"Parameter Estimation and Inverse Problems, 2/e" provides geoscience students and professionals with answers to common questions like how one can derive a physical model from a finite set of observations containing errors, and how one may determine the quality of such a model. This book takes on these fundamental and challenging problems, introducing students and professionals to the broad range of approaches that lie in the realm of inverse theory. The authors present both the underlying theory and practical algorithms for solving inverse problems. The authors' treatment is appropriate for geoscience graduate students and advanced undergraduates with a basic working knowledge of calculus, linear algebra, and statistics. "Parameter Estimation and Inverse Problems, 2/e" introduces readers to both Classical and Bayesian approaches to linear and nonlinear problems with particular attention paid to computational, mathematical, and statistical issues related to their application to geophysical problems. The textbook includes Appendices covering essential linear algebra, statistics, and notation in the context of the subject. This book includes a companion website that features computational examples (including all examples contained in the textbook) and useful subroutines using MATLAB. It: includes appendices for review of needed concepts in linear, statistics, and vector calculus; features a companion website that contains comprehensive MATLAB code for all examples, which readers can reproduce, experiment with, and modify; offers an online instructor's guide that helps professors teach, customize exercises, and select homework problems; and, is accessible to students and professionals without a highly specialized mathematical background.

/pdf/parameter-estimation-and-inverse-problems-370q4164ci.pdf

Parameter estimation and inverse problems

of uncertainty for the OTD global totals represents primarily the uncertainty (and variability) in the flash detection efficiency of the instrument The OTD measurements have been used to construct lightning climatology maps that demonstrate the geographical and seasonal distribution of lightning activity for the globe An analysis of this annual lightning distribution confirms that lightning occurs mainly over land areas, with an average land/ocean ratio of 10:1 The Congo basin, which stands out year-round, shows a peak mean annual flash density of 80 fl km 2 yr 1 in Rwanda, and includes an area of over 3 million km 2 exhibiting flash densities greater than 30 fl km 2 yr 1 (the flash density of central Florida) Lightning is predominant in the northern Atlantic and western Pacific Ocean basins year-round where instability is produced from cold air passing over warm ocean water Lightning is less frequent in the eastern tropical Pacific and Indian Ocean basins where the air mass is warmer A dominant Northern Hemisphere summer peak occurs in the annual cycle, and evidence is found for a tropically driven semiannual cycle INDEX TERMS: 3304 Meteorology and Atmospheric Dynamics: Atmospheric electricity; 3309 Meteorology and Atmospheric Dynamics: Climatology (1620); 3324 Meteorology and Atmospheric Dynamics: Lightning; 3394 Meteorology and Atmospheric Dynamics: Instruments and techniques;

/pdf/global-frequency-and-distribution-of-lightning-as-observed-1zf7bt3rws.pdf

Global frequency and distribution of lightning as observed from space by the Optical Transient Detector

Observations have been made of a new terrestrial phenomenon: brief (approx. millisecond), intense flashes of gamma rays, observed with space-borne detectors. These flashes must originate at altitudes in the atmosphere above at least 30 km in order to be observable by orbiting detectors aboard the Compton Gamma-Ray Observatory (CGRO). At least a dozen events have been detected over the past 2 years. The photon spectra from the events are very hard and are consistent with bremsstrahlung emission from energetic (MeV) electrons. The most likely origin of these high energy electrons, while speculative at this time, is a rare type of high altitude electrical discharge above thunderstorm regions.

/pdf/discovery-of-intense-gamma-ray-flashes-of-atmospheric-origin-58ej84so5v.pdf

Discovery of Intense Gamma-Ray Flashes of Atmospheric Origin

Physics of shock waves and high-temperature hydrodynamic phenomena - Ya.B. Zeldovich and Yu.P. Raizer (translated from the Russian and then edited by Wallace D. Hayes and Ronald F. Probstein); Dover Publications, New York, 2002, 944 pp., $34.

A simple parameterization has been developed to simulate global lightning distributions. Convective cloud top height is used as the variable in the parameterization, with different formulations for continental and marine thunderstorms. The parameterization has been validated using two lightning data sets: one global and one regional. In both cases the simulated lightning distributions and frequencies are in very good agreement with the observed lightning data. This parameterization could be used for global studies of lightning climatology; the earth's electric circuit; in general circulation models for modeling global lightning activity, atmospheric NO(x) concentrations, and perhaps forest fire distributions for both the present and future climate; and, possibly, even as a short-term forecasting aid.

A simple lightning parameterization for calculating global lightning distributions

A GPS-based system has been developed that accurately locates the sources of VHF radiation from lightning discharges in three spatial dimensions and time. The observations are found to reflect the basic charge structure of electrified storms. Observations have also been obtained of a distinct type of energetic discharge referred to as positive bipolar breakdown, recently identified as the source of trans-ionospheric pulse pairs (TIPPs) observed by satellites from space. The bipolar breakdown has been confirmed to occur between the main negative and upper positive charge regions of a storm and found to be the initial event of otherwise normal intracloud discharges. The latter is contrary to previous findings that the breakdown appeared to be temporally isolated from other lightning in a storm. Peak VHF radiation from the energetic discharges is observed to be typically 30 dB stronger than that from other lightning processes and to correspond to source power in excess of 100 kW over a 6 MHz bandwidth centered at 63 MHz.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999GL010856

A GPS‐based three‐dimensional lightning mapping system: Initial observations in central New Mexico

[1] The location accuracy of the New Mexico Tech Lightning Mapping Array (LMA) has been investigated experimentally using sounding balloon measurements, airplane tracks, and observations of distant storms. We have also developed simple geometric models for estimating the location uncertainty of sources both over and outside the network. The model results are found to be a good estimator of the observed errors and also agree with covariance estimates of the location uncertainties obtained from the least squares solution technique. Sources over the network are located with an uncertainty of 6–12 m rms in the horizontal and 20–30 m rms in the vertical. This corresponds well with the uncertainties of the arrival time measurements, determined from the distribution of chi-square values to be 40–50 ns rms. Outside the network the location uncertainties increase with distance. The geometric model shows that the range and altitude errors increase as the range squared, r2, while the azimuthal error increases linearly with r. For the 13 station, 70 km diameter network deployed during STEPS the range and height errors of distant sources were comparable to each other, while the azimuthal errors were much smaller. The difference in the range and azimuth errors causes distant storms to be elongated radially in plan views of the observations. The overall results are shown to agree well with hyperbolic formulations of time of arrival measurements [e.g., Proctor, 1971]. Two appendices describe (1) the basic operation of the LMA and the detailed manner in which its measurements are processed and (2) the effect of systematic errors on lightning observations. The latter provides an alternative explanation for the systematic height errors found by Boccippio et al. [2001] in distant storm data from the Lightning Detection and Ranging system at Kennedy Space Center.

Accuracy of the Lightning Mapping Array

Sources of charge for the individual strokes of four multiple-stroke flashes to ground have been determined, using measurements of the electrostatic field change obtained at eight locations on the ground beneath the storm. The resulting charge locations have been compared to 3-cm radar measurements of precipitation structure in the storm. The field changes of individual strokes were found to be reasonably consistent with the lowering to ground of a localized or spherically symmetric charge in the cloud. The centers of charge for successive strokes of each flash developed over large horizontal distances within the cloud, up to 8 km, at more or less constant elevation between the −9° and −17°C environmental (clear air) temperature levels. Comparison with the radar measurements has shown that the discharges developed through the full horizontal extent of the precipitating region of the storm and appeared to be bounded within this extent. In one instance where cellular structure of the storm was apparent, the strokes selectively discharged regions where the precipitation echo was the strongest. Vertical extent of the stroke charge locations was small in comparison with the vertical extent of the storm. The field changes in the intervals between strokes have been found to exhibit many of the features which Malan and Schonland used to infer that ground flashes discharge a nearly vertical column of charge in the cloud. This and other evidence is used to show that their observations, which were made at a single station, could instead have been of horizontally developing discharges. The interstroke field changes have been analyzed using a point dipole model and found to correspond to predominantly horizontal charge motion that was closely associated with the ground stroke sources for the flashes. The interstroke activity served effectively to transport negative charge in the direction of earlier stroke volumes and often persisted in the vicinity of an earlier stroke volume, while subsequent strokes discharged more distant regions of the cloud. Long-duration field changes that sometimes preceded the first stroke of a flash have been analyzed and found to correspond to a series of vertical and horizontal breakdown events within the cloud, prior to development of a leader to ground. These events were associated in part with the negative charge region that became the source of the first stroke and effectively transported negative charge away from the first stroke charge volume and from the charge volumes of subsequent strokes. Several continuing current discharges were found also to progress horizontally within the cloud and sustained currents in the range of 580 A to less than 50 A. The continuing current field changes were consistently better fitted by the monopole charge model than the field changes of discrete strokes within the same flash.

An analysis of the charge structure of lightning discharges to ground

A detailed study of the complete sequence of VHF radiation events during intracloud (IC) flashes in Florida has shown that IC flashes often have a bilevel structure connected by a single upward channel. The lower and upper level channels appear to correspond to the main negative and upper positive charge regions of the storm, respectively. The IC flashes are characterized by active and final stages, each comprising about half of the overall flash duration. During the active stage, negative charge is transported upward in the cloud, first as a result of initial breakdown, which establishes the upward channel, and subsequently as a result of repeated breakdown events from the lower to the upper level. For the flashes of this study, the initial breakdown lasted 10 to 20 ms and propagated upward at a speed of 1.5 to 3×10 5 m s -1 . It also established the beginning of one or more horizontal branches in the upper positive charge region, which were extended by the subsequent breakdown. Little or no radiation was detected along the upward channel during the subsequent breakdown, indicating that the channel remained more or less continuously conducting following the initial breakdown. After a time delay, the subsequent breakdown extended the lower-level channels in a retrograde manner horizontally away from the flash origin. At the end of the active stage the upward channel ceases to be conducting and the remaining, final stage of a flash is characterized by fast (10 6 to 10 7 m s -1 ), well-defined K-type streamers that begin at successively greater distances along the lower-level channel and transport negative charge toward or into the base of the upward channel. The streamers are identical to those observed during the interstroke intervals and final stages of cloud-to-ground (CG) flashes. Typically, a few late-flash K streamers continue through the flash origin into and along an upper-level channel. The overall results are consistent with other observations of IC flashes and explain several features of the observations. Similar results are obtained for IC flashes in New Mexico storms. Comparison of the initial breakdown of IC and CG flashes shows that the IC breakdown is more intermittent than that of CG flashes, even though both have the same polarity and propagate at about the same average speed.

Paul R. Krehbiel

Papers

A GPS‐based three‐dimensional lightning mapping system: Initial observations in central New Mexico

Accuracy of the Lightning Mapping Array

An analysis of the charge structure of lightning discharges to ground

Accuracy of the Lightning Mapping Array

The spatial and temporal development of intracloud lightning