## 4 Analysis and Results
### Sensitivity Analysis  
We now examine how variations in the input parameters affect the model-predicted electric field waveforms. The following model input parameters will be considered: observation point height, return-stroke peak current, leader channel length, and leader speed. Model formulation described in Sections 2 and 3 is used throughout this section. Results of the sensitivity analysis are shown in Figures 6-9. In each Figure, we vary one of the input parameters, while keeping the other three at their nominal values (those used in Section 4.1).  
Figure 6: Electric field waveforms at points located 150 m, 250 m, and 750 m above ground as the leader passes through each of them. In all three cases, \(t=0\) corresponds to \(z_{tip}=z_{p}\). The return-stroke peak current \(I_{p}=12\) kA, \(H=5\) km, and the leader speed \(v=10^{7}\) m/s. Full time scale is 2.5 \(\upmu\)s.  
Figure 7: Electric field at a point located 250 m above ground as the leader passes through it, computed for prospective return-stroke (RS) peak currents of 12 kA, 25 kA, and 55 kA. Other model input parameters were as follows: \(H=5\) km, and \(v=10^{7}\) m/s. Full time scale is 2.5 \(\upmu\)s.  
In Figure 6, one can see how the electric field peak increases with decreasing the height of observation point above ground (\(z_{p}\)). Both the highest peak and the largest half-peak-width correspond to \(z_{p}=150\) m. Note that the influence of this parameter is relatively small. The E-field peak does not exceed the value (\(\sim\)4 MV/m) required for ambient electrons to run away in warm channels that are traversed by subsequent lightning leaders.  
In Figure 7, we see that as the peak current increases from 12 kA to 55 kA the electric field peak increases from 2.2 MV/m to 10 MV/m. Thus, there appears to be a linear relationship between the prospective return-stroke peak current, which can be viewed as a proxy for leader tip potential, and the electric field peak. For higher peak currents (25 kA and 55 kA), the E-field peak exceeds the value (\(\sim\)4 MV/m) required for ambient electrons to run away in warm channels that are traversed by subsequent lightning leaders.  
Figure 8: Electric field waveforms at a point located 250 m above ground as the leader passes through it, computed for maximum leader channel lengths of 3 km, 5 km, and 7.5 km. The return-stroke peak current is \(I_{p}=12\) kA and the leader speed is \(10^{7}\) m/s. Full time scale is 2.5 \(\upmu\)s.  
Figure 9: Electric field waveforms at a point located 250 m above ground as the leader passes through it, computed for leader speeds of \(0.5\times 10^{7}\) m/s, \(1.0\times 10^{7}\) m/s, and \(1.5\times 10^{7}\) m/s. Other model input parameters were as follows: \(I_{p}=12\) kA and \(H=5\) km. Full time scale is 5 \(\upmu\)s.  
The maximum leader channel length does not significantly influence the electric field (see Figure 8), as expected, since the largest contributions to the electric field come from the charges located near within 200 m or so of the leader tip. The E-field peak does not exceed the value (~4 MV/m) required for ambient electrons to run away in warm channels that are traversed by subsequent lightning leaders.  
In Figure 9, one can see that the leader with a \(0.5\times 10^{7}\) m/s speed has the highest electric field peak and largest half-peak width, and the leader with \(1.5\times 10^{7}\) m/s speed has the lowest electric field peak and smallest half-peak width. There appears to be an inverse relationship between the leader speed and the electric field peak, with slower leaders producing higher electric fields. For the slowest leader (\(0.5\times 10^{7}\) m/s), the E-field peak exceeds the value (~4 MV/m) required for ambient electrons to run away in warm channels that are traversed by subsequent lightning leaders.