In this paper, we present an analysis of lightning return stroke currents along an elevated strike object. We derive closed form expressions, both in the frequency domain and in the time domain, to calculate the lightning current at any height along a strike object taking into account reflections at the top and at the bottom. The frequency dependence of reflection coefficients is taken into account. We also derive an expression to calculate the reflection coefficient as a function of frequency at the bottom of the lightning strike object from two currents measured at different heights along the strike object. We show that, if the current and its time derivative overlap with reflections at the top or bottom of the strike object, it is impossible to derive the reflection coefficient at the top of the strike object exactly from any number of simultaneous current measurements. We propose an extrapolation method to estimate this reflection coefficient. We propose a second method to calculate the top reflection coefficient from just one current derivative measurement if the current derivative and its reflections do not overlap significantly. We apply the proposed methodology to experimental data obtained on Peissenberg Tower (Germany) consisting of lightning currents measured at two heights. The results suggest that the reflection coefficient at ground level can be considered as practically constant in the frequency range 100 kHz to 800 kHz. Although the estimated ground and top reflection coefficients are in good overall agreement with values found in the literature, the estimated values for the top reflection coefficient from the extrapolation method are somewhat lower than those found employing the current derivative method; the differences are discussed.
 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  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%.
 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 (ρt and ρg, 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.  proposed a formula, corrected by Rachidi et al. , as explained in section 2, that involves an infinite summation in the time domain, assuming that the reflection coefficients, ρt and ρg, are constant and known. Gavric  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.
Janischewskyj et al.  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.  and will be extended in this paper.
Rakov  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.
Rachidi et al. , 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.
 In this paper, we obtain a closed form expression for the infinite summation formula of Rachidi et al.  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.
 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.  and Fuchs [1998a].
2. Model of a Vertically Extended Strike Object
 The geometry of the tower is shown in Figure 1a. We begin with the same assumptions made in recent studies [e.g., Guerrieri et al., 1998; Janischewskyj et al., 1996; Rachidi et al., 2001; Rakov, 2001; Rachidi et al., 2002]. A debate has recently arisen [e.g., Kordi et al., 2002; Thottappillil et al., 2001, 2002] concerning the validity of a lossless transmission line assumption for the case of a vertical structure above a ground plane. This controversy is not yet settled. However, on the basis of the measurements on reduced-scale models of towers, in which it is shown that the errors incurred in are small for the first few reflections [e.g., Gutierrez et al., 2002], we consider in this paper the strike object as a lossless uniform transmission line of length h with a propagation speed equal to the speed of light c. We further assume that the reflection coefficients, defined for the currents propagating in the tower, are constant (this last assumption will be relaxed later). We also disregard any upward connecting leader and any reflections at the return stroke wavefront. At the onset of the return stroke at the tower top, the return stroke current depends only on the impedances of the lightning channel and of the tower top, until information about the ground gets to the top of the tower in the form of ground reflections. We model the ground plane as a frequency-dependent lumped impedance at the bottom of the tower.
Figure 1b presents a generalized equivalent circuit adapted from Rachidi et al. . In Figure 1b, the current source represents the lightning channel with its associated equivalent impedance, Zch. The amplitude of the current source, equal to 2 io(h, t), was chosen so that the current injected into the top of the tower equals the so-called “undisturbed current” io(h, t) when both reflection coefficients ρt and ρg are equal to zero. The reflection coefficients are zero when the equivalent impedance of the channel, Zch, is identical to the characteristic impedance of the tower, Zt, and to the grounding impedance, Zg. Using that model, we can obtain the following expression describing the spatial-temporal distribution of the current along the strike object [Rachidi et al., 2002]
where ioT(h, t) is the current transmitted into the top of the tower, z is the height of the measurement point along the strike object, h is the total strike object's height, and ρt and ρg are the reflection coefficients for current at the top and bottom of the object, respectively. Equation (1) differs from the expression used by Guerrieri et al.  in that the current ioT(h, t) is used instead of the undisturbed current io(h, t) to take into account the impedance discontinuity at the tower top as explained by Rachidi et al. , where the current injected into the top of the tower is expressed as (1 − ρt) io(h, t).
 Using equation (1) and assuming that both reflection coefficients, ρt and ρg, are known, it is possible to write an expression to extract the “undisturbed” current from the current measured at the top of the tower as proposed by Guerrieri et al. . This reads
 Let us now relax the assumption that the reflection coefficients are constant and independent of frequency and let us transform equation (1) into the frequency domain. The time domain versions of the frequency dependent reflection coefficients in equation (1) become impulse response functions (ρt(t) and ρg(t)), and the multiplications become convolution products. Making these changes, we obtain
where the current transmitted into the top of the tower is now ioT(h, t) = (δ(t) − ρt(t)) * io(h, t).
 In equation (3), the asterisk represents a convolution product and the powers “n” and “n + 1” in the reflection coefficients ρt(t) and ρg(t) are implicitly carried out using convolution products. In terms where the exponential variable n equals zero, the expression ρi0(t) = δ(t) (where the subindex i = t or g) should be used.
 Now, transforming equation (3) into the frequency domain, we obtain
Equations (6a) and (6b) are closed form expressions for the current at any point z along the strike object taking into account all the reflections at the bottom and at the top.
 If the reflection coefficients ρt(ω) and ρg(ω) are known, the current transmitted into the tower IoT(h, ω) and the “undisturbed” current Io(h, ω) = IoT(h, ω)/(1 − ρt(ω)) can be directly inferred from the measured current I(z, ω) using equation (6a) or similarly in the time domain using equation (6b).
3. Determination of the Ground Reflection Coefficient From Two Simultaneous Current Measurements
 In this section, we derive an expression to calculate the reflection coefficient ρg(ω) from currents measured simultaneously at two different heights along the strike object for the general case in which no conditions are imposed on the height of the tower. Let us assume that I(z1, ω) and I(z2, ω) are the currents measured simultaneously at heights z1 and z2 in the frequency domain. Making use of equation (6a), we can write expressions for each of the currents as follows
Solving for the ground reflection coefficient ρg(ω) in equation (9), we obtain the result sought:
Interestingly, the ground reflection coefficient can be found without prior knowledge of the reflection coefficient at the top of the strike object. Equation (10) allows us to infer the ground reflection coefficient at any frequency from two simultaneously measured currents at two different heights along the strike object. The application of equation (10) is illustrated and numerically validated in Appendix A.
Equation (10) can be expressed in terms of current derivatives by multiplying the numerator and the denominator of the right-hand side by “jω”,
in which İ(z, ω) = jω I (z, ω) represents the Fourier transform of ∂i(z, t)/∂t. The new equation is better suited for instrumented towers where the current derivative is measured directly using, for example, magnetic loops or Rogowski coils; this is, for example, the case of the well known Peissenberg Tower, the CN Tower, and the Saint Chrischona Tower. In section 5, we will apply equation (11) to recover the ground reflection coefficient in the frequency domain for current derivatives measured at the Peissenberg Tower, using simultaneous measurement at two different heights.
4. Estimation of the Top Reflection Coefficient of the Strike Object
 Either equation (3) or equations (6) give the measurable (disturbed) current at any height along the strike object. The variables that appear in equation (6a) are listed in Table 1. In the second column of that table, we have identified the variables that are directly known from a current measurement at a given height z.
The second column indicates which variables are known from direct current measurements.
Strike object length h
Measurement height z
Measured current i(z, ω)
Ground reflection coefficient ρg(ω)
Top reflection coefficient ρt(ω)
Injected current ioT(h, ω)
 Three of the variables in Table 1, the current transmitted into the tower, ioT(h, ω), the reflection coefficient at the ground, ρg(ω), and the reflection coefficient at the top, ρt(ω), are unknown. As already mentioned in section 2, Guerrieri et al.  assumed values for two of the three parameters (the reflection coefficients) and used equation (3) to find the only remaining unknown, the current ioT(h, ω). The following question arises: Is it possible to make three independent measurements to obtain from them all three unknown parameters? We now attempt to answer that question.
 In section 3, we particularized equation (6a) for two different heights (equations (7) and (8)), and we were able to solve for the ground reflection coefficient. As we are about to show, once the ground reflection coefficient is known, measurements at other heights do not provide any additional information and therefore do not allow us to calculate the top reflection coefficient or the “undisturbed” current.
The factor K(ω) is independent of the height of the measurement system along the tower, and it can be determined from two simultaneous measurements of the current at different heights as follows: First, the ground reflection coefficient can be found from equation (10) using the two simultaneous current measurements. Then, to find K(ω), we rewrite equation (12) as follows,
We can now substitute one of the two measured currents and the ground reflection coefficient into equation (14) to obtain K(ω).
 Now, since K(ω) and the ground reflection coefficient are known from two current measurements for a given strike, we can use equation (12) to find the current at any other height z along the elevated strike object without prior knowledge of the top reflection coefficient or the injected current. The implication is that a current measurement at a third height does not supply any new information, and it is therefore impossible, under the current assumptions, to find exactly the frequency dependent reflection coefficient at the top or the undisturbed current from any number of simultaneous current measurements.
 Nevertheless, if the strike object is long enough that the undisturbed current or its time derivative falls to zero before any reflections arrive, it would be possible to measure the reflection coefficients both at the top and at the bottom using just one current measurement in the time domain. For practical tower heights, only the derivative may be narrow enough. In sections 4.1 and 4.2, we propose two approximate methods to estimate the top reflection coefficient.
4.1. Extrapolation Technique Using Measured Current Waveforms at the Top of the Tower
 The current at any given height along the tower is composed of the original current transmitted into the tower plus multiple reflections coming from mismatched impedances at its top and bottom. In this first method to calculate ρt(t), we will employ a current waveform measured at the top of the tower, although the method can be extended to currents measured anywhere along it. Figure 2a reproduces the components of current at the top of the strike object, z = h. The choice of the polarities for the components of the current in Figure 2a is based on the following observation made by Rakov : “The effective grounding impedance of the tower is much smaller than its characteristic impedance and the latter impedance is appreciably lower than the equivalent impedance of the lightning channel.” The observation implies that the current reflection coefficient at the ground is positive and that the top reflection coefficient is negative. The total current is the addition of all the components.
 Let us observe, in Figure 2a, the terms composing the measured current for times ranging from t = 2h/c to t = 4h/c. Three terms make up that current: (1) ioT(h, t), (2) ρg(t) * ioT(h, t − 2h/c), and (3) ρt(t) * ρg(t) * ioT(h, t − 2h/c). The total current in that time range is therefore given by
Solving for ρt(t), we obtain
where the division operation represents an inverse convolution.
 The known and unknown terms in equation (16) are included in Table 2. From Table 2 we can see that the only unknowns are ρt(t) (which we are trying to estimate) and ioT(h, t) for 2h/c < t < 4h/c. The term ioT(h, t − 2h/c) for 2h/c < t < 4h/c is identical to ioT(h, t) for 0 < t < 2h/c, and it is directly measurable. If we extrapolate it for times into the range 2h/c < t < 4h/c, we can obtain an estimate for ρt(t). In section 5, we will illustrate the application of equation (16) to estimate the top reflection coefficient in the time domain for a current measured at the Peissenberg Tower (Germany), using simultaneously measured currents at two different heights but assuming that both the top and the bottom reflection coefficients are constant.
4.2. No-Overlap Current Derivative Components Method
 This second method to estimate ρt(t) in the time domain makes use of (1) the ground reflection coefficient obtained using, for example, the techniques presented in section 3, and (2) a directly measured current derivative. We will employ the current derivative waveform measured at the top of the tower, although the method can be extended to currents measured anywhere along it. We now show that, given a tower height, it is possible to calculate the reflection coefficients at the top of the strike object if the derivative of the current injected at the top of the tower is narrow enough that none of the reflections overlap with it. For convenience we reproduce equation (3) here,
Let us take the derivative of equation (3) with respect to time:
where we have made use the property of convolution products that d[f(t) * g(t)]/dt = df(t)/dt * g(t).
 In equation (17), we can identify a “transmitted current derivative” into the tower top dioT(h, t)/dt. This transmitted current derivative propagates down the tower, and it is reflected at the bottom, then reflected again at the tower top and so on.
 Let us consider the current derivative at the top of the tower, z = h. The first reflection from the bottom arrives 2h/c after the onset of the injected current derivative, dioT(h, t)/dt. The arrival of this reflection triggers a reflection from the top. No further reflections arrive until t = 4h/c. For 0 < t < 4h/c, equation (17) reduces therefore to three terms, and we can write
As can be seen from Figure 2b, the first term on the right-hand side of equation (18), dioT(h, t)/dt, can be measured directly since it corresponds to the measured current derivative for 0 < t < 2h/c. Note that, although lightning current derivatives are bipolar, we have used a unipolar waveforms in Figure 2b for clarity. Once dioT(h, t)/dt is known, and assuming that the reflection coefficient at the ground has been obtained from two simultaneous measurements at two heights, it is possible to calculate the reflection coefficient at the top as follows:
 Observe that the current for the interval 2h/c < t < 4h/c in Figure 2b is the sum of the second and third terms on the right-hand side of equation (18). One of those two terms, ρg(t) * dioT(h, t − 2h/c)/dt, can be readily calculated since it equals the convolution of two known quantities. Subtracting that term from the measured current, we are left with ρt(t) * ρg(t) * dioT(h, t − 2h/c)/dt, where only ρt(t) is unknown. It is now easy to obtain ρt(t) by dividing (convolutionally) by the known functions ρg(t) and dioT(h, t − 2h/c)/dt.
 In section 5, we will apply the present method to estimate the top reflection coefficient in the time domain for current measurements at the Peissenberg Tower (Germany).
5. Application of the Proposed Methodology to Peissenberg Tower
5.1. Peissenberg Tower Measurement Setup and Experimental Data
 The 168 m tall Peissenberg telecommunication tower (Figure 3) was used from 1978 until 1999 to study lightning currents and the associated electromagnetic fields. Two current derivative measurement systems were installed, one near the top of the tower at approximately 167 m and a second one near the bottom at 13 m. The systems were able to measure return stroke currents and their derivatives with a time resolution of 10 ns, a vertical resolution of 10 bits, and record duration of 50 μs [Fuchs, 1998b].
 In this section, current derivatives measured simultaneously at the two heights will be employed to evaluate the expressions found for the ground and top reflection coefficients. These reflection coefficients have been estimated in the past considering that they are constant and frequency-independent. Heidler et al.  analyzed 117 samples and reported average values for the ground and top reflection coefficients of 0.7 and −0.53, respectively. Using 13 samples from strikes to the same tower, Fuchs [1998a] estimated average values for the ground and top reflection coefficients of 0.698 and −0.529, respectively. Fuchs estimated additionally maximum and minimum values of 0.805 and 0.638 for the ground reflection coefficient and −0.684 and −0.392 for the top reflection coefficient.
Figure 4a shows the first 10 μs of current derivatives, near the top and bottom of the tower, for a return stroke recorded by the Peissenberg Tower system on 6 January 1998. Figure 4b shows the associated currents after numerical integration. The presence of multiple reflections is clearly discernible in the current waveforms.
5.2. Application of the Methodology in the Frequency Domain to Recover the Ground Reflection Coefficient
 We will now apply equation (11) to recover the ground reflection coefficient in the frequency domain from two simultaneously measured return stroke current derivatives. To do that, we will consider three sets of experimentally measured current derivative waveforms presented in Figures 5a, 5b, and 5c.
Figure 6 presents the ground reflection coefficient determined using equation (11) in the frequency range of 6 kHz up to 940 kHz. These two frequencies correspond approximately to 1/(π tmax) and 1/(π tr), where tmax = 50 μs and tr = 0.34 μs are the duration and the average rise time, respectively, of the current samples employed in this analysis.
 In Figure 6a, the fact that the values of the reflection coefficients are comparable to those obtained by Heidler et al.  and Fuchs [1998a], and the similarity in the behavior for the absolute value of ρg(ω) obtained for all three cases examined supports the validity of the methodology. The obtained ρg(ω) decreases slowly with frequency from 0.8 to 0.6 for intermediate frequencies. Note that, in Figure 6b, the real part of ρg approaches −1 at DC. This value, however, is to be taken with caution since at very low frequencies, no traveling waves are present and concepts such as reflection coefficients lose their significance. The imaginary part of ρg(ω) is basically negligible for a large interval of frequencies (Figure 6b), implying an essentially resistive behavior of the grounding impedance.
5.3. Application of the Methodology to Recover the Top Reflection Coefficient in the Time Domain
 In this section, we use the two methods introduced in section 4 to estimate the top reflection coefficient ρt(t) by applying them to currents and current derivatives measured at the Peissenberg Tower.
5.3.1. Extrapolation Technique Using Measured Current Waveforms at the Top of the Tower
 Using the technique based on linear extrapolation described in section 4, we will now estimate the top reflection coefficient from the current waveform presented in Figure 4b. Although the measurement was made one meter below the top of the tower, we will assume for our calculations that it was measured exactly at the top and that the ground reflection coefficient has a constant value of 0.7, independent of frequency. This value is the average of the maximum and minimum obtained in section 5.2. We will disregard for this validation the frequency dependence of the top reflection coefficient and assume it constant.
 Let us go back to the time domain expression introduced in section 4 (equation (16)). With constant reflection coefficients, the convolution terms in equation (16) disappear, and they are replaced by standard multiplications:
Figure 7a shows the three components of the current employed to calculate the top reflection coefficient from equation (19). Figure 7b presents the behavior of the top reflection coefficient as a function of the time instant at which the reflection coefficient is estimated. It is interesting to observe that, under the current assumptions, the average value for the top reflection coefficient tends to a constant value of −0.43 ± 0.02. This result is in agreement with the values for the top reflection coefficient reported by Fuchs [1998a] using 13 samples and mentioned in section 5.1.
5.3.2. Current Derivative Method
 The second method to obtain ρt(t), introduced in section 4.2, employs the current derivative waveforms observed at the tower top, z = h. In section 4.2., we assumed a transmitted current derivative dioT(h, t)/dt sharp enough for subsequent reflections produced in the tower not to overlap with it. In practice, however, the reflected terms do overlap with the transmitted current derivative dioT(h, t)/dt. Since the overlap is small compared with the peak values attained by the current derivatives, it can be disregarded for the evaluation of ρt. To minimize the error, we will use only peak values of the current derivative as shown in Figure 8.
 The values obtained for the top reflection coefficient ρt are equal to −0.59 ± 0.05. This result is somewhat higher than that obtained using the first technique, but it is in excellent agreement with the estimates of −0.53 and −0.529 given by Heidler et al.  and Fuchs [1998a], respectively. Moreover, estimated values for the top reflection coefficient are in agreement with estimates of return stroke channel impedance given by Rakov  (a value of about 570 Ω for frequencies ranging from 10 kHz to 10 MHz) and assuming a reasonable value of about 100–300 Ω for the tower impedance [e.g., Gavric, 2002; Rakov, 2001].
 The difference between the results obtained from the two methods can be explained, at least in part, by errors in the first method due to the approximation involved in the extrapolation process.
 Another possible explanation for the discrepancy between the values obtained using the two methods is the fact that the first method uses an extrapolation of the low frequency tail of ioT(h, t) whereas, for the current derivative method, we have used the peak value of the dioT(h, t)/dt, which is associated with high frequencies, suggesting a dependence of the top reflection coefficient with frequency, in agreement with the observations of Janischewskyj et al. .
 For practical tower heights and typical currents, the current derivative rarely (if ever) decays to zero by the time its reflection off the bottom of the tower arrives at the top. As done in this section, the current derivative method can still be applied if the current derivative has decayed to negligible amplitudes by the time the reflection from the bottom arrives. To minimize the error due to the overlap, the peak amplitudes of the current derivative and of its first reflection were used. The use of peak values greatly simplifies the calculations, but it presents the disadvantage of disregarding the frequency dependence information of the top reflection coefficient. The importance of the extrapolation method lies in the fact that, although it is inherently less reliable than the current derivative method, it is applicable even if the overlap between the derivative and its reflection is not negligible.
 In this paper we have developed a closed form expression in the frequency domain to calculate the lightning current at any height along a strike object taking into account frequency-dependent reflection coefficients at the top and at the bottom. We have derived an expression to calculate the reflection coefficient as a function of frequency at the bottom of the lightning strike object from two currents measured simultaneously at different heights along the strike object. We found that the ground reflection coefficient can be found without prior knowledge of the reflection coefficient at the top of the strike object.
 We showed that, unless the tower is tall enough that the current injected at the top of the object or its derivative drop to zero before the arrival of reflections, it is impossible, at least under our assumptions, to derive either the reflection coefficient ρt(ω) at the top of the strike object or the “undisturbed” current from any number of simultaneous current measurements. We proposed two methods to estimate the top reflection coefficient.
 The proposed methods were applied to experimental data obtained on Peissenberg Tower where lightning currents were measured simultaneously at two heights. It was found that the reflection coefficient at ground level can be considered as practically constant over a relatively wide range of frequencies from 100 kHz to 800 kHz.
 The estimated top reflection coefficients are in good agreement with values found in the literature. Nevertheless, we found that the estimated values for the top reflection coefficient from the extrapolation method are lower that those found employing the current derivative method. The difference might be due to possible experimental errors and also to the fact that the extrapolation method provides values for the top reflection coefficient calculated from the low-frequency tail of the current waveforms, while the current derivative method uses values associated with the faster parts of the waveform. This observation suggests that the top reflection coefficient is dependent of frequency.
Appendix A:: Numerical Validation and Illustration of the Application of Equations (6) and (10)
where Io is the amplitude of the channel base current, τ1 is the front time constant, τ2 is the decay time constant, η is the amplitude correction factor and N is an exponent (ranging from 2 to 10).
 The Heidler current parameters used are given in Table A1. In Figure A1, we compare, for the first 20 μs, the disturbed current expression defined by Guerrieri et al.  and the new expression introduced by Rachidi et al. , which includes the term (1 − ρt) to take into account the impedance discontinuity at the tower top.
Table A1. Parameters of Two Heidler Functions That Reproduce the “Undisturbed” Current Waveshapes io(t) at the Top of the Elevated Object
 The factor (1 − ρt) introduced by Rachidi et al.  increases the amplitude of the current waveform flowing down into the tower for all the cases where the top reflection coefficient ρt has negative values (ρt has been found to have negative values from experimental measurements in instrumented towers available today [Rakov, 2001]).
 Indeed, Guerrieri et al.  did not take into account the treatment of the impedance discontinuity at the tower top in the attachment process; the Guerrieri et al. equation disregards the mentioned factor considering, implicitly, ρt equal to 0 for the first contact to the tower (Zch = Zt). A comparison of two cases of currents using equation (1) [Rachidi et al., 2002] in the time domain and equation (6a) in the frequency domain is presented in Figures A2 and A3, for currents at z1 = 160 m and z2 = 5 m. The tower height (h = 168 m) and the observation points correspond to those of the Peissenberg tower, and the undisturbed current was obtained using the parameters given in Table A1. The frequency domain formula is transformed back into the time domain to the effect of comparisons. The reflection coefficients were assumed to be constant and independent of frequency.
 As can be seen from Figures A2 and A3, our results obtained in the frequency domain using equation (6a) are indistinguishable from those obtained in the time domain by equation (1). Inaccuracies due to the numerical Fourier transforms are negligible.
A2. Ground Reflection Coefficient Formula
 In this section, we use equation (10) to recalculate the constant reflection coefficients that were used to generate the currents generated in the last section for h = 168 m, z1 = 160 m and z2 = 5 m (shown in Figures A2 and A3). The results are presented in Figure A4. Both curves in Figure A4 are in agreement with the original values of the ground reflection coefficients at the ground.
 We will now use equation (10) to recover a frequency-dependent ground reflection coefficient. The objective here is to apply equation (10) numerically and not to investigate the actual frequency behavior of the reflection coefficient at the ground. We will proceed in a manner similar to that used for the constant reflection coefficients above: We will first generate the currents, and we will then use equation (10) to reextract the ground reflection coefficient.
 Let us first define a frequency dependent function for ρg using, as sole criterion, the existence of an analytical form of its Fourier transform:
which, in the time domain, can be expressed as
To find the currents, we can proceed two different ways: (1) We can substitute equation (A2) into equation (6a) and then apply the inverse Fourier transform, or (2) we can substitute equation (A3) into equation (3) or equation (6b), which involves convolution operations. We have chosen the first method. The currents at z1 = 160 m and z2 = 5 m are plotted in Figure A5 for the case where ρt = −0.5.
 Note that as reported by Guerrieri et al. , the reflections are not readily discernible from the waveshapes. We have extracted the ground reflection coefficient as a function of frequency using these currents in equation (10), and we have plotted it in Figure A6. The original reflection coefficient ρg(ω) (equation (A2)), is also plotted for comparison. Clearly, the original reflection coefficient has been accurately recovered.
 This research has been financially supported by the Swiss National Science Foundation (grant 20-56862.99). The authors would like to express their thanks to C. A. Nucci and. V. Rakov and to the three reviewers whose comments and suggestions allowed us to improve the manuscript.