## 1. Introduction

[2] Present knowledge of lightning current parameters (peak value, front steepness, and duration) comes essentially from direct measurements obtained using instrumented towers [e.g., *Beierl*, 1992; *Berger et al.*, 1975; *Montandon and Beyeler*, 1994b] or triggered lightning [e.g., *Leteinturier et al.*, 1990; *Willett et al.*, 1988]. More recently, lightning current and current-derivative measurements have been observed to exhibit reflections in the lightning waveform produced at the elevated strike object's top and bottom[*Beierl*, 1992; *Janischewskyj et al.*, 1996; *Montandon and Beyeler*, 1994a]. The implication of this observation is that the measured current parameters are “disturbed” by the presence of these reflections and the degree of disturbance depends on the physical and electrical characteristics of the strike object.

[3] The measured lightning currents using an instrumented tower whose height exceeds about one tenth the minimum significant wavelength associated with lightning current may be affected by propagation effects along the tower. This appears to be the case even for the towers used by Berger and co-workers [*Berger et al.*, 1975], on which a considerable fraction of the lightning statistics applied to lightning protection are based today (considering that the frequency spectrum of the lightning current has significant components at frequencies up to a few MHz corresponding to a minimum wavelength of about 100 m). *Rakov* [2002] estimated that, for subsequent strokes, the difference in the peak current measured (1) at an ideally grounded object of negligible height (*h* = 0) and (2) at the top of Berger's tower (*h* = 70 m) would be about 10%.

[4] It is desirable to obtain statistics on the “primary” current exempt from the disturbances introduced by the transient processes along the tower. Some workers [*Beierl*, 1992; *Guerrieri et al.*, 1996; *Guerrieri et al.*, 1998] obtained this undisturbed current (which they call “decontaminated” current) by assuming constant, frequency independent reflection coefficients at the top and the bottom of the strike object (ρ_{t} and ρ_{g}, respectively). In those studies, the authors inferred the value of the reflection coefficients from a reduced experimental set of current waveforms found in the literature [*Beierl*, 1992; *Montandon and Beyeler*, 1994b; *Willett et al.*, 1988]. To decontaminate the current, *Guerrieri et al.* [1998] proposed a formula, corrected by *Rachidi et al.* [2002], as explained in section 2, that involves an infinite summation in the time domain, assuming that the reflection coefficients, ρ_{t} and ρ_{g}, are constant and known. *Gavric* [2002] proposed an iterative method based on the Electromagnetic Transient Program (EMTP) to remove superimposed reflections caused by a strike tower from digitally recorded lightning flash currents.

[5] *Janischewskyj et al.* [1996] derived reflection coefficients at the Canadian National (CN) Tower in Toronto and stated that the values depend on the initial rise time of the measured current, although the limited number of points in their plots render the drawing of conclusions difficult. A dependence on the risetime would suggest that at least one of the reflection coefficients is a function of the frequency. They also proposed a method to extract the reflection coefficients from the measured current waveform. However, their method is applicable only assuming a simplified current waveform (double ramp) and neglecting any frequency dependence for the reflection coefficients. The last consideration was relaxed in a first approximation by *Bermudez et al.* [2001] and will be extended in this paper.

[6] *Rakov* [2001] reviewed experimental data showing the transient behavior of tall objects struck by lightning. He concluded that the peak current measured at the bottom of the strike object is more strongly affected by the transient process in the object than the peak current at the top.

[7] *Rachidi et al.* [2002], on the basis of a distributed-source representation of the lightning channel, generalized the mathematical formulations of the so-called engineering lightning return stroke models to take into account the presence of a vertically extended strike object. The distributed-source representation of the lightning channel adopted in their study allowed for more general and straightforward formulations of the generalized return-stroke models than the traditional representations implying a lumped current source at the bottom of the channel, including a self-consistent treatment of the impedance discontinuity at the tower top.

[8] In this paper, we obtain a closed form expression for the infinite summation formula of *Rachidi et al.* [2002] both in the time domain and in the frequency domain considering the possible frequency dependence of the reflection coefficients. Further, we show how the reflection coefficient at the ground can be obtained from lightning current measurements at two different heights along the elevated strike object.

[9] Although it is possible to obtain the reflection coefficient at the bottom of the strike object from two simultaneous current measurements, we show that, unless the tower is tall enough that the current or its time derivative do not overlap, the exact calculation of the reflection coefficient at the top is impossible from any number of lightning current measurements along the strike object. We propose two methods to estimate this reflection coefficient. One of the methods is based on an extrapolation technique. The second method is based on the fact that the waveshape of the time derivative of the current is much narrower than that of the current itself. The proposed methods to infer the ground and top reflection coefficients are tested versus experimental data obtained at the Peissenberg Tower and compared with estimated values found by *Heidler et al.* [2001] and *Fuchs* [1998a].