we perform a theoretical investigation aimed at removing such restriction. As a result, we first derive a new set of integral equations that relate the channel-base current to the radiated fields during a generic time-window, then we obtain the corresponding equations in the frequency domain via Fourier Transform, which represent the analytical formulation of an inverse problem to be numerically solved through appropriate regularization techniques.