Abstract
A “dimensional reduction” (“DR”) method is introduced for analyzing lightning field changes (ΔEs) whereby the number of unknowns in a discrete two-charge model is reduced from the standard eight (x, y, z, Q, x′, y′, z′, Q′) to just four (x, y, z, Q). The four unknowns (x, y, z, Q) are found by performing a numerical minimization of a chi-square function. At each step of the minimization, an overdetermined fixed matrix (OFM) method is used to immediately retrieve the best “residual source” (x′, y′, z′, Q′), given the values of (x, y, z, Q). In this way, all eight parameters (x, y, z, Q, x′, y′, z′, Q′) are found, yet a numerical search of only four parameters (x, y, z, Q) is required. The DR method has been used to analyze lightning-caused ΔEs derived from multiple ground-based electric field measurements at the NASA Kennedy Space Center (KSC) and U.S. Air Force Eastern Range (ER). The accuracy of the DR method has been assessed by comparing retrievals with data provided by the lightning detection and ranging (LDAR) system at the KSC-ER, and from least squares error estimation theory, and the method is shown to be a useful “stand alone” charge retrieval tool. Since more than one charge distribution describes a finite set of ΔEs (i.e., solutions are nonunique), and since there can be appreciable differences in the physical characteristics of these solutions, not all DR solutions are physically acceptable. Hence, an alternative and more accurate method of analysis is introduced that uses LDAR data to constrain the geometry of the charge solutions, thereby removing physically unacceptable retrievals. The charge solutions derived from this method are shown to compare well with independent satellite- and ground-based observations of lightning in several Florida storms.
The maximal function and conditional square function control the variation: An elementary proof