This paper analyzes the possibility of using single station measurements of low-frequency electromagnetic waves to locate lightning occurring around much, and possibly all, of the world. The electromagnetic waves generated by lightning, which propagate through the Earth-ionosphere waveguide, are known as sferics. We combine a direction finding algorithm to compute the arrival azimuth of the recorded sferic along with a distance of propagation estimate to deduce the location of the lightning. While the methodology used for estimating the distance of propagation is described by Mackay and Fraser-Smith (2010), this paper analyzes the effectiveness of the method. We examine the limitations including those independent of station location and performance, and those which are station dependent. We analyze the geographic coverage attained from a single station along with the percentage of lightning emitting sferics that are recorded at the station and available for computing the lightning locations.
 The purpose of this paper is to analyze the global coverage and effectiveness of a single station lightning location method. The single station method analyzed is described in detail by Mackay and Fraser-Smith . For this method, magnetic fields are recorded on the Earth's surface at frequencies in the extremely low frequencies (ELF; 30–3000 Hz) and very low frequencies (VLF; 3–30 kHz). Two crossed loop antennas are employed to measure the magnetic field in the north-south and east-west directions. Lightning can be modeled as a vertical dipole with the current passing through it resembling an impulse waveform. A broad range of frequencies of electromagnetic waves are excited by this impulse. The higher frequencies dissipate into the ionosphere, while the lower frequencies, such as the VLF and ELF waves, are trapped between the Earth's surface and the lower ionosphere and propagate along through this waveguide. Remote stations record the electromagnetic waves emitted by the lightning which forms a signal known as a sferic. In the work of Mackay and Fraser-Smith  the ELF portion of the sferic, or the slow tail, is used to compute the distance between the causal lightning and the recording station using the propagation model described by Wait [1960a]. With a direction finding algorithm and this computed distance, the lightning location can be deduced from a single station. With recorded data from a single site, we hope to cover a large portion of the world's lightning. We discuss the limitations of this method along with the lightning location results attained from two different sites: Søndrestrøm, Greenland (66.99°N, 50.95°W) and Palmer, Antarctica (64.77°S, 64.05°W).
 To discuss lightning location algorithms, it is helpful to know where lightning is occurring around the globe. In Figure 1 [Christian, 1999], we see the average number of lightning per hour per km2 occurring around the world. Lightning occurs much more frequently over land than water, with central Africa having the highest frequency of lightning year-round. The global distribution of lightning changes with the seasons. Lightning occurs more in the northern hemisphere during the northern summer months and more in the southern hemisphere during the northern winter. There is also a diurnal variation. Figure 2 from Mackerras et al.  shows that over land, where most of the lightning falls, there is an increase during the local afternoon. With an abundance of signals from lightning to record, we hope to observe the distinct geographic and diurnal variation by analyzing data recorded from a single station.
2. Existing Lightning Detection Methods
 There exist a variety of methods to locate lightning remotely. Most involve measuring either the optical emissions from lightning, or the electromagnetic waves propagating along the Earth-ionosphere waveguide. The data from Figure 1 were acquired by two satellite instruments. The satellite data are from the Optical Transient Detector, OTD, and the Lightning Imaging Sensor, LIS. OTD, being the predecessor to LIS, is no longer recording data. Both systems detect lightning occurring below the orbital path by measuring the optical emissions from the lightning. This results in very accurate and complete lightning detection in the limited field of view of the satellite. With many years in orbit, satellite lightning detection is very well suited for calculating the general distribution of lightning and lightning storm flash rate rather than real time global lightning detection for individual lightning. The optical sensor also has the advantage of sensing cloud-to-ground along with cloud-to-cloud lightning. Thomas et al.  showed the cloud-to-ground discharges that were confined to mid and lower altitudes were less well detected by the LIS system than intracloud discharges.
 There also exist different ground based methods for lightning detection. The ground based methods limit the sensitivity to cloud-to-ground, or vertical, lightning. Some methods are based on electromagnetic data recorded at multiple stations. Many previous methods require VLF measurements from multiple stations [e.g., Lee, 1986; Cummins et al., 1998; Dowden et al., 2002; Said et al., 2010]. Multiple station measurements at ELF have also been used, as described by Füllekrug and Constable . With data from multiple stations, the methods for locating lightning can vary. Some of these methods include using the difference in arrival times of sferics at different sites, or triangulation involving direction measurements of the incoming signal. Using multiple stations is an effective way to locate lightning but applying this method requires access and maintenance to simultaneous data from multiple station, with these stations having precise timing and calibration.
 Previous work has also tackled lightning location using electromagnetic measurements from a single station. J.R. Wait developed a theory to approximate the propagation distance of the sferic, ρ, using a single station measurement by modeling the slowtail [Wait, 1960a]. The separation time between the start of the sferic and the first quarter cycle of the slow tail is known as the slow tail separation, ts. Wait showed theoretically that is linearly dependent on ρ by the following equation:
where σi is the ionosphere conductivity, h is the height of the ionosphere, and δ is a factor describing the pulse width (in seconds) of the lightning's current moment. When measurements of the electric field are available, distance approximation from a single station is possible by measuring the wave impedance, or the ratio of the electric and magnetic field [Burke and Jones, 1995]. A different method to locate lightning with a single station is known as the Kharkov method [Brundell et al., 2002; Rafalsky et al., 1995a, 1995b]. This method examines the propagation in the TE1 mode, from around 1850 Hz to 3400 Hz. This requires a strong signal in the specified frequency range, limiting the distance of propagation of the sferic when applying this method. In the work of Mackay and Fraser-Smith , we deduced the distance of propagation by analyzing the slow tail of the sferic. This method can be used for sferics propagating over a wide variety of distances.
3. Distance Estimation
 In this section we will review the methodology for estimating the distance of propagation between lightning and recording station. A more thorough demonstration of the method is available in the work of Mackay and Fraser-Smith . We begin by applying the modal equations described by Wait [1960a] which represent the propagation of ELF and VLF electromagnetic waves through a model of the Earth-ionosphere waveguide. In this model, the Earth is a perfect conductor and the ionosphereis a uniform isotropic layer with a sharp lower boundary. The method analyzes the propagation of the slow tail or ELF portion of lightning induced sferic which lies in the lowest-order mode. The equations representing this mode are below:
where ps is the current moment at the lightning source, ρ is the distance of propagation, h is the height of the ionosphere, σi is the conductivity of the ionosphere, and hr,0 is the magnetic field in the lowest waveguide mode at the receiver. In the work of Mackay and Fraser-Smith  we used the inverse of these equations to compute an average current moment ps calculated from several sferics corresponding to lightning with known locations. To deduce the distance of propagation for a sferic originating from a lightning with an unknown location, we use the recorded slow tail. We then compute different possible current moments, ps, by iterating over a full range of possible distances of propagation. ρ is estimated to be the distance corresponding to the computed source with the highest correlation to the average source.
 The modal equations describe a uniform waveguide, where h and σi are assigned different values during daytime or nighttime. For sferics propagating across the day-night terminator, Mackay and Fraser-Smith  applies adjusted values for h and σi to the uniform waveguide model. In the work of Mackay and Fraser-Smith , we analyzed the accuracy of this method and found the errors in the estimation of ρ to have a standard deviation of approximately 9% of the actual distance for sferics propagating in day, night, and mixed paths.
4. Direction Estimation
 For this paper, we find the azimuth of the sferic by analyzing the VLF portion of the sferic as in the work of Said et al. . To do so, we begin by taking a band-pass filter around the VLF frequencies for the north-south and east-west recordings of a sferic. We then isolate the data containing the peak of the VLF and plot the north-south data against the east-west data as shown in Figure 3. We find a best fit line to fit the north-south and east-west points and compute its slope. From the slope we can deduce the arrival azimuth of the signal.
5. Station Independent Limitations
 In sections 3 and 4, we discuss locating lightning in two sections: distance estimation and direction estimation. Each of these impose limitations on the technique. First, both sections of the methodology are limited in accuracy. Second, for the distance estimation we require a clean slow tail recorded at the station. Last, for the direction estimation, there remains a 180° ambiguity in the direction estimate. These limitations exist regardless of the station used to collect the data.
5.1. Limitations in Accuracy
 The error analysis from Mackay and Fraser-Smith  compared lightning located from sferics recorded at Søndrestrøm station to lightning locations from NLDN. Figure 4 shows the lightning location attained using the two different methods. The NLDN median location error is approximately 1 km for lightning occurring in the United States [Rakov and Uman, 2003]. World lightning location networks using multiple stations achieve lightning location with an uncertainty of around 5–10 km [Dowden et al., 2002; Said et al., 2010]. Though the method described in this paper improves on range and accuracy of other existing single station lightning detection methods, it cannot achieve the same accuracy as a large multiple station network. For the lightning shown in Figure 4, the median error for direction estimation is 2°, and the median distance estimation error is 270 km. The distance estimation accuracy is limited by the assumption that the source for the slowtail, or the current moment, has the same form for all lightning. As demonstrated in Figure 4, a single station is capable of accurately locating lightning storm regions by combining the measurements of each individual lightning within the storm.
5.2. Percentage of Sferics With Clean Slow Tails
 In the work of Mackay and Fraser-Smith  we mention using only “large and clean” sferics. The reason for specifying “large” sferics is to ensure that the sferic is strong enough to be recorded at the station. The “clean” sferics are specified to exclude sferics where there is interference with other recorded sferics during the slow tail signal used to compute the distance estimate. In this section we examine the percentage of recorded sferics that are deemed useable for our distance estimation method.
 We will begin by calculating the percentage of useable sferics out of all the recorded sferics from a section of recorded data. Then we will look at multiple segments of data throughout a whole day to arrive at an average percent of useable sferics. For each length of recorded data, we start by picking out all the recorded sferics. To do this, we pass the data through a band-pass filter to select the VLF frequency range. Then, we take the envelope of the VLF data. Each peak of the envelope over the noise level is considered it's own sferic. Now that we have picked out all the sferics, we can measure the difference in recorded time between adjacent sferics. Sferics arriving within 5 ms will have overlapping slow tails. The interference caused by the overlap causes unreliable results when applying our distance estimation. This calculation was done for 48 time series of recorded data from Søndrestrøm station. Each sample was taken at a half an hour interval with a duration of 14 s. In Figure 5, we see adistinct daily variation in the number of sferics recorded at Søndrestrøm, however the percent of useable sferics does not follow the same variation and averages 70% rather consistently throughout the day.
5.3. Arrival Azimuth Ambiguity
 In section 4, we explained the method of estimating the direction of propagation of a recorded sferic. However, this method leaves us with a 180° ambiguity. There are different options to either resolve or accommodate for the ambiguity.
 The first option to resolve the ambiguity is to use data from a second station. With only the azimuth estimate from two different stations, we can draw the two great circles which cross each of the stations. These two tracks intersect at two points. With distance estimates calculated from each station we can resolve which of two intersecting points is the lightning location. As this paper describes single station lightning location, we will not use a second station to resolve the azimuth ambiguity. However if a second station is available, the additional data will resolve the problem.
 Previous papers have resolved the ambiguity when an electric dipole antenna is available to record the vertical electric field [Brundell et al., 2002]. When the electric field is recorded the azimuth calculation is done differently from section 4. Instead of calculating the slope of the two magnetic field measurements, you calculate the slope of the computed Poynting's vector. By calculating Poynting's vector, P = E × H, the electric field will be found to be either in or 180° out of phase with the magnetic field, moving all the Poynting's vector points into a single quadrant and resolving the 180° ambiguity. The data we are using for this paper does not include electric field measurements and so we chose to deal with the azimuth ambiguity in a different manner.
 The simplest solution when no additional data are available is to make an educated guess. If the two possibilities has the first location fall in a higher lightning frequency occurrence area than the second location, the first location is more likely. From Figure 1, we can calculate the probability distribution of lightning occurring over the world and calculate the probability that the lightning occurred at each location.
 When we estimate the distance of propagation, we require a single recording of the slow tail. To obtain a strong slow tail, we multiply the north-south recording by the cosine of the estimated azimuth, and the east-west recording by the sine of the azimuth and combine the two. For this process, we assume the azimuth is between −90° to 90°. Negative cloud-to-ground lightning belonging to that hemisphere will give computed current moments with positive correlations to our average current moment (which was calculated using lightning from the same hemisphere), while positive cloud-to-ground lightning from the same hemisphere will produce negative correlations. If the lightning is from the opposite hemisphere, we will have a negative correlation for a negative flash, and a positive correlation for a positive flash. Therefore if we know a sferic propagated from a negative or positive lightning, we can resolve the 180° ambiguity by examining whether the current moment correlation is positive or negative. It is thought that less than 10% of cloud-to-ground lightning are positive cloud-to-ground lightning [Rakov and Uman, 2003]. The slow tails produced by both types of lightning undergo the same dispersion in the Earth-ionosphere waveguide, and therefore the distance estimation technique described in section 3 can be used for both types. We can include the 90% chance that a sferic is from a negative cloud-to-ground lightning when calculating the probability the sferic originated from both possible locations to improve our 180° ambiguity resolution.
6. Station Dependent Limitations
 With the model described in section 3, a lightning location can be pinpointed only if the corresponding sferic was recorded by a station. Depending on a station location and its receiver's signal-to-noise ratio, different stations will be able to record different sferics. A quiet station with high signal-to-noise ratio will be able to record sferics propagating a longer distance and originating from weaker signals than a noisier receiver at the same location. The location of the station also affects which sferics are recorded. Two stations with the same signal-to-noise ratio may not cover the same area if sferics must propagate over higher attenuating surfaces to reach one of the stations.
6.1. Attenuation to Location
 The signals propagating within a waveguide attenuate at a rate given by exp(−αρ), where α is the attenuation factor. Equation (3) shows that the attenuation rate for the model waveguide we are using is α = ℜ (iωS0/c). In section 3, we assume the ground is a perfect conductor. This assumption is valid when the ionospheric conductivity is much lower than the ground conductivity. However, propagation over Antarctic ice has been shown to greatly attenuate VLF signals [Chevalier et al., 2006; Rogers and Peden, 1975]. If we remove the assumption that the ground conductivity is significantly higher than the ionosphere conductivity, S0 from equation (4) can be described as in the work of Wait [1960b] by:
 To see the effects of different ground surfaces on our sferics, we create a very general model of the Earth. We break up the Earth's surface into four classes of materials: seawater, Earth's crust, glacial ice, and sea ice. The four materials have the approximate conductivities shown in the Table 1.
Table 1. Conductivities, σg, and Attenuation Values, αn, for Earth Surface
3.3 × 10−4
3.4 × 10−4
3.63 × 10−4
1.37 × 10−3
 In Figure 6, we show a world map with the different ground conductivities. A signal of a certain frequency propagating for ρn kilometers over a surface of type n, will be attenuated by exp(−αnρn). In calculating αn we approximate h and σi as averages of the day and night values. Modeling a signal propagating over multiple surfaces, or a nonuniform waveguide, is complicated. To simplify, we will approximate the total attenuation to be exp(−αnρn). We can now calculate an approximate, representative attenuation from any point in the world to the station for a single frequency. Figure 7 shows the attenuation of a 1 kHz signal to two different stations. Note how the Antarctic ice greatly reduces the sensitivity of the Palmer station to lightning over Asia and Australasia.
6.2. Noise Threshold Limitation
 The receiver at any particular station will have a threshold noise level, and the sferic power must be higher than the noise level for that sferic to be detectable. In general, each receiver will have a different threshold. We set this noise threshold of a station to be the measured VLF power from recorded data containing no sferics. By using a data set consisting of lightning with known locations, time of occurrences and peak currents, we can compare the VLF power of the corresponding recorded sferics to the computed noise threshold level. The comparison allows us to measure the sensitivity of the station. For our purposes, we will use the National Lightning Detection Network, or NLDN, data set [Cummins et al., 1998]. We then measure the VLF power from the segment of data corresponding to when the sferic would have, or has arrived, at the station. The lightning corresponding with VLF power around or below the noise threshold are considered undetectable. In Figure 8, we plot the VLF power recorded at the Palmer and Søndrestrøm sites for every NLDN lightning and compare it to the peak current and the attenuation between the lightning's location and the station. The attenuation is computed as in section 6.1.
 The VLF noise power for the Palmer and Søndrestrøm stations are very different. Many of the sferics at Søndrestrøm fall below the noise cutoff, while most of the sferics at Palmer are above the cutoff. From equation (3), we expect the magnitude of the magnetic field to be proportional to Ipe−αρ, where Ip is the peak current recorded by NLDN. The magnitude of the electric field follows the same form. Therefore, we expect: Power = E × H ∝ I2e(−2αρ). In Figure 8, we draw a line in each plot to approximate the noise power cutoff in terms of peak current and attenuation. With the attenuation maps computed in Figure 7, we can compute the peak current required for lightning from different locations to emit sferics strong enough to be recorded at that station. Using the NLDN data as an average distribution of peak current for lightning occurring around the world, we can take any location on the world map and convert the cutoff peak current into the percent of lightning at or above the cutoff. In Figure 9 we show the coverage maps for the two stations. The lower noise threshold for Palmer gives it much better coverage than Søndrestrøm despite Figure 7 showing that sferics undergo higher attenuation from more locations before being recorded at Palmer.
 To analyze the limitations discussed in sections 6.1 and 6.2, we will compare the computed location of lightning from a single station, using the methodology from sections 3 and 4, with a map of the expected lightning distribution visible from that station. The expected distribution is computed by modifying the worldwide lightning distribution from Figure 1 to account for each of the limitations. We examine lightning data recorded from two stations, Søndrestrøm and Palmer.
 Søndrestrøm data were recorded in 14 s increments every half hour. We combine the sferics recorded in 48 consecutive data files which span the whole day of 29 July 2004 and plot the computed location of the lightning in Figure 10. Figure 11 shows the distribution of lightning we expect to be “visible” from Søndrestrøm station over the same time period. Figure 11 is obtained by combining the coverage in Figure 9 with the satellite data from Christian  showing the world distribution of lightning in the month of July. We then reduce the expected overall lightning by 30% to take into account the unusable sferics described in section 5.2. Rakov and Uman  estimates that approximately three-quarters of lightning discharges do not contact the ground, therefore we also reduce the total number of expected detected lightning by 75% to eliminate the cloud-to-cloud lightning visible with the satellite and not with our method. We can now compare the computed geographical distribution in Figure 10 to the expected distribution shown in Figure 11. We find the computed locations of the lightning along with the total number of detected lightning match quite closely with the expected distribution.
 For Palmer, we analyze 1 min segments of data recorded throughout the same day, 29 July 2004. This allows us to observe the diurnal variation described in section 1. In Figure 12 we show the location of lightning computed from the data recorded at Palmer, along with the expected distribution of lightning for a July day at the corresponding time. The expected distribution for Palmer data was computed in the same way as for Søndrestrøm, except that the average July lightning distribution is modified using the curve from Figure 2 to include the expected diurnal variation. We find the variance in the location and the quantity of lightning detected by the Palmer station at different times in the day follows our expected distribution. The total number of lightning detected is higher than expected but with such short increments of data the difference is not significant.
 Using slow tail measurements, a single ELF/VLF recording station can locate lightning over a large portion of the globe, as seen with data from Palmer station in Antarctica. Even Søndrestrøm station, which has a more limited range, covers approximately 15% of the globe. The signal-to-noise ratio along with the location of the station is critical in determining the global coverage of that station. Irrespective of the station used, we can locate lightning from roughly 70% of the sferics recorded. Some recorded sferics cannot be used due to interference from other sferics, with overlapping slow tails. As might be expected, these interfering sferics are especially common during active storms which have a large number of lightning occurring in close proximity with one another. However, these storms can still be tracked using the single station method with the remaining clean sferics. The single station method can be applied to sferics originating from positive or negative lightning, and propagating day or night, with distances varying from 2000 km up to 20000 km. Approximately half the world can be covered by a single station and a very small number of stations strategically placed can provide worldwide coverage.
 This work was supported by the Office of Naval Research through grant N00014-04-1-0748 and grant N00014-10-1-0378 and by the National Science Foundation through grant ANT-0637005. Logistics support for the Søndrestrømfjord measurements was provided by the National Science Foundation through grant ATM-0334122. The authors thank Vaisala, Inc. for providing the NLDN data used in this study. We would also like to thank Ryan Said for providing the ELF/VLF data from Palmer Station.