Two approaches are used to characterize how accurately the north Alabama Lightning Mapping Array (LMA) is able to locate lightning VHF sources in space and time. The first method uses a Monte Carlo computer simulation to estimate source retrieval errors. The simulation applies a VHF source retrieval algorithm that was recently developed at the NASA Marshall Space Flight Center (MSFC) and that is similar, but not identical to, the standard New Mexico Tech retrieval algorithm. The second method uses a purely theoretical technique (i.e., chi-squared Curvature Matrix Theory) to estimate retrieval errors. Both methods assume that the LMA system has an overall rms timing error of 50 ns, but all other possible errors (e.g., anomalous VHF noise sources) are neglected. The detailed spatial distributions of retrieval errors are provided. Even though the two methods are independent of one another, they nevertheless provide remarkably similar results. However, altitude error estimates derived from the two methods differ (the Monte Carlo result being taken as more accurate). Additionally, this study clarifies the mathematical retrieval process. In particular, the mathematical difference between the first-guess linear solution and the Marquardt-iterated solution is rigorously established thereby explaining why Marquardt iterations improve upon the linear solution.
A one-speed Boltzmann transport theory, with diffusion approximations, is applied to study the radiative transfer properties of lightning in optically thick thunderclouds. Near-infrared (X = 0.7774 p.m) photons associated with a prominent oxygen emission triplet in the lighting spectrum are considered. Transient and spatially complex lightning radiation sources are placed inside a rectangular parallelepiped thundercloud geometry and the effects of multiple scattering are studied. The cloud is assumed to be composed of a homogeneous collection of identical spherical water droplets, each droplet a nearly conservative, anisotropic scatterer. Conceptually, we treat the thundercloud like a nuclear reactor, with photons replaced by neutrons, and utilize standard one-speed neutron diffusion techniques common in nuclear reactor analyses. Valid analytic results for the intensity distribution (expanded in spherical harmonics) are obtained for regions sufficiently far from sources. Model estimates of the arrival-time delay and pulse width broadening of lightning signals radiated from within the cloud are determined and the results are in good agreement with both experimental data and previous Monte Carlo estimates. Additional model studies of this kind will be used to study the general information content of cloud top lightning radiation signatures. IntroductionSatellite observations of lightning have been gathered for several years [Sparrow and Ney, 1968; Vorphal et al., 1970; Sparrow and Ney, 1971; Turman, 1978; Orville and Henderson, 1986] and the requirements for a comprehensive space-based lightning detection system has been addressed by Christensen et al. [1980]. These requirements have been more clearly defined in subsequent studies [Few et al., 1980; Davis et al., 1983] and the details of a proposed lightning mapper system (LMS) are described by Christian et al. [1989]. The LMS would detect and locate lightning from geosynchronous orbit, allowing for a broader areal coverage of world-wide lightning and more continuous thunderstorm monitoring. Similar benefits will be gained by a related, low Earth orbiting lightning imaging sensor (LIS) [Christian et al., 1992]. Recently, it has been shown that the lightning flashing rate is an indicator of total rain volume [Goodman and MacGorman, !986; Goodman and Buechler, 1990] and the total Maxwell current density above thunderstorms [Blakeslee et al., 1989]. This suggests that staring lightning imagers might be used to remotely infer a broad range of meteorological/electrical phenomena, however, the usefulness of other types of space-based observations of lightning, including spectral, temporal, and polarization measurements has yet to be studied in great detail. In order to fully benefit from spaceborne observations of lightning it is desirable to investigate the general information content of cloud top lightning signals radiated to space and the patterns of cloud top illumination due to lightning. Lightning optical waveform data obtained from high-altitude U2 missions above ...
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.
A retrieval method is introduced for estimating the fraction of ground flashes in a set of N flashes observed from either a low earth-orbiting or geostationary satellite lightning imager. The methodology exploits the fact that mean optical characteristics of ground and cloud flashes differ, and hence a properly posed equation set for mean conditions of a set of N observed flashes can be mathematically inverted to estimate the ground flash fraction (and hence the cloud flash-to-ground flash ratio). Explicit analytic expressions for the retrieval errors are derived, and numerical tests of the retrieval method are provided to quantify retrieval accuracy. It has been found that the retrieval method works best when only one optimum optical parameter is used (the singlecharacteristic solution approach) rather than a mixture of optical parameters (the multiple-characteristic solution approach); that is, the suboptimum optical parameters in the mix degrade retrieval accuracy. Since the retrieval method uses conterminous United States (CONUS)-averaged values of the lightning optical measurements, retrieval errors tend to be smallest in geographical regions whose specific mean lightning optical measurements are closest to the CONUS mean values. The rms ground flash fraction retrieval errors for 52 widely distributed regions across CONUS ranged from as low as 0.061 to 0.111, depending on the true ground flash fraction sought.
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