Sprites and halos in the mesosphere are produced electrostatically by lightning ground flashes whose polarity is positive, by a margin of at least 1000 to 1 in collected observations. The initiation of these events is controlled by the vertical charge moment change of the flash. Schumann resonance ELF methods have been used to measure the charge moments of millions of flashes worldwide. The bipolar distributions of these events show stronger positive than negative tails, consistent with the predominance of “positive” sprites, but the negative tail of supercritical events is still of the order of 10% of the total supercritical population, more than 1 order of magnitude larger than the observed fraction of “negative” sprites. This juxtaposition constitutes a paradox. The suggested resolution of the paradox is that the more impulsive population of supercritical negative flashes is producing dim halos that are not readily detected in conventional video imagery. Additional sensitive, high-resolution, and high-speed imager (<1 ms) studies of halos and their lightning parents are needed to verify this hypothesis.
 The dark horse in the race to understand TLEs in general is the so-called “halo,” a quasi-uniform electrical breakdown in the mesosphere. Discovered by Barrington-Leigh et al. , the halo, like the sprite, was interpreted as arising from electrostatic stress induced by lightning. The layered nature of the halo has led to some confusion in the optical observations [Barrington-Leigh et al., 2001] with the elve [Inan et al., 1996a], now widely believed to originate from the radiation field of lightning ground flashes, without pronounced polarity asymmetry [Barrington-Leigh and Inan, 1999]. Bering et al.  have recently suggested the occurrence of numerous halos associated with negative polarity ground flashes from stratospheric balloon photometry measurements. This circumstance may provide a possible explanation for the sprite polarity paradox.
2. Polarity of Lightning Ground Flashes Causal to Sprites
 Coordinated field campaigns to study sprites and their parent lightning have been underway for a decade using video camera observations and lightning detection methods at ELF/VLF [Lyons, 1994; Sentman et al., 1995; Boccippio et al., 1995; Bering et al., 2004; Thomas et al., 2005]. For studies within the United States, the National Lightning Detection Network (NLDN) has served a key role in providing the locations and return stroke times of flashes to ground, over large areas. Detailed comparisons in numerous studies have shown that the lightning type causal to sprites is almost invariably the positive ground flash. The compilation in Table 1 of verified pairings of optical (sprite) and lightning observations show several thousand cases of positive CG-induced sprites to date.
Table 1. Summary of Case Studies of Sprite Parent Lightning Polarity
 Remarkably few observations associate sprites with negative polarity ground flashes. The most notable study is that of Barrington-Leigh et al.  in which two (and possibly three) sprites are convincingly linked with negative ground flashes documented by the NLDN. The sprites were also well documented in video and with photometers, with a timing uncertainty in the range of 5 ms. The negative ground flashes showed exceptional peak current (−97 kA and −93 kA), exceptional continuing current, and exceptional ELF-determined charge moments (−1380 C-km and −1550 C-km). The latter values are well above any criteria of the C. T. R. Wilson type, given the observed initiation altitudes for these sprites.
 Other studies have associated sprites with negative polarity lightning, but with substantially less convincing documentation on a physical pairing. Winckler  compared video camera images of sprites with NLDN events and found 3 of 38 cases with suspected negative polarity lightning. The NLDN timing resolution was only 1 second however, raising considerable uncertainty in the linkage, particularly for an active frontal system with an order of magnitude more negative than positive lightning. Sao Sabbas , in comparisons of a similar nature within the NLDN, found 11% of video-observed sprite events associated with negative ground flashes. Her timing uncertainty however was 60 ms and her chosen width of acceptance window for matches with the (accurately timed) NLDN events was 420 ms. This specification is nearly an order of magnitude less stringent than the timing of Barrington-Leigh et al. , and as in the Winckler  study, casts doubt on the reliability of the lightning-sprite linkages for the negative events. Sao Sabbas et al.  report 16 ms timing uncertainty in a later study, and then the number of sprite “finds” with negative lightning polarity was much reduced, as shown in Table 1. The positive/negative event statistics of Sao Sabbas  with the larger search window are clearly inconsistent with the collection of other studies in Table 1, and are suspect. The single negative sprite of Neubert et al.  also fell outside their nominal 100 ms window for intercomparisons. A summary of all sprite-lightning comparisons judged to be reliable leads to the important result that the ratio of “positive” to “negative” sprites in these observations is more than 1000 to 1. Stated in another way, fewer than 0.1% of observed sprites (mostly over land) have been attributed to lightning with negative polarity.
3. Bipolar Distributions of Lightning Charge Moment on a Global Basis
 As stated in section 1, the lightning charge moment is the quantity of interest in assessing the initiation of sprites and halos in the mesosphere. The critical charge moment depends on the height of initiation of the sprite or halo, but generally values exceeding 500 C-km are needed [Huang et al., 1999; Hobara et al., 2001; Hu et al., 2002; Williams et al., 2007]. Charge moments of this magnitude are exceptional among the global lightning population. As a consequence, these special lightning flashes stand well above all other lightning for short periods of time in exciting the Schumann cavity.
 The determination of the vertical charge moment of lightning flashes by single-station Schumann resonance ELF methods is now widely recognized [Burke and Jones, 1995; Huang et al., 1999; Hobara et al., 2006]. The vertical charge moment (C-km) is the product of the total charge transferred to ground by lightning and the vertical distance over which the charge is transferred. The determination of the polarity of the charge transferred to ground (and hence the polarity of the charge moment change) with the ELF method is highly reliable and is based on thousands of event comparisons (with common GPS timing) with the National Lightning Detection Network. Comparisons of charge moment change [Lyons et al., 2003] by the ELF method and by the classical electrostatic method [Wilson, 1916; Krehbiel et al., 1979] show good agreement, thereby validating the ELF methods on a global basis. Williams et al.  have recently measured charge moments for sprite-producing lightning in Australia at 16 Mm range from the Rhode Island station, and the values are comparable to those needed for sprites in North America at 2 Mm from the same receiver [Huang et al., 1999].
 The Schumann resonance receiving station in West Greenwich, Rhode Island has been in near-continuous operation since 1993. Procedures have been automated for extracting charge moments for individual lightning flashes on a global basis. Since the large lightning flashes are geolocated with accuracies of the order of 0.5–2 Mm [Boccippio et al., 1998], it is possible to organize the events by region in providing charge moments. The polarity of the lightning and its charge moment can be determined reliably on the basis of the initial excursion of the vertical electric field in the ELF sferic, and on this basis, bipolar distributions of charge moment can be extracted from the observations. Such distributions are shown in Figure 1 for North America, Figure 2 for South America, Figure 3 for Africa, and Figure 4 for the Atlantic Ocean, for the period 1995–2000. These regional plots integrate the charge moment determinations for 500,000 flashes in North America, 730,000 in South America, 220,000 in Africa, and 230,000 in the Atlantic Ocean.
 These regional lightning totals are influenced in part by the source-receiver distances and by signal-to-noise considerations. Despite Africa's greater distance from the Rhode Island receiver than South America, the numbers of supercritical events of both polarities are larger in Africa (Figure 3) than in South America (Figure 2). This result finds a tentative explanation in recent findings during the African Monsoon Multidisciplinary Analysis (AMMA) campaign showing an abundance of squall lines in Africa with large regions of trailing stratiform precipitation where large positive ground flashes are common [Williams, 1998].
 In the center of all these distributions at zero charge moment, the counts also vanish, because the lightning waveforms overlap to such an extent that individual flashes cannot be resolved [Huang et al., 1999; Füllekrug et al., 2002]. With increasing charge moment the numbers of events increase, with larger numbers on the negative side than positive side. This finding is consistent with the well-known prevalence of negative ground flashes over positive ground flashes, by a typical 10 to 1 margin. As the charge moment increases further beyond the threshold value for sprites, one enters the important tails of the distribution. Here the polarity prevalence reverses, and positives begin to dominate. It is important to note however that substantial numbers of events are found above the nominal sprite threshold (“supercritical” events) for both negative and positive polarity.
 The quantitative behavior in the tails of the bimodal distribution may be studied further by examining the percentage of all flashes with supercritical charge moments, as a function of the threshold chosen for criticality. This behavior is plotted in Figures 5, 6, 7, and 8, also for the four regions of Figures 1–4. The result shows that the relative prevalence of positive and negative supercritical events is roughly independent of the selected threshold for charge moment (and by inference, roughly independent of the altitude at which sprites initiate), above a value of 750–1000 C-km. The percentage of total supercritical flashes with negative polarity is of order 10%, and is never less than 5%.
 The key finding here is that roughly 10% of all lightning flashes globally believed capable of making sprites are negative in polarity. This result is much the same in different land regions. This statistic stands in marked contrast with the 0.1% of sprites linked with negative lightning in the observations organized in Table 1. Not a single study among the 30 odd studies in Table 1 with stringent timing shows as many as 10% of the observations with negative polarity lightning. This contrast constitutes the quantitative paradox addressed in this study. If 10% of all large lightning flashes have negative polarity and possess charge moments sufficiently large to make sprites, then why are less than 0.1% of observed sprites linked with negative polarity lightning?
4. Comparison of the Impulsiveness of Positive and Negative Lightning Flashes
 One additional distinction can be drawn between positive and negative flashes on the basis of the ELF observations, and this distinction may ultimately aid in the resolution of the paradox. The slope of the frequency spectrum of current moment is a measure of the impulsiveness of the flash [Sentman, 1996]. When the duration of the lightning current is short in comparison to the time required for light to circle the Earth (∼130 ms), then the flash is impulsive and the current moment spectrum is flat with frequency (i.e., “white”). When a more persistent current flows, the spectral slope becomes negative (i.e., the spectrum “reddens”).
 The vertical current moment IdS (ω) of a lightning flash may be obtained directly from the normal mode equations for the uniform Earth-ionosphere cavity [Burke and Jones, 1995; Huang et al., 1999]. The current moment can be computed from the measured frequency spectrum in either the vertical electric or the horizontal magnetic field. In this case, use is made of the magnetic field because it is better calibrated [Huang et al., 1999]. As before, the vertical electric field is used to determine the polarity of the flash.
Figure 9 compares the least squares fit slopes of the current moment spectra for a large number of lightning flashes worldwide, as recorded in Rhode Island. Although a considerable spread of slopes is observed for both polarities, the negative flashes have a clear tendency for mean spectrum close to “white” (zero slope), whereas the positive flashes show a significant tendency for negative slopes. This ELF evidence for more persistent currents in positive is consistent with the general knowledge for ordinary lightning that return strokes in positive flashes are almost invariably followed by continuing current, whereas negative flashes are often characterized by discrete strokes without continuing current [Rakov and Uman, 2003; Heckman, 1992; Williams, 2006]. A minority of events in Figure 9 show positive slopes, indicate of “blue” current moment spectra. No particular physical significance is attached to these events because of their infrequent occurrence and the inevitable presence of noise in the analysis of slopes.
 The quantitative contrast between sprite/halo observations in section 2 and the ELF-observed populations of charge moment change in section 3 presents a paradox. The polarity asymmetry of sprites is pronounced, and has suggested explanations in other kinds of polarity asymmetry. These are discussed in turn below.
5.1. Polarity Asymmetry in Vertical Charge Moments
 The potent nature of positive ground flashes in many quantities (total charge transfer [Rakov and Uman, 2003], peak current [Rakov and Uman, 2003], lateral extent in the cloud [Williams, 1998]) have led to speculation that this polarity asymmetry in lightning ground flash characteristics might explain the polarity asymmetry in sprites. Indeed, the results presented in Figures 1–4 show a predominance of supercritical charge moments for positive flashes. Barrington-Leigh et al. [1999, p. 3608] in concluding their study on anomalous sprites, state: “This suggests that the rarity of negative sprites may result from a rarity of large (charge moment changes) caused by −CGs”. In the same vein, Pasko et al. [2000, p. 503] state: “The removal of large quantities of charge is required (for sprites)…, which is extremely rare for negative CG lightning.” The observations in Figures 1–4 show however that large numbers of negative flashes possess charge moments sufficient to cause sprites by the quasi-electrostatic mechanism of C. T. R. Wilson. The hypothesis advanced by Barrington-Leigh et al.  is therefore not by itself an adequate resolution of the paradox.
Cummer and Lyons  and Lyons et al. [2006b], focusing on sprites production on three days in the Great Plains with video and ELF methods, conclude that no “negative” sprites are observed because no supercritical negative charge moments are evident. This provides a satisfactory resolution of the polarity paradox for this relatively short observing period, but the global maps of the same Rhode Island observations of Hobara et al.  extending over a 5 year period show abundant evidence for supercritical negative charge moments in the Great Plains at other times. The Rhode Island ELF observations were not available for the storms studied by Cummer and Lyons , but in another study when their ELF data were available for comparison [Lyons et al., 2003], Rhode Island detected twice as many events as did Cummer. Detection efficiency is another consideration in judging the results of lightning/TLE comparisons. For reasons still unclear, ELF transients with positive polarity tend to process for charge moment by Schumann resonance methods [Burke and Jones, 1995; Huang et al., 1999] more cleanly than events of negative polarity.
5.2. Polarity Asymmetry in Sprite Detectability
 The global observations reported here for a single ELF station and used to compute the 10% statistic in section 3 gather events from both land and ocean and over all times of day and night. Sprites require local nighttime conditions for optical detection and documentation. Furthermore, the great majority of sprites have been observed over land areas where storms most favorable to sprite production are found [Williams, 1998], and where video cameras are more easily operated. Could these sampling biases be responsible for the apparent sprite polarity paradox?
 Regarding the oceans, our global maps of supercritical flashes located from Rhode Island [Hobara et al., 2006] (and earlier analysis over oceans by Füllekrug et al. ) show a relatively richer population (relative to regional maps of land areas (Figures 1–4) and global maps of ordinary (optical) lightning [e.g., Christian et al., 2003] of negative supercritical events over oceans. Consistent with this finding are the abundances of both halos and elves over oceans in ISUAL observations (Y. Takahashi and T. Adachi, personal communication, 2005; S. Cummer, personal communication, 2006), dominated by negative polarity ground flashes. Still, the great majority of supercritical events worldwide are land based, where sprite observations are most frequently made. The ocean alias in these observations cannot account for the quantitative sprite polarity asymmetry.
 The North American analysis received more attention than other areas because of the disproportionate emphasis on sprite studies in this region in earlier studies [Lyons, 1994; Sentman et al., 1995; Lyons, 1996; Nelson, 1997; Huang et al., 1999; Bering et al., 2004]. Toward identifying the population of supercritical events over land only in North America, a conservative filter was applied to include only those events more than 300 km inland from all sources of water (including the Great Lakes). This had the effect of rejecting a substantial population of negative events over adjacent oceans from the analysis. With this filter, the percentage of all supercritical events with negative polarity is never less than 5%, regardless of chosen threshold in the charge moment. On this basis, we are unable to account for the sprite polarity asymmetry in North America on the basis of a predominance of supercritical positive flashes.
 To investigate the possibility of diurnal (day-night) aliasing, the local time variations of supercritical events in Africa were examined. Figure 10 shows the diurnal variation of events with charge moment greater than 500 C-km for both positive and negative flashes. Consistent with the notion that “mesoscale” lightning flashes occur later in the diurnal cycle than ordinary lightning [Williams and Yair, 2006], the maximum for negative events is in late afternoon (when sprites would not be easily detectable, to be sure). The peak for positive events is even later (1900 local time (LT)), and is well past the time of nominal maximum afternoon lightning activity in Africa [Whipple, 1929]. Despite the presence of maximum activity at times of marginal sprite detectability, the numbers of supercritical events in nighttime hours is still abundant. The number of supercritical negative events at night (2000–0500 LT) is still a healthy 27% of the daily total, on average. The missed sprite events in the daytime cannot account for the order-of-magnitude contrast in the quantitative polarity paradox address in section 3.
5.3. Polarity Asymmetry in the Threshold for Streamer Propagation
Pasko et al.  and Thomas et al.  have both suggested that the appropriate threshold for sprite initiation in the mesosphere is the propagation threshold for streamers, rather than the dielectric strength for air [Wilson, 1925; Huang et al., 1999; Williams, 2001]. This suggestion is not unreasonable, so long as there is some “seed” for ionization. These authors go on to speculate that the well-recognized polarity asymmetry in streamer propagation threshold [Bazelyan and Raizer, 2000] is responsible for the polarity asymmetry in sprites. Positive streamers in air propagate in electric fields weaker by a factor of two than negative streamers. A “positive”/“negative” sprite would then involve a positive streamer propagating downward/upward from the initiation location in the mesosphere. It is not clear why the threshold for positive streamers propagating downward will be substantially less than the threshold for propagation in the opposite direction. On this basis, this explanation for the sprite polarity paradox does not seem viable, though a full test of this idea requires a quantitative prediction involving the vertical gradient in air density.
5.4. Polarity Asymmetry in an Electron Runaway Mechanism
5.5. Polarity Asymmetry of the Global Electrical Circuit
 The global circuit possesses its own polarity asymmetry—the Earth is negative and the atmosphere is positive—a result whose origin most likely lies in the still poorly understood aspects of ice microphysics [Williams, 2006]. D. L. Jones (personal communication, 2000) has suggested that the more potent lightning ground flashes with positive polarity behave in this way because they are drawing down the stored electrical energy of the global circuit. (The positive flashes cannot draw it all down because the positive charge in the global circuit does not reside on a single conductor, but is instead distributed throughout a dielectric.) In contrast, the more common negative flashes to ground add to the energy of the global circuit by increasing the negative charge on the Earth. While this idea deserves more attention, it does not forbid the existence of supercritical negative ground flashes, the key observations in the sprite polarity paradox.
5.6. Polarity Asymmetry in the Forcing of Sprites and Halos by Lightning
 The results on slopes in Figure 9 for the current moment spectra in Schumann resonance observations for positive and negative flashes show a clear statistical asymmetry. The linkage of this ELF result with earlier observations throughout the lightning literature showing that positive ground flashes almost invariably show long-continuing current; whereas negative flashes often exhibit discrete strokes is reasonable. A theoretical reason for this polarity asymmetry in lightning is given by Williams .
 The favored working hypothesis in the present study is then related to the asymmetrical nature of the forcing of TLEs by positive and negative flashes. The lightning-like filamentary structure of sprites requires a sustained electric field whereas a brief overvolting of the mesosphere with a more impulsive negative ground flash is more favorable to isolated halos. Support for these ideas in the atmosphere comes from observations by Barrington-Leigh et al.  and Bering et al. , both showing evidence for halos with short delays (∼1 ms) from the parent ground flash, and in the case of Miyasato et al.  and Bering et al. , evidence that the parent lightning flashes for halos are predominantly negative polarity. Many students of sprites registered surprise when Bering et al.  published their findings, including the present authors. In more recent ISUAL observations (K. Yamamoto et al., personal communication, 2006), lightning of negative polarity is dominating the production of observed halos over oceans by amore than 2 to 1. Lyons et al.  noted preferentially larger peak currents with negative polarity over the ocean (values > 200 kA). A possible reason for the polarity asymmetry in the impulsiveness of the “final jump” prior to return strokes from the sea has been given by Williams . These results support a short but strong electrostatic forcing by negative ground flashes over oceans.
 Support for these ideas in the laboratory comes from observations on the necessary conditions to create a uniform discharge in air at atmospheric pressure, an unusual situation. The recipe: fast application of an overvolted electric field [Roth et al., 2005]. Departures from this procedure result in the highly nonlinear and localized filamentary discharges more akin to sprites than to halos.
 The searches for NLDN events parent to TLEs in observations with video camera in early days of sprite research provide indirect evidence for the halo hypothesis. Ten years ago, only sprites and elves had been identified in the “zoo” of TLEs; the halo identification came later [Barrington-Leigh et al., 2001]. Extensive tabulations by Walt Lyons show numerous events identified with “[E],” then believed to be elves. Positive ground flash pairings from the NLDN with these events are conspicuously absent. As specific examples, on 4 August 1996, 84 TLEs were tabulated and 10 of them had no corresponding +CGs. Half of these were labeled “[E].” On 5 August 1996, 66 TLEs were tabulated, 9 without +CGs, with 5 of them in the [E] category. In the large storm on 24 July 1996, studied by Huang et al. , 29 of 327 events (9%) had no associated +CGs. Seven of these had [E] identifications. As a group, the events labeled [E] are the ones least likely to have corresponding positive ground flashes in the NLDN observations. Verified elves have been linked unambiguously with both positive and negative ground flashes [Barrington-Leigh and Inan, 1999] and this mixed result is consistent with the radiation field origin for elves [Inan et al., 1996a]. Is it possible that many or most of the [E] events are in fact halos, with negative ground flashes as the primary cause? Unfortunately, this suggestion remains difficult to test given the timing uncertainties in conventional video resolution and the simultaneous presence of large numbers of negative ground flashes in these storms. Some of the [E] events had no corresponding negative ground flash as possible parent either, and the generally accepted explanation is that the parent, whatever the polarity, escaped detection by the NLDN because the waveforms were too complex.
 If the existence of “negative” halos is to resolve the sprite polarity paradox, then it would seem that some halos associated with supercritical negative events are missed in the optical observations. This would seem to require that halos are dimmer than sprites, and hence more difficult in general to detect. Observations by Miyasato et al.  on a few strong events do not show the halos to be substantially dimmer than elves and sprites, though this sample data set is small. In contrast, recent (and still preliminary) ISUAL observations for halos over oceans indicate that halos are dimmer than sprites by a factor of 10 (T. Adachi and Y. Takahashi, personal communication, 2006), making halos substantially more difficult to identify in CCD imagery.
 Finally, one reviewer made comment about sprites with halos, an increasingly recognized phenomena in high-speed video camera observations. It is likely that the majority of the events previously dubbed “sprelves” by W. A. Lyons are in this category. Previous unpublished analyses suggest that these events are caused primarily by positive ground flashes that are impulsive in their beginnings (to make the halos) but also have a subsequent strong component of continuing current (to sustain the sprite).
 The substantial supercritical tail of the charge moment distributions, regionally and globally, is inconsistent with the tiny fraction of all sprites and halos associated with lightning of negative polarity. The favored explanation—that dim halos, not easily documented in conventional video imagery are produced by negative flashes—is consistent with the more impulsive nature of negative lightning. Detailed quantitative evidence is still lacking, and requires further observations over large negative charge moments with sensitive, time-resolved video imagery.
 Discussions with T. Adachi, E. Bering, S. Cummer, U. Ebert, M. Füllekrug, A. Gurevich, U. Inan, D. L. Jones, V. Mazur, J. Mrazek, V. Pasko, O. Pinto, R. Roussel-Dupré, F. Sao-Sabbas, D. Sentman, M. Stanley, Y. Takahashi, M. Taylor, Y. Yair, and Z. Kawasaki are greatly appreciated. This work was supported by grant ATM-0337298 from the U.S. National Science Foundation to study Schumann resonances. Partial support of FMA Research was also provided by the National Science Foundation (ATM-0221215).