Mathematical constraints on the use of transmission line models to investigate the preliminary breakdown stage of lightning flashes

The initial stage of in-cloud lightning development is characterized by a series of initial breakdown pulses (IBPs), observed as abrupt electric field changes at a remote sensor. Recent studies have attributed this process to the stepwise elongation of a negative lightning leader channel and used transmission line (TL) models to interpret the observed electromagnetic emission. In these models a current pulse is injected at the base of the channel, propagates along it, and the current parameters are adjusted to fit the measured IBPs. In this paper we explore the limitations of TL models by comparing four of its variants: the classic TL (with no attenuation) and three modified transmission line models that have different current attenuation characteristics, i.e., following linear, exponential, and Gaussian functions. For a compact channel (less than a few hundred meters long) all four models tend to produce similar electromagnetic signatures, and nearly identical waveforms can be obtained by simply varying the channel length. It is impossible to simultaneously identify, with confidence, both channel length and the speed of current wave propagation at the source by matching a far-field waveform. The field-to-current conversion factor for in-cloud sources is not a constant and can vary by as much as 2 orders of magnitude depending on channel length and current pulse risetime. This conclusion contrasts with the assumption used by the National Lightning Detection Network, that the field-to-current conversion can be performed by a multiplicative constant, as it is done for return strokes in cloud-to-ground lightning.