1 Introduction

Lightning is the third leading source of storm deaths in the USA, killing an average of 51 people per year from 1984 to 2013 (NWS 2015). Lightning had been the second leading of storm deaths in the USA for many years, but tornadoes recently took second place (Roeder 2012). Lightning is also a significant source of storm deaths worldwide with an estimated average number of fatalities of up to 24,000 per year (Cardoso et al. 2011; Holle 2008; Holle and López 2003).

The geographical distribution of lightning fatalities in the USA is well known. The distribution by state has been extensively studied (Curran et al. 2000; Holle 2009, 2012; Roeder and Jensenius 2012) as well on a 60 × 60 km resolution map (Ashley and Gilson 2009). Such information is very useful in lightning safety education since tuning lightning safety to the local population is important (Roeder et al. 2011, 2012). Unfortunately, the geographical distribution of lightning fatalities in developing countries is not as well known, mostly due to fatalities in remote locations not being reported, identifying lightning as the cause after the fact, and lower-quality record keeping. Therefore, the purpose of this paper is the development of a new method to estimate the spatial lightning fatality distribution that may be useful in guiding lightning safety initiatives in developing countries. This method uses GIS software to multiply annual cloud-to-ground (CG) lightning flash density by population density and display the results on a map for easy visualization. This multiplication is assumed to be a first approximation for the risk of lightning fatalities. A lightning fatality risk map was created and verified for the contiguous United States (CONUS). This work was inspired by Geographical Information System (GIS) applications of lightning data by Gijben (2012). Other factors impacting the risk of lightning fatality are not included and are listed in Table 1. The availability of locations safe from lightning should also be considered, such as the quantification of urbanization by Gomes and Ab Kadir (2011). In addition, these results are annual averages, and the diurnal and seasonal distributions of lightning are not considered.

Table 1 Factors not considered in the lightning fatality risk map
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2 Lightning fatality risk map

The lightning fatality risk map for the CONUS was created by multiplying the annual CG lightning flash density (flashes/km2 year) by the population density (people/km2). A representative annual lightning flash density for the CONUS is shown in Fig. 1, and the population density is shown in Fig. 2. The CG lightning flashes are from the National Lightning Detection Network™ (NLDN) which reports more than 90 % of all CG lightning flashes with a median location accuracy of better than 500 m (Cummins et al. 2006; Cummins and Murphy 2009) for the time period employed here. In this study, the lightning flashes from January 2003 through December 2012 were initially accumulated into 0.1 by 0.1 degree regions normalized by the number of observation years. For some analyses, these data were accumulated into larger grids and spatially smoothed using a 3 × 3 Gaussian filter to filter out noise and more easily see overall patterns. The population density is from the 2000 National Historical Geographic Information System (http://www.nhgis.org), accumulated into the same 0.1 × 0.1 degree grids that were then employed for the flash density analysis. The resulting lightning fatality risk map is shown in Fig. 3, and some technical details are in Table 2. The results are normalized by grid area, which decreases with increasing latitude, to allow direct comparison of the gridded polygons. For comparison purposes, the observed lightning fatality locations (1959–2006) are included in Fig. 3 as dots.

Fig. 1
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CG lightning flash density (1997–2010) for the USA from the National Lightning Detection Network (Cummins et al. 2006; Cummins and Murphy 2009). The NLDN is owned and operated by Vaisala, Inc

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Fig. 2
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2000 Population density for the continental USA from the 2013 US Census (www.census.gov)

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Fig. 3
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Lightning fatality risk for the CONUS. Lightning fatality risk is the product of CG mean lightning flash density (2003–2012) and population density (2000). Details of the map are in Table 2. The black dots are the individual lightning fatalities (1959–2006), which are included for visualizing spatial correlation

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Table 2 Technical details of the new CONUS calculated lightning fatality risk map (Fig. 3)
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The concept of the lightning fatality risk map seems reasonable: Lightning fatalities should be directly related to CG lightning flash density and population density. Others have made the same assumption but did so only for an entire country or state (Gomes and Ab Kadir 2011), not on a high-resolution grid. Another study identified the factors important for lightning fatalities in a region: number of flashes, number of people, and area (Gao et al. 2014). These factors can be combined into lightning flash density and population density, equivalent to the assumption in this study.

3 Verification

Although reasonable, the concept of the lightning fatality risk map is new and verification is required. Fortunately, a dataset of observed lightning fatalities is available (Ashley and Gilson 2009). Some details of this observed lightning fatality map are in Table 3 and shown in Fig. 4. The period of the lightning fatality data is older and longer than the lightning flash density. This was done to provide a sufficient sample size of lightning fatalities, and the gridded lightning fatality database was the only one available. However, while differences would result, especially from population shifts during this period, the general pattern should be representative. It should be noted that 6.6 % of the lightning fatalities had no location and were not plotted. In addition, 15.5 % had uncertain locations, e.g., being recorded at the county seat even though the fatality may have occurred anywhere in the county, and introduce a small amount of variability into the map that affects verification of the lightning risk map.

Table 3 Technical details of the previous CONUS observed lightning fatality map (Fig. 4; Ashley and Gilson 2009) used as part of the verification of the calculated lightning fatality risk map
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Fig. 4
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Number of observed lightning fatalities in the USA (1959–2006) smoothed on a 60 × 60 km grid (Ashley and Gilson 2009). This is the ground truth for verification of the new lightning fatality risk maps presented here. Details of map in Table 4

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The verification included both subjective and objective components. The subjective verification was a visual comparison of the lightning fatality risk maps with the known lightning flash density and population density across the USA. The subjective verification showed the lightning fatality risk maps behaved as expected, including discussion of six state and regional maps for parts of the USA (see Roeder et al. 2014 for details).

Subjective analysis of the Florida map shown here (see Fig. 5) demonstrates how one might use lightning fatality risk maps and how small-scale features can be revealed using a map with fine grid resolution. Florida has the highest lightning fatality rate in the USA, the highest lightning flash density, and some of the sharpest gradients of population density. The highest flash rates in the USA are across central Florida (Fig. 1). Population centers in the Florida lightning risk map are Tampa/St. Petersburg and Orlando, both with more than 10 M people flashes/year km4. Similarly the high population density of the Miami area is evident, even though that area has less lightning activity than central Florida. The city of Jacksonville is also apparent, although it lies in an area of relatively lower lightning activity. Port Charlotte in southwest Florida can be seen as a region with more than 400 K annual people flashes/year km4. All of these areas show one or more spatially proximate fatalities. The areas of low lightning fatality risk in Florida are also apparent. There is a rapid decrease in population density southeast of Orlando due to rural areas and swamps. Although the lightning flash rate remains high, the drop in lightning fatality risk due to the much lower population density is shown in Fig. 5. In addition, the extremely rapid decrease in population west of Miami/Ft. Lauderdale is also indicated.

Fig. 5
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Lightning fatality risk map for Florida. The black dots are the observed lightning fatalities

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Fig. 6
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Lightning fatality density versus lightning fatality risk. The linear regression is shown for the best linear regression (1.0° grid resolution, 1.5 grid smoothing)

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The objective verification is a grid-by-grid comparison of the calculated lightning fatality risks with the observed annual lightning fatality density across the CONUS. As discussed previously, fatalities are from Ashley and Gilson (2009). Although the periods of record for the datasets differ, we assume the general conclusions are still valid (2003–2012 for the lightning fatality risk, 1959–2006 for the lightning fatalities, 2010 for the population). Since the map of lightning fatalities in Fig. 4 had extensive smoothing, this objective verification was done using Ashley and Gilson’s original data set of observed lightning fatalities. The lightning fatality risk (population density × CG annual lightning flash density) and annual lightning fatality were placed on the same grid spacing and the same smoothing applied. A Gaussian smoothing function was used with the scale factor based on various numbers of grid spaces. Eight combinations were analyzed: two grid spacings with four smoothing functions each (Table 4). Various regression analyses were done to determine the behavior between the calculated lightning fatality risk and the observed lightning fatalities. The correlation coefficients for the best-fit linear regression and best-fit quadratic regression are shown for each of the eight cases. A linear regression of observed annual lightning fatality on calculated lightning fatality risk was then performed on each of the eight grids (Table 4). The best linear regression was the 1.0° latitude/longitude grid with the Gaussian smoothing of 1.5 grid spaces. The equation for this regression is at (1) and is shown graphically in Fig. 6.

$$ \begin{aligned} y & = 3.21 \times 10^{ - 6} x - 8.95 \times 10^{ - 5} \\ r^{2} & = 0.820 \\ \end{aligned} $$
(1)

where x = lightning fatality risk (people flashes/year km4), y = annual lightning fatality density (fatalities/km2).

Table 4 Correlation coefficients (r 2) from the linear regressions and quadratic regressions of observed lightning fatality versus calculated lightning fatality risk for eight combinations of grid spacing and smoothing functions
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Linear regression through the origin was considered but not used. While one might assume that zero lightning flash density or zero population density would lead to zero lightning fatalities, that assumption does not consider people traveling to outdoor areas for recreation, which would not be assessed in the population density since people are counted where they live. Since the assumption of intersection at the origin cannot be made a priori, regression through the origin is not justified.

The linear regression with the best correlation coefficient (grid spacing of 0.5°, 1.5 grid Gaussian smoothing; Fig. 6) appears to have a systematic bias. At lower lightning fatality risk, the fatalities appear to trend toward being above the linear regression. At higher risk, the fatalities appear to occur more frequently below the regression and with larger differences from the regression than the points above the regression. This suggests that a nonlinear regression may give better results, perhaps a best-fit log-linear or quadratic polynomial.

A log-linear regression was tried since it would have the desired properties to match the plotted data: approaches zero fatalities toward zero risk, monotonically increasing fatalities and higher risk, asymptotically approaches an upper limit. Unfortunately, the log-linear regression was dominated by the large number of lower risk values and so yielded a r 2 of only 0.650, significantly worse than the linear regression (r 2 = 0.820). Of course, risk values of zero had to be excluded to allow the log-linear regression, reducing the number of data pairs to 1204.

A quadratic regression yielded an r 2 of 0.864, slightly better than the linear regression (r 2 of 0.820) for this grid spacing and smoothing. More importantly, the quadratic regression did not have the systematic bias of the linear regression. This quadratic regression equation is at (2) and shown graphically at Fig. 7.

$$ \begin{aligned} y & = 5 \times 10^{ - 9} x^{2} + 5 \times 10^{ - 6} x + 6 \times 10^{ - 5} \\ r^{2} & = 0.864 \\ \end{aligned} $$
(2)

where x and y are as in (1).

Fig. 7
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Lightning fatality density versus lightning fatality risk. The quadratic regression is for the 1.0° grid resolution, 1.5 grid smoothing

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However, care must be taken in using the quadratic regression to higher values of lightning fatality risk. At risk values higher than about 475 people flashes/year km4, the predicted lightning fatalities will decrease at higher risk, which is contrary to expectation. It should be noted that care must also be taken with the linear regression since it tends to underestimate the lightning fatalities at lower risk and significantly overestimate the fatalities at higher risks.

Fig. 8
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Lightning fatalities versus lightning fatality risk. The hybrid quadratic/log-linear regression is shown (1.0° grid resolution, 1.5 grid smoothing). The correlation coefficient, weighted by number of data point, is 0.827

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A hybrid quadratic/log-linear regression was carried out in order to overcome the shortfall of the quadratic regression noted above. This hybrid uses a quadratic regression at lower lightning fatality risk and a log-linear regression at higher risk values. The risk threshold for changing the regressions was manually selected to avoid discontinuities in value and slope between the two regressions. A lightning fatality risk threshold of 300 people flashes/year km4 was selected based on visual examination of the graphs. The log-linear regression was adjusted slightly to completely avoid discontinuities. The combined r 2 for both regressions, weighted by the number of data pairs in each regression, was 0.827. This weighted combined correlation coefficient was done to allow traditional interpretation, i.e., 1.0 is perfect correlation. The equation of the hybrid regression is provided below and shown graphically in Fig. 8.

$$ \begin{aligned} I{\text{f}}\;x & \le 300\;{\text{people}}\;{\text{flashes/year}}\;{\text{km}}^{4} : \\ y & = - 5 \times 10^{ - 9} x^{2} + 5 \times 10^{ - 6} x + 6 \times 10^{ - 5} \\ \end{aligned} $$
(3)
$$ \begin{aligned} {\text{If}}\;x & > 300\;{\text{people}}\,{\text{flashes/year}}\,{\text{km}}^{4} : \\ y & = 6.16 \times 10^{ - 4} \ln \left( x \right){-}2.45 \times 10^{ - 3} \\ & \quad {\text{combined }}r^{2} = 0.827 \\ \end{aligned} $$

where x and y are as in (1).

The hybrid quadratic/log-linear regression has a slightly higher correlation coefficient than the best linear regression, but a slightly lower correlation coefficient than the quadratic regression. It overcomes the systematic bias of the linear regression and avoids the undesired behavior of the quadratic regression of decreasing fatalities at higher risk. Finally, it asymptotically approaches an upper limit as is consistent with visual inspection of the data. Therefore, this hybrid quadratic/log-linear regression is recommended over the other regressions, even though it does not have the highest correlation coefficient.

4 Future work

The lightning fatality risk map presented here is a preliminary attempt to establish and verify the new method. Since it verifies well in the USA, the most important future work is to extend the method to other countries. This would first be done preferably where the pattern of lightning fatalities is already well known. It would also be useful if those countries had a wide range of CG flash density and a wide range of population density. These additional verifications would be beneficial in justifying the application of this method in developing countries where lightning fatalities may not be reported well. This is especially important since the main motivation for this work was to help guide lightning safety efforts in developing countries. Once some further verification of the method is made, it will be straightforward to create a lightning fatality risk map for developing countries. A global lightning fatality risk map could also be developed that can be used to inform lightning safety efforts anywhere in the world.

The lightning fatality risk map is for annual lightning risk. It may be useful to apply the same method to monthly or seasonal maps. Likewise, studying diurnal patterns of lightning fatality risk may be useful. The hybrid quadratic/log-linear regression had some subjectivity in its development. A more objective method might be pursued that could be more easily applied in contexts where verification data are not readily available.

The CG lightning flash density was used in constructing the lightning fatality risk map; however, the density of ground contact points would be more appropriate. This is not the same as the stroke density, since in flashes with multiple strokes, the subsequent strokes often strike the same point and represent little additional risk. However, the subsequent strokes also often contact the ground elsewhere (Valine and Krider 2002), often a few km away, and so represent substantial additional risk. Unfortunately, the number of ground strike points is not reported by most lightning detection systems. However, it can be inferred from stroke detection systems, as demonstrated by Cummins (2012). This was not feasible for a study on this spatial geographical scale, but may be important for a more geographically focused study.

Another important factor in lightning fatalities is the behavior of the local population. People that spend more time outside, especially during lightning activity, or cannot or will not seek safety when lightning threatens have a greater likelihood of becoming a lightning fatality. If the data were available or inferable, variations in behavior could be included as another multiplicative factor in the construction of lightning fatality maps, perhaps as a percent of time spent at risk. However, in areas without lightning-safe locations such as some parts of the developing world, the variations in behavior would not be important.

Finally, the lightning fatality risk map presented here assumes that the population density is constantly at the reported grid point. However, there are areas with significant population change throughout the year, such as those related to tourism. In addition, local populations may move out of the immediate area during lightning season. In some developing countries, migration may also be an important factor. Future research should engage social scientists to develop sources and databases that may assist with such research.

5 Summary

A new method to estimate the risk of lightning fatality was developed. This method uses a GIS to multiply annual CG lightning flash density and population density to estimate the risk of lightning fatality and visualize the results on a map. This method was applied to the contiguous US and verified against the observed lightning fatalities. The method verifies well with the preferred hybrid quadratic/log-linear regression model having a correlation coefficient as high as r 2 = 0.827 for the 1.0° lat/lon grid with 1.5 grid point Gaussian smoothing. The hybrid quadratic/log-linear regression is preferred since it overcomes many of the problems of the other regressions even though one of those other regressions had a slightly higher correlation coefficient. Further refinements to the lightning fatality risk maps are possible and discussed.

The main motivation for developing the lightning fatality risk method is to assist lightning safety efforts in developing countries. Since the method works well for the USA, it may be useful in some developing countries where the geographical distribution of actual lightning fatalities may not be well documented. Given that the distribution of CG lightning can be reasonably well determined from the various global lightning detection networks, or other sources if available, and if the distribution of population density is also known, then a GIS can be used to create lightning fatality risk maps for those countries. These maps could then be used to guide lightning safety efforts in those countries to be more cost-efficient and more effective by spending funds on areas where it is most needed and by tailoring the efforts to the people living in that area.