We propose a new formula to calculate the planetary rate of sprite events, based on observations with sprite detectors. This formula uses the number of detected sprites, the detection efficiency and the false alarm rate of the detector and spatial and temporal effectiveness functions. The role of these elements in the formula is discussed for optical and non-optical recordings. We use the formula to calculate an average planetary rate of sprite events of ∼2.8 per minute with an accuracy of a factor ∼2–3 by use of observations reported in the literature. The proposed formula can be used to calculate the occurrence rate of any physical event detected by remote sensing.
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 Sprites are Transient Luminous Events (TLEs) occurring above thunderstorm clouds (Figure 1a). Sprites are nearly always initiated by particular intense positive cloud to ground (+CG) lightning (≃90% of the cases): a charge ≳+100 C is lowered to the ground. Optical sprite observations were first reported in the early 90's [e.g., Franz et al., 1990; Sentman et al., 1995]. Sprites are so dim (the luminosity of sprites is 1–30 MR which is ∼500–2000 times smaller than that of daylight) such that their observation by optical measurements is practically limited to night time. Optical observations define the occurrences of sprites. However, the occurrence of a sprite can also be inferred without the use of optical recordings. We call “sprite signatures” all non-optical events which indicate sprite occurrences. Up to date, 4 different sprite signatures have been reported in the literature. (1) Earth-ionosphere cavity resonances [Füllekrug and Reising, 1998]: (Figure 1b). (2) Extremely Low Frequency (ELF) transients [Reising et al., 1996, 1999]: (Figure 1c). (3) Very Low Frequency (VLF) perturbations [Haldoupis et al., 2004]: (Figure 1d). (4) Infrasound chirps [Farges et al., 2005]: (Figure 1e).
 A reliable assessment of the planetary rate of sprite events is necessary to assess the global impact of sprites on the properties of the atmosphere (e.g., chemical composition, mesospheric conductivity, global electrical circuit, response of sprites to climate change). However, even 15 years after the first sprite observation, the rate of sprite occurrences on Earth is yet to be determined. Several estimates of the planetary rate of sprite events have been made. These estimates range from 0.5 to 33 sprites per minute [Sato and Fukunishi, 2003; Yair et al., 2004; Blanc et al., 2004]. In this work, we propose a new formula to calculate the planetary rate of sprite events. We use an approach similar to that used by [Brooks, 1925] to estimate the planetary rate of lightning. Brooks estimated that ∼100 lightning occur every second on Earth. This estimate differs by a factor of ∼2–3 from estimates of the global lightning rate made with modern measurement technology in space [Christian et al., 2003]. Brooks counted the number of lightning occurrences in a single thunderstorm for 28 minutes and performed a global scaling via the spatial density of thunderstorms on Earth per year. Similarly, the Planetary Rate Of Sprite Events (PROSE) formula introduced in this paper, derives the global rate of sprite occurrences by upscaling limited observations. The scaling procedure uses the intrinsic property of the detector together with limitations of the observations. We use the PROSE formula to calculate the planetary rate of sprite events from optical recordings during 2 different experiments [Neubert et al., 2005; Reising et al., 1999], and radio wave recordings of Earth-ionosphere cavity resonances [Füllekrug and Constable, 2000].
2. The PROSE Formula
 In this contribution, we assess the yearly average planetary rate of sprite events. This average rate is calculated from the number of sprite occurrences on Earth (NE) in one year. A formula is needed to calculate NE from the observations of any sprite detector. This formula must address two important issues which relate (1) to the intrinsic properties of the detector and (2) to space and time limitations of the observations. (1) The detection efficiency (DE) of the sprite detector may be <1 ≡ 100% (some sprites are not detected) and the false alarm rate (FA) of the sprite detector may be >0 ≡ 0% (some detected events are not sprites). In this case, the imperfections of the detector must be considered. (2) The detector's observations may be limited to some part of the Earth's surface, to some limited time interval, and/or by data availability.
 According to the PROSE formula, the number of sprites occurrences on Earth in one year is:
ND is the number of events observed by the sprite detector. The ratio (1 − FA)/DE is the detector effectiveness. The product of this factor and ND is the number of sprite occurrences. The spatial effectiveness Sɛ and the temporal effectiveness Tɛ scale spatially and temporally the observations of the sprite detector. The spatial effectiveness depends on the operational surface So: the part of the Earth's surface covered by the observations. The operational surface So cannot be larger than the maximum detection range Ro of the sprite detector. If the operational surface is the entire Earth's surface, the spatial effectiveness is one. The temporal effectiveness depends on the operational time To: the time interval covered by the observations of the sprite detector. The operational time is determined by intrinsic limitations of the sprite detector and/or by limitations in data availability. If the operational time is an entire year, the temporal effectiveness is one.
 Note that equation (1) can be applied to calculate the occurrence rate of any physical event detected by remote sensing: just substitute the words sprite, Earth and year with the appropriate words.
 A zeroth order approximation for the spatial and temporal effectiveness can be obtained by considering sprites to be distributed uniformly in space and time. A first order approximation for the spatial and temporal effectiveness can be obtained by considering a linear dependence between the number of sprite and lightning occurrences. A second order approximation may be obtained considering a linear dependence between sprites and positive CG lightning occurrences or considering a linear dependence between sprite occurrences and Mesoscale Convective Systems (MCSs). Positive CG lightning initiate sprites [Boccippio et al., 1995], while MCSs are prominent in sprite production [Lyons et al., 2003] because they provide the positive charge reservoir needed for the sprite producing lightning [Williams, 1998].
3. Sprite Detectors
 The PROSE formula derives the planetary rate of sprite events from the observations of any sprite detector. To use the PROSE formula, it is necessary to know the detection efficiency and the false alarm rate of the sprite detector. The detection efficiency DE (the percentage of sprites detected) and the false alarm rate FA (the percentage of detected events which are not sprites) of a sprite detector are defined as
NSp is the number of sprite occurrences (number of optical sprites recordings) and ND is the number of events detected by the sprite detector. NES is the number of detected events that correspond to sprite occurrences and NENS is the number of detected events that do not correspond to sprite occurrences. All these quantities depend on the distance of the sprites from the camera and from the sprite detector. However, the estimated detection efficiency and false alarm rate cannot take in account this dependence since the probability density function for sprite luminosity is not yet known.
 In the case of sprite detectors based on sprite signatures one new observation is needed to calculate the planetary rate of sprite events via the PROSE formula, after the detection efficiency and the false alarm rate have been estimated. Otherwise, in virtue of equations (1) and (2), the calculated rate would be the same of that obtained with the optical observations alone. Finally, the maximum distance at which an event can be detected to determine the operational range Ro for calculation of the operational surface So needs to be considered.
3.1. Optical Sprite Recordings
 Optical observations are used to define sprite occurrences. Optical recordings undergo human scrutiny for the determination of the total number of sprite occurrences. We consider the detection efficiency to be 1 and the false alarm rate to be 0. Yet, even human scrutiny may lead to a detection efficiency smaller than 1 (e.g., DE=0.9) and a false alarm rate larger than 0 (e.g., FA=0.05). Such inaccuracies will result in a small (∼5%) correction in the calculated planetary rate of sprite events (equation (1)) in comparison to an overall accuracy of a factor ∼2–3 (Section 4).
 Optical observations above thunderstorm clouds in search for sprites can be performed from Earth, air or space. The field of view of a low-light TV camera (∼20°–50°) placed on a mountain top and the Earth's curvature usually limit the observations to a single thunderstorm or a single mesoscale convective system. However, observations from space could cover, in theory, the entire night time zone of the Earth with a network of geostationary, sprite observing satellites.
3.2. Earth-Ionosphere Cavity Resonances
Füllekrug and Reising  consider a sprite detector based on counting the number of discrete excitations of Earth-ionosphere cavity resonances (signal to noise ratio larger than 2) due to +CG lightning. Using optical observations and Earth-ionosphere cavity resonance measurements related to the same sprite-producing thunderstorm [Füllekrug and Reising, 1998], we find a false alarm rate of 0.2 and a detection efficiency of 0.36 (equation (2)). The electromagnetic radiation in the frequency range 5–90 Hz produced by lightning can propagate with little attenuation around the globe within the Earth-ionosphere cavity. Thus, the particularly intense +CG lightning which usually initiate sprites will also excite Earth-ionosphere cavity resonances detectable everywhere on the globe. The operational range Ro of a sprite detector based on Earth-ionosphere cavity resonances is the entire Earth's surface.
3.3. Other Sprite Signatures
 Sprite detectors based on ELF transients and infrasound chirps are already available [Reising et al., 1999; Ignaccolo et al., 2006]. However a second independent measurement for calculating the planetary rate of sprite events is necessary (as explained at the beginning of this section) but not yet available. The operational range of a single ELF station is approximately one half of the Earth's surface [Uppenbrink, 1999], while the range of a single infrasound station extends up to a reported maximum of ∼1000 km [Neubert et al., 2005] depending on the strength and direction of stratospheric winds. No sprite detector based on VLF perturbations has been proposed yet. The operational range of a single VLF transmitter-receiver pair is limited to a narrow strip (∼150 km) [Haldoupis et al., 2004] along the great circle path connecting the two stations.
4. The Planetary Rate of Sprite Events
 We calculate the planetary rate of sprite events using measurements of Earth-ionosphere cavity resonances, and optical observations from low-light TV cameras.
 The Earth-ionosphere cavity resonances data cover the time interval from 7th April to 4th May 1998. During this time interval 33,804 excitations of Earth-ionosphere cavity resonances due to +CG lightning were detected [Füllekrug and Constable, 2000] by the sprite detector described in Section 3.2. This detector has a detection efficiency of 0.36 and a false alarm rate of 0.2. The operational surface is the entire Earth's surface, thus the spatial effectiveness is Sɛ=1. The duration of the operational time is 28 days. We calculate the temporal effectiveness using the zeroth order approximation: Tɛ=365/28. According to the PROSE formula, the planetary rate of sprite events is ≃1.86 per minute (2,682 per day). The seasonal variability of the planetary sprite rate is expected to be on the same order of magnitude as the mean value (a conservative estimate given that the annual variability of the lighting rate is ≈10% [Christian et al., 2003]). Therefore, the accuracy of the calculated planetary rate of sprites is on the same order of magnitude (≈1).
 The optical observations we use to calculate the planetary rate of sprite events are those on the 21st of July 2003 performed by the low-light TV camera located at Pic du Midi (42.9°N; 0.01°E; 2877 m above sea level), and those on the 1st of August 1996 performed by the low-light TV camera located at Yucca Ridge field station (40.40°N; 104.56°W; 1690 m above sea level). 28 sprites were observed [Haldoupis et al., 2004; Farges et al., 2005] from 0200 UT to 0315 UT at Pic du Midi and 98 sprites were observed [Reising et al., 1999] from 0644 UT to 0907 UT at Yucca Ridge field station. In both cases, the detection efficiency is 1 and the false alarm rate is 0. The operational range of a low-light TV camera is an arc of radius ρ spanning the field of view of the camera (20° for Pic du Midi, 50° for Yucca Ridge field station). The radius ρ is determined by the altitude of the camera, by the Earth's curvature and by the typical vertical extension of a sprite (∼45 km starting from ∼40 km up to ∼85 km). Considering all these variables, we obtain ρ ≃ 800–1000 km for both cameras. The operational range of both cameras cover a small portion of the Earth's surface (≈1/10,000). Therefore we cannot consider the spatial effectiveness to be 1. The spatial effectiveness is calculated using a linear dependence between sprite and lightning occurrences (first order approximation). A detailed calculation of the second order approximation (+CG lightning or MCS) is beyond the scope of the present study. Given the rate ℛ(ϑ, ϕ) of lightning flashes per year and per km2 at latitude ϑ and longitude ϕ, the spatial effectiveness is:
where SE is the Earth's surface and So is the operational surface. The accuracy of this approximation can be estimated by comparing the rate of all lightning with the rate of +CG lightning. The ratio between local maxima of total lightning activity and +CG lightning activity is ∼7–31 with an average of ∼18 and a standard deviation of ∼8 [Orville and Silver, 1997]. Thus, the resulting accuracy of the calculated planetary rate of sprite events compared to the mean planetary rate of sprites is 18/8 ≈ 2. Using a value of ρ = 900 km, the field of view of the cameras and the lightning density function ℛ(ϑ, ϕ) (ℛ(ϑ, ϕ) ≃ 11.68 for Pic du Midi and ℛ(ϑ, ϕ) ≃ 14.14 for Yucca Ridge field station), we estimate the spatial effectiveness Sɛ ≃ 466 for the observations at Pic du Midi and Sɛ ≃ 154 for those at Yucca Ridge field station. The operational time To for both the Pic du Midi and Yucca Ridge field station are short (1h15m and 2h23m respectively) and fully occupied by thunderstorm activity. In this case, the zeroth order approximation for the operational time will lead to a biased (too large) planetary rate of sprite events. Thus, we consider 28 and 98 as the total number of sprites occurrences in the operational range of the cameras at Pic du Midi and Yucca Ridge field, respectively, during the entire night (∼10 hours). Moreover, we assume that the thunderstorm activity in the region (a circle of ∼900 km radius) around the locations of both cameras is essentially limited to the summer period. During the EuroSprite2005 campaign a frequency of one thunderstorm every ∼3–4 nights was observed with the camera located at Pic du Midi. This frequency and the limitation of the thunderstorm activity to the summer lead to a total of ∼30 thunderstorm days per year. With these assumptions, the zeroth order approximation for the temporal effectiveness Tɛ=30days/10hours=72 for both optical observations has the same accuracy as the approximation for the Earth-ionosphere cavity resonances measurements (Tɛ = 365/28) as explained at the beginning of this section. The resulting planetary rate of sprite events is ≃3.02 per minute (4,392 per day) for the observations at Pic du Midi and ≃3.5 per minute (5,040 per day) for the observations at Yucca Ridge field station.
 The good agreement between all the three different calculated rates makes us confident that the PROSE formula provides physical meaningful results. The average rate from all the three results is 2.8 per minute (4,032 per day).
 We propose a new formula (PROSE formula) for calculating the planetary rate of sprite events by use of optical observations and/or sprite signatures. The applicability of the formula is not limited to sprites only. The PROSE formula can be used to upscale the number of any physical events detected by remote sensing.
 We calculate a planetary rate of sprite events of ∼2.8 per minute. Due to the approximation used for the spatial and temporal effectiveness (Section 4), we expect an accuracy on the same order of magnitude of the rate itself (i.e., ∼2–3 as in the case of Brooks estimate for the global rate of lightning occurrences). A better accuracy can be obtained using the second order approximation for both the spatial and temporal effectiveness.
 The use of the PROSE formula together with new measurements will allow to calculate a more precise planetary rate of sprite events in the future.
 This research was sponsored by the European Commission under contract CAL HPRN-CT-2002-00216. The magnetic field measurements at Hollister were kindly provided by the Department of Materials Science and Mineral Engineering, and the Seismological Laboratory at UC Berkeley, CA, USA. The rate ℛ(ϑ, ϕ) of lightning flashes per year and per km2 at latitude ϑ and longitude ϕ was kindly provided by the Global Hydrology Resource Center at the Global Hydrology and Climate Center, Huntsville, AL, USA and by the DLR-Institut für Physik der Atmosphäre, Oberpfaffenhofen, Germany. Support for the Sondrestromfjord data acquisition was provided by the US National Science Foundation through cooperative agreements ATM-9813556 and ATM-0334122. The first author wishes to thank his wife Jennifer for her patience and her parents for the beautiful Dallas Stars jersey: GO STARS! Also thanks to E. Arnone for his helpful comments. The senior author wishes to thank the first author for following him all the way through Europe up to rainy Bath.