## 2 Leader Model Description  
The model presented by us is meant to apply to any subsequent (dart, dart-stepped, or chaotic) leader developing in a warm (reduced air density) channel, although the stepping process is not included in the model. Only the charges deposited on the leader channel (mostly in the corona sheath) are considered in computing electric fields. We assume that the leader path is straight and vertical, and we place the observation point (the point where the electric field is computed) on this path, with the leader tip approaching the point, traversing it, and moving beyond that point. [PERSON] et al. (2012, 2012) [6; 7] experimentally showed that lightning leader tips are primarily responsible for X-ray emissions; thus, special attention has been paid to the electric field calculations near the leader tip. In all the electric field equations presented in this paper, \(t=0\) corresponds to the time at which the leader tip passes the observation point. The leader model described below follows the ideas forming the basis of the model proposed by [PERSON] et al. (2009, 2010) [8; 9]. We consider the leader propagating downward at a constant speed along the vertical channel that was created by the preceding return stroke, decayed, but is still warm. The travelling current source model used here is similar to the ones previously described by [PERSON] et al. (2009, 2010) [8; 9], [PERSON] (1995) [10], and [PERSON] et al. (1997) [11]. It employs an approach in which the leader tip acts as downward moving current source launching a current wave moving upward, with the associated positive charge leaking radially outward to form the leader corona sheath. Each segment of the leader channel injects radial streamer current charging the corona sheath, the radius of which increases with height (see Figure 2). This is different from the distributed-source return-stroke models [PERSON] et al. (2007) [12] where the distributed current sources (modeling the corona sheath charged during the leader stage) are progressively tapped by the upward-moving return-stroke front. In [PERSON] et al. (2009) [8] and [PERSON] et al. (2010) [9], the dipole approximation was used in the field calculations and this had limited the numerical accuracy of the calculation. In the present work, the monopole approximation was used. According to this formulation, the electric field at any given point along the channel can be divided into static field produced by stationary charges, velocity field generated by moving charges and radiation field produced by accelerating charges (See [PERSON] et al. (2010) [13]). The results obtained show that the tip electric field is mainly determined by the static field term alone, while the contributions from the velocity and radiation fields can be neglected. For this reason, only the static field term is used in the calculations. Furthermore, contributions from the charges inside the cloud are assumed to be insignificant in the present study. The presence of ground is also neglected. We computed the electric field due to the leader charges at different points along its axis, both ahead and behind the leader tip. In other words, the longitudinal electric fields were computed at any point (see point P in Figure 2) along the leader axis aligned with the \(z\)-axis (directed upward with \(z=0\) at the ground level) of the cylindrical coordinate system. We assumed that at time \(t_{0}\), the tip of the leader is located at height \(z_{tip}\) above the ground. We introduce the equations for the static component of the electric field in Section 3. In order to compute the electric field produced by the dart leader channel, first, we have to establish the leader channel geometry and the electric charge distribution along the channel during the leader propagation toward the ground. The maximum leader charge per unit length (\(\rho_{L0}(z)\)--maximum line charge density) at height \(z\) of the leader channel can be related to the prospective return-stroke current peak \(I_{P}\). [PERSON] et al.