On the reconstruction of the attenuation function of a return-stroke current from the Fourier Transform of finite-duration measurements

A correct representation of the lightning current is crucial when the electromagnetic field radiated to a point of interest has to be computed. Based on the engineering models of Transmission Line type, such representation involves the knowledge of the return-stroke speed, the channel-base current, the channel height and the attenuation function. Whereas the first three quantities can be measured in different ways, no measurement technique can directly provide reliable information on the attenuation function. In the past decades, researchers have applied various strategies to address this problem. These strategies are all based on a common feature, i.e., the unknown function is postulated a priori and then validated through the comparison of the computed electromagnetic fields at one or more observation points with the corresponding measured waveforms. In this paper, we propose an alternative approach for the identification of the attenuation function: starting from appropriate measurements of the return-stroke speed, the channel-base current, the channel height and the radiated electromagnetic field, we first formulate an algebraic inverse and ill-posed problem, obtained from the discretization of integral equations relating the source to the radiated field in the frequency domain, and then we solve it by means of a Tikhonov regularization technique. The proposed framework is preceded by a detailed theoretical analysis, with special emphasis on the description and filtering of measurement noise and on the minimum duration of the measurement time-windows ensuring reliable results.