On the retrieval of lightning radio sources from time‐of‐arrival data

Abstract

We examine the problem of retrieving three‐dimensional lightning locations from radio frequency time‐of‐arrival (TOA) measurements. Arbitrary antenna locations are considered. By judiciously differencing measurements that are related to the location of the antennas and their excitation times, the problem is converted from the initial spherical nonlinear form to a system of linear equations. In the linear formalism, the source location and time‐of‐occurrence is viewed geometrically as an intersection of hyperplanes in the four‐dimensional Minkowski space (x, y, z, t). The linear equations are solved to obtain explicit analytic expressions for the location and time variables. Retrieval errors are not interpreted with conventional Geometrical Dilution of Precision (GDOP) arguments as discussed by Holmes and Reedy [1951], but with more recent inversion analyses considered by Twomey [1977]. Measurement errors are propagated analytically so that the specific effect of these errors on the solution is clarified. The sensitivity of the solution on the number of antennas used, antenna network geometry, source position, and measurement differencing schemes are discussed in terms of the eigenvalues of the linear system.

Locating initial breakdown pulses using electric field change network

“…Most of the early TOA work has been conducted following this method and solving non‐linear equations numerically to get the retrieval locations. However Koshak and Solakiewicz [] introduced a method to linearize these non‐linear equations and introduced analytical solutions to get the location as well as the errors for each coordinate. In the perfect situation, five stations produce four linear equations, and getting analytical solutions to four unknowns ( x , y , z , t ) would be trivial.…”

“…In the perfect situation, five stations produce four linear equations, and getting analytical solutions to four unknowns ( x , y , z , t ) would be trivial. These authors also indicated that this linear system “can be taken as an under‐ or over‐determined system of equations that can be solved using the general theory of constrained linear inversion.” Koshak and Solakiewicz [] applied this method for theoretical sensor configurations: a square network having a sensor at each corner, a triangular network having a sensor at each corner plus a fourth sensor at the center, and a symmetric seven‐antenna network having sensors at each corner of a hexagon and one at the center. They determined that the mean location errors were smallest for the symmetric seven‐antenna network.…”

“…For classic bipolar IBPs, we locate the negative peak for negative CG flashes and the positive peak for normal intracloud (IC) flashes, since these are the leading peaks in the bipolar waveform and also since they are sharper than the opposite‐polarity overshoot [ Weidman and Krider , ]. Arrival times (good to the nearest 0.2 μs, restricted by the 5 MHz sampling rate) and sensor site locations (given by the GPS to within 5 m) are then used with the Koshak and Solakiewicz [] TOA method to retrieve ( x , y , z , t ) coordinates. Our coordinate system is centered at the LDAR2 origin and has x oriented east‐west, y north‐south, and z upward (altitude relative to mean sea level).…”

Error analysis for a long‐range lightning monitoring network of ground‐based receivers in Europe

“…Besides the measurement characteristics of the system (direction finders, GPS time stamps, etc.) another differentiation is in the number of dimensions (e.g., resolving the 3D structure of a lightning stroke versus just its spatial location) involved in the location retrieval, which varies according to the operation frequency bandwidth [ Koshak and Solakiewicz , 1996].…”

Statistical estimation of locations of lightning events

“…The problem of detection of locations of lightning events on the basis of ground-based measurements has been studied extensively within the last three decades [see, e.g., Cianos et al, 1972;Hager and Wang, 1995;Koshak and Solakiewicz, 1996;Lee, 1986;Lewis et al, 1960;MacClement andMurty, 1978;Murty and MacClement, 1973;Panyukov and Strauss, 1996;Proctor, 1971; Rustan et al, 1980;Taylor, 1978;Thompson et al, 1994]. All of the authors derive the location of a lightning event from the times of arrival of electromagnetic radiation at several locations.…”