For ELF stations using hardware notch-filters to suppress power-grid emissions, the amplitude and the phase of the recorded sferics will inevitably be distorted. The phase shift causes the event time to move and hinders cross-comparing of events between different platforms. Notch-filtering also create fictitious dips in the current moment spectrum of the ELF sferics that lead to error in deducing the charge moment change (CMC) of the source discharge. To alleviate the effects of notch-filtering, we use an elementary signal processing method to reconstruct the source signals and use lab-generated ELF-like signals to check the limitations of the reconstruction. The results indicate that the delay of signals due to the notch-filtering module can be reliably restored, while the amplitude is less; but the reconstruction does restore the low frequency components of the sferics that are important for the CMC determination. Using the corrected event time, 20 associated sprites recorded by the ISUAL/FORMOSAT-2 experiment during June to September of 2008 were found. Comparing with the notch-filtered sferics, the reconstructed sferics are found to increase the current moment amplitude Iods by 69 ± 55%, reduce the time constant τ by 52 ± 15%, and lowers the CMC by 22 ± 21%, respectively. From the linear-correlation of the sprite brightness and the CMC of the sprite-producing positive lightning, a threshold CMC of ∼900 C-km for sprite initiation is inferred based on the reconstructed sferic data; this threshold value is also 25% lower than that inferred using the notch-filtered sferics.
 The sprite, which was first discovered in 1989 [Franz et al., 1990], is a type of optical transient luminous events (TLEs) above active thunderstorms [Sentman et al., 1995; Chen et al., 2008; Pasko, 2010, and references therein]. Soon after its discovery, sprite was found to be mostly associated with positive cloud-to-ground lightning (+CG) [Boccippio et al., 1995], which also radiates intense electromagnetic energy in the ELF (extreme-low-frequency; 3Hz to 3 kHz) and the very-low-frequency (VLF; 3 kHz to 30 kHz) bands.
 Two methods have been proven useful to extract the CMC from the recorded ELF sferics. One is a time domain deconvolution method introduced by Cummer and Inan , which is usually applied for relatively nearby events (∼5000 km or less). The other is the frequency-dependent normal mode equations commonly used for distance events [Huang et al., 1999; Sato and Fukunishi, 2003].
 Based on 7 analyzable sprite events, Huang et al.  reported that the relative brightness of sprites shows a linear dependence with the CMC of the causative lighting. Takahashi et al.  using 14 sprites recorded by the ISUAL experiment [Chern et al., 2003] to demonstrate that the N21P brightness, the N22P brightness, and the optical energy of sprites are all proportional to the lightning CMC. The inferred N2 1P total time-integrated total photons from sprites was reported to range from 0.2 × 1024 to 8.9 × 1024 photons.
 The work reported in this manuscript is based on the ELF events recorded by the Lulin station and the associated sprite events recorded by the ISUAL experiment. The Lulin ELF recording system [Wang et al., 2005] in Taiwan was set up as an auxiliary ground supporting equipment for the global survey of TLEs onboard the FORMOSAT-2 satellite [Chern et al., 2003; Chen et al., 2008]. However, even at the remote Lulin site, the interference from the power grid emissions is still significant and a hardware filter was installed in the system to prevent the background signals from swarming the system. Therefore, cross-comparison of events from the ISUAL experiment and the Lulin ELF station are often not trivial. In this paper, we use an elementary signal processing method to reconstruct the ELF sferics that have been recorded by the Lulin notch-filtered ELF system. Synthetic waveforms from a signal generator are used to demonstrate that the elementary method indeed produces reconstructed waveform with correct attributes. We then proceed to find the associated sprites, to compute the brightness of the sprites, and to infer the CMC of the causative lightning. The correlation between the brightness of the sprites and the CMC of the sprite-producing lightning is also explored.
 In this work, data from two instruments are analyzed. The first data set was from the Lulin ELF recording system based on a pair of EMI-BF4 magnetic coils that are located in the Yushan National Park of Taiwan. The coil orientations are parallel (H; north south) and perpendicular to (D; east west) the geo-magnetic field [Wang et al., 2005]. The antennas are sensitive in the radio frequency band of 0.3 Hz to 500 Hz. Hence the power grid emissions at 60 Hz and the higher harmonics present considerable impacts to the ELF measurement and hardware filters were deployed to suppress these background noises. A Lulin signal modulator contain a 1-Hz 2-pole Butterworth-type high pass filter, two 100-Hz 4-pole Butterworth-type low-pass filters, and two Butterworth-type notch-filters (60 Hz and 120 Hz with quality factor, Q = 6). The block diagram of the signal modulators is shown in Figure 1. If the normalized amplitude response of the notch-filters and the phase of the input sine waves are taken to be unity and zero respectively, the frequency-dependent output magnitude and phase responses of the Lulin signal modulators are represented by the blue and the red lines in Figures 2a and 2b. The near-perfect overlap of the response curves for the H- and the D-channel signal modulators is a good indication that the basic characteristics of them are near identical.
 The second data set consisted in data of sprites recorded by the ISUAL payload onboard the FORMOSAT-2 satellite operating from an 891 km sun-synchronous orbits [Chern et al., 2003; Chen et al., 2008], from June to September of 2008. This set of sprite images, which were taken through a N21P (623–750nm) band-pass filter and with a 29-ms exposure time, is used to compute the sprite brightness [Kuo et al., 2005, 2008; Lee et al., 2010].
3. Reconstruction of the ELF Sferics
3.1. The Elementary Signal Reconstruction Method
 It has been a well-known fact that the hardware filters in a signal chain can result in the phase and the amplitude of the original signals to be altered [e.g., Huang, 1998; Hobara et al., 2000]. The hardware-dependent phase shift causes an event time delay that can hinder comparing of events between different stations and platforms. Notch removal of the power grid emissions can also produce fictitious dips in the current moment spectrum of the ELF sferics that can lead to large error in deducing the CMC of the parent lightning. To accurately infer the physical parameters of the parent discharges, the unaltered waveforms are needed. Therefore, for the Lulin ELF system, the unknown sferics have to be reconstructed from the notch-filtered signals.
 If the normalized amplitude and the phase responses of the linear filter are Mfilter(f) and ϕfilter(f), respectively, the elementary signal process theory [e.g., O'Shea et al., 2010] requires that the input signal Φin(t) relates to the output signal Φout(t) via the following equation:
 However for the Lulin signal modulator, as the input frequency approaches 60 Hz the Mfilter drops to zero, as shown in Figure 2. Since Mfilter is in the denominator of equation (1), small white noises in the output signals near this frequency or numerical errors will be exaggerated to an abnormal proportion after the signal reconstruction. Because noises are inevitably present in any signals, the signal reconstruction is not expected to be perfect. Nevertheless, good reconstruction can be achieved if special attention is paid to constrain the reconstruction to remain within the signal frequency domain(s) of the output signal. For the Lulin ELF system, the wave components of the recorded sferics below 1 Hz and beyond 100 Hz should be excluded since these frequencies have been filtered out by the signal modulator. Also the reconstructed signal component near 60 Hz should not be significant, since most of the signal components near 60 Hz were removed while passing the notch-filters. Hence after performing the signal reconstruction in the frequency domain using FFT, equation (1), we pass the reconstructed signals through a 3 Hz high-pass, a 120 Hz low-pass, and a 60 Hz (2.4 Hz full-width-at-half-maximum width) digital filters to remove the out-of-band wave components.
3.2. Limitation of the Signal Reconstruction
 To characterize the alterations of the original sferics after passing of the notch-filters in the Lulin ELF system and to learn the limitations of the simple signal reconstruction, a Hewlett Packard 33120A Function Generator was used to compose various input signals for the notch-filtering module, including a quick-rise and exponential decay waveform (Figure 3, black line) that resembles a typical ELF sferics from positive cloud-to-ground discharges (+CGs). The notch-filtering module outputs of the test signals, including that depicted in Figure 3 (blue line), all indicated that the module consistently caused an event delay of ∼9 ms in relative to the inputs. Also, the amplitude of the modulator output is notably smaller and contains a significant amount of ringing. The time delay due to the hardware filter can be critical at times, especially when the TOA (Time of Arrival) method with multiple ELF stations is employed to triangulate the location of the source discharges. Obviously, if all the stations have the same hardware filters, the recorded sferics will suffer nearly identical time delay from the notch-filters. Therefore in geo-locating the sferic sources, the relative time delay will be canceled out and the inferred source location will be correct. However, for a network consisting of stations with and stations without notch-filters, the inferred event locations can contain substantial offsets. For a 9 ms delay, the geo-locating offset can be greater than 2000 km.
 For the notch-filtered output of this example input signal, the reconstructed signal, via equation (1), is represented by the red line in Figure 3. The reconstruction has restored the timing of the main peak, reduced the ringing, but, as expected, failed to re-gain all the original signal amplitude. The reason is that the components below 1 Hz, near 60 Hz, and above 100 Hz in the quick rise-exponential waveform were removed while passing through the notch-filtering module. However, if one compares the reconstructed signal with the 100 Hz digital low-pass version of the original signal (Figure 3, green dashed line), the similarity is high, except for the small ringing due to the notch-filtering. Evidently, the elementary reconstruction has restored most of the below 100 Hz wave components that were lost in passing through the hardware filters. Therefore, it is expected that the reconstructed sferics from this simple method will also produce more reliable CMC values, comparing with their filtered counterparts.
4. The Physical Characteristics of the Sprite-Producing Discharges
4.1. The Associated ELF Sferics of the ISUAL Sprites
 Partly due to the lack of adequate supports and damages from lightning, the Lulin notch-filtering ELF station was out of service for extended periods of time. Only during June to September of 2008, the Lulin station and the FORMOSAT-2/ISUAL were able to operate concurrently. In this period, 29 TLEs (14 pure sprites and 15 sprites with halo and/or elve) were screened out as potentially coincident events based on the proximity of the event time. After further considering the bearing angles from the magnetic Lissajous figures pointing of the ELF sferics, only 20 events are considered to be coincident TLE events. For these twenty ELF sferics, the average directional deviation with the sprites is 9.7 ± 7.9 degrees. The other nine events are ruled out due to the bearing angle pointing of the ELF source deviated more than 30 degrees from the TLEs [Füllekrug and Sukhorukov, 1999], or there were too many ELF sferics in the timing uncertainty window of both systems.
 For the twenty coincident TLEs, the event distance ranges from 1600 km to 15600 km and the average is ∼10000 km. After the reconstruction of the ELF sferics for these 20 coincident events, the average event time shifts back by 9.2 ± 0.4 ms, in relative to that of the notch-filtered sferics. The result is consistent with the 9.5 ms peak shift found in the lab-experiment (section 3.2). As a further check, we also computed the bearing angles pointing of the notch-filtered and the reconstructed sferics. The average difference between the bearings from each set of ELF sferics is found to be 0.39 ± 3.69 degrees. This difference is minimal and can safely be treated as no difference. Hence it can be further concluded that the notch-filtering process affects the signal amplitude, the phase, but not the directional bearing.
4.2. Formulae for Extraction Current Moment Amplitude (Iods), Time Constant (τ), and CMC
 In equation (2), Hϕ is the complex amplitude spectrum of the magnetic field projected into the ϕ direction, I(f)ds is the complex vertical current moment spectrum of the discharge, a is the radius of the Earth (6378 km), h is the height of the ionosphere (80 km), n is an integer, ν is a complex modal eigenvalue which describes the propagation and dissipation of the waves [Ishaq and Jones, 1977], pn1(cosθ) is the associated Legendre function of order n and the first degree, and θ is the angular great circle distance between the discharge source and the recording system.
 Under the assumption of a single exponential decay current moment form [Sentman, 1996] and using the current moment amplitude (Iods) and the time constant (τ) estimation methods described by Huang et al. , the following equation can be derived:
 From equation (2), the reconstructed Hϕ can be used to infer the current moment spectrum I(f)ds, then the time constant (τ) and the CMC can be computed via equation (3).
4.3. Current Moment Amplitude (Iods), Time Constant (τ), and CMC Inferred From the Reconstructed Sferics
 Among the 20 sprite-associated ELF sferics, the procedures and methods discussed in section 3.1 and 4.2 can be reliably applied to 18 of them. It is worth to mention that we apply the frequency components below 100 Hz to estimate the CMC. Ideally, frequency components in the entire radiation band are required to accurately estimate the shape of the lightning source current. Missing high frequency components may potentially results in an underestimate of Iods and an overestimate of τ since a narrower but taller pulse may produce the same waveform after passing through the 100 Hz low-pass filter. However, this does not change the estimated CMC due to the fact that any low-pass filtering does not change the time integration of the signal. Moreover, the high frequency components are hard to measure due to their high attenuation rates. For example, frequency components beyond 100 Hz can be too small to be reliably measured for events more than 10,000 km away. The inferred current moment amplitudes, the time constants of the sferics, and the CMCs are tabulated in Tables 1 and 2. Most notably, the CMCs of the sprite-producing discharges are found to range from 1500 C-km to 14000 C-km; whereas those computed from the reconstructed sferics falls between 1300 C-km and 10000 C-km. When the reconstructed sferics is used in place of the notch-filtered counterparts, on average the current moment amplitude Iods increases by 69 ± 55%, the time constant τ reduces by 52 ± 15%, and the CMC lowers by 22 ± 21%. Also as shown in section 3.2 and Figure 3, the signal reconstruction suppresses ringing in the notch-filtered signals and recovers some of the signal amplitude lost in passing through the modulator. Hence comparing with notch-filtered sferics, the reconstructed sferics are expected to have larger Iods and smaller τ. The combined influence of these two factors reduces the over-estimation of CMC, and produces a smaller value that is closer to the true CMC of the causative discharge.
Table 1. The Current Moment Amplitude (Iods), the Time Constant (τ), and the Charge Moment Change (CMC) of Six Coincident Pure Sprite Events
Event (Date in 2008)
Current Moment Amplitude, Iods (A-km)
Time Constant, τ (ms)
Charge Moment Change, CMC (C-km)
Sprite (22 Jul)
Sprite (27 Jul)
Sprite (12 Aug)
Sprite (16 Aug)
Sprite (21 Aug)
Sprite (24 Aug)
Table 2. The Current Moment Amplitude (Iods), the Time Constant (τ), and the Charge Moment Change (CMC) of the 12 Coincident Sprites With Halo and/or Elve
Event (Date in 2008)
Current Moment Amplitude, Iods (A-km)
Time Constant, τ (ms)
Charge Moment Change, CMC (C-km)
Sprite+halo+elve (17 Jun)
Sprite+elve (29 Jun)
Sprite+halo (1 Jul)
Sprite+halo (3 Jul)
Sprite+halo+elve (15 Jul)
Sprite+halo (22 Jul)
Sprite+elve (23 Jul)
Sprite+halo (24 Jul)
Sprite+halo+elve (9 Aug)
Sprite+elve (31 Aug)
Sprite+halo (8 Sep)
Sprite+halo+elve (16 Sep)
 To lend further support to the above assertion, quantitative comparisons of the model sferics computed using equations (2) and (3), the notch-filtered sferics, and the reconstructed sferics are performed. The inferred parameters, Tables 1 and 2, from the notch-filtered and the reconstructed sferics are used as the inputs to the equation (3). The derived current moment I(f)ds is then plugged into equation (2) to obtain the model sferics for the notch-filtered and the reconstructed sferics, respectively. Figure 4 shows the results for the associated ELF sferics of the sprite recorded on 12 August 2008. For this particular example, the correlation coefficients (R) between the models, the notch-filtered, and the reconstructed sferics improve from 0.78 to 0.94. For the twenty sferics analyzed in the work, the averaged R advances from 0.80 (notch-filtered sferics) to 0.85 (reconstructed sferics). The results indicate that, owing to the reconstructed sferics having a higher degree of similarity to the true sferics, the CMCs computed from the reconstructed sferics are closer to the true values comparing with those inferred from the notch-filtered sferics.
5. Time-Integrated Total Photons of the Coincident ISUAL Sprites
 From previous studies, the brightness of sprites is known to be correlated to the CMC of the causative lightning [Pasko et al., 1997; Huang et al., 1999; Takahashi et al., 2010]. As an additional illustration of the effects of sferics reconstruction, the N21P brightness (in units of mega- Rayleigh) of eight ISUAL sprites, that were not contaminated by lightning emissions, is computed using the method described by Kuo et al. [2005, 2008] and Lee et al. . Since the image data were recorded through a N21P –filter, the N21P brightness of sprites in units of photons therefore has to be corrected for the known transmission of the ISUAL imager N21P band (14% at 70 km altitude) [Mende et al., 2005; Kuo et al., 2008]. The sprite N21P emissions in units of photons, therefore, is the mega-Rayleigh brightness (106 × 106) multiplied the imager integrating time (29 ms) and the size of the sprite in cm2. For the eight analyzable ISUAL sprites, the brightness is found to be between 0.2 × 1024 and 8.9 × 1024 photons. The resulting sprite brightness is plotted against the CMC of the causative lightning in Figure 5. As shown in Figure 5, the data point scatter is substantially lower for the set of CMC inferred from the reconstructed sferics. The correlation coefficient (R) for the straight line fit of these two sets of data (CMC versus imager N21P intensity) is 0.76 for the notch-filtered sferics and is 0.92 for the reconstructed sferics. Arguably, the assumed linear sprite brightness versus CMC relationship is not well mirrored by the data points, since the linearity seems to depend heavily on the outlier point at 12,000 C-km. With the outlier point removed, the linear correlation would drop down to 0.4, which clearly cannot be used to support a linearity claim. However, a clear linear relation between the causative lightning CMC and the relative sprite brightness [Huang, 1998] has long been reported in the early years of the TLE research. More recently, Takahashi et al.  have also found the linear relationship between the absolute luminous intensity of fourteen ISUAL sprites and the charge moment of parent lightning. Therefore, even the data set reported in this work is relative small, but the result is consistent with previous works.
 In Figure 5, the interception of the straight line fit of the reconstructed CMC data to the x axis is ∼900 C-km, which is 25% lower than that inferred using the notch-filtered sferics. This threshold value seems to be significant higher than the ∼600 C-km reported by Takahashi et al.  and Hu et al. ; a value believed to be needed for a +CG discharge to induce sprites. However, due to the sferic sources studied in this works are all far away events (∼10000 km), the relative uncertainty of the inferred CMC may range from 10% to more than 100% [Williams et al., 2010]. Also the number of data points available to perform the straight line fitting is low. Hence the uncertainty in the zero-crossing point of the straight line is expected to be high, but the value reported here is still in a reasonable range.
 The linear correlation between the sprite brightness and the CMC of the causative lightning is consistent with the QE field model simulation reported by Pasko et al.  and the observations reported by Huang et al.  and Takahashi et al. . This linear relation can be very useful since the CMC of the sprite-producing lightning can be inferred from it and vice versa, provided the sprite brightness (N21P, and/or N22P), the optical energy, or the CMC has been precisely determined.
 For ELF receivers that employ hardware notch-filters to remove the 50/60 Hz power grid emissions, the content of the recorded sferics will be altered. In this article, a simple signal reconstruction method was used to perform the reconstruction. Using a known waveform from a function generator, this method has been proved to restore both the signal amplitude and the phase of the input signal below 100 Hz to a satisfactory level. Applying this reconstruction method to the Lulin notch-filtered ELF sferics, the event timing can be restored and some of the lost amplitude can be recovered. For the Lulin notch-filtered ELF station, the recorded event time is typically delayed by 9 ms in passing through the signal modulator. In general, the timing correction will make cross-platform event comparison more reliable. By using the reconstructed sferics in place of the notch-filtered sferics, the inferred current moment amplitude (Iods) increases by average of 69 ± 55%, while both the time constant (τ) and the charge moment change (CMC) decrease by averages of 52 ± 15% and 22 ± 21%, respectively. The threshold of CMC to produce sprites estimated from the reconstructed sferics is ∼900 C-km, which is also 25% lower than that infers using the notch-filtered sferics. With closer-to-true CMC values, a tighter linear correlation between the sprite N21P brightness and the CMC of the causative lightning was obtained. Hence notch-filtered ELF systems can still be very useful, if a proper signal reconstruction has been performed on the recorded sferics.
 We thank Mitsuteru Sato and Steve A. Cummer for the fruitful discussions regarding various aspects of ELF recording systems. We also would like to express our gratitude to the Lulin Observatory, National Central University, for hosting our ELF station and for the logistic supports. This work was supported in part by grants NSPO-S-100010, NSC 99-2112-M-006-006-MY3, NSC 97-2111-M-006-001-MY3, and NSC 97-2111-M-006-004-MY3.