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Spatially adaptive probabilistic computation of a sub-kilometre resolution lightning climatology for New Zealand
Author’s Accepted Manuscript Spatially adaptive probabilistic computation of a sub-kilometre resolution lightning climatology for New Zealand Thomas R. Etherington, George L.W. Perry www.elsevier.com/locate/cageo
PII: DOI: Reference:
S0098-3004(16)30444-7 http://dx.doi.org/10.1016/j.cageo.2016.09.010 CAGEO3841
To appear in: Computers and Geosciences Received date: 15 April 2016 Revised date: 9 August 2016 Accepted date: 25 September 2016 Cite this article as: Thomas R. Etherington and George L.W. Perry, Spatially adaptive probabilistic computation of a sub-kilometre resolution lightning climatology for New Zealand, Computers and Geosciences, http://dx.doi.org/10.1016/j.cageo.2016.09.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Spatially adaptive probabilistic computation of a sub-kilometre resolution lightning climatology for New Zealand Thomas R. Etheringtona,b,c*, George L.W. Perryc a
Royal Botanic Gardens Kew, Wakehurst Place, West Sussex, UK
b c
Department of Zoology, University of Oxford, Oxford, UK
School of Environment, The University of Auckland, Auckland, New Zealand
*
Corresponding author. [email protected]
Abstract Lightning is a key component of the Earth’s atmosphere and climate systems, and there is a potential positive feedback between a warming climate and increased lightning activity. In the biosphere, lightning is important as the main natural ignition source for wildfires and because of its contribution to the nitrogen cycle. Therefore, it is important to develop lightning climatologies to characterise and monitor lightning activity. While traditional methods for constructing lightning climatologies are suitable for examining lightning’s influence on atmospheric processes, they are less well suited for examining questions about biosphere-lightning interactions. For example, examining the interaction between lightning and wildfires requires linking atmospheric processes to finer scale terrestrial processes and patterns. Most wildfires ignited by lightning are less than one hectare in size, and so require lightning climatologies at a comparable spatial resolution. However, such high resolution lightning climatologies cannot be derived using the traditional cell-count methodology. Here we present a novel geocomputational approach for analysing lightning data at high spatial resolutions. Our approach is based on probabilistic computational methods and is capable of producing a sub-kilometre lightning climatology that honours the spatial accuracy of the strike locations and is adaptive to underlying spatial patterns. We demonstrate our methods by applying them to the mid-latitude oceanic landmass of New Zealand, an area with geographic conditions that are under-represented in existing lightning climatologies. Our resulting lightning climatology has unparalleled spatial resolution, and the spatial and temporal patterns we observe in it are consistent with other continental and tropical lightning climatologies. To encourage further use and development of our probabilistic approach, we provide Python scripts that demonstrate the method alongside our resulting New Zealand lightning climatology. Keywords: Getis-Ord; GIS; Python; spatial autocorrelation; smoothing; wildfire
1. Introduction Lightning is intrinsically linked with Earth’s climate system (Williams, 2005), and global warming and drying are predicted to result in an increase in lightning activity (Price, 2009; Romps et al., 2014). Increased lightning activity may further warm the climate as lightning increases atmospheric nitrogen oxide that affects production of ozone and hence the Earth’s radiation balance (Schumann and Huntrieser, 2007). In the biosphere increased lightning activity may also affect climate indirectly by increasing levels of atmospheric carbon dioxide and aerosols by igniting wildfires (Bowman et al., 2009), which may also become more common as the warmer and drier global climate increase fuel-bed flammability (Macias Fauria et al., 2011). Given the potential for positive feedbacks between a changing climate and increased lightning activity in both the atmosphere and biosphere, there is a pressing need for lightning climatologies that quantify and monitor variations in lightning density across space and frequency across time.
Lightning climatologies are developed from data collected by a lightning location system (LLS) that can detect lightning optically via satellites or by a network of sensors that detect lightning on the basis of the electromagnetic energy lightning produces (Cummins and Murphy, 2009). Global coverage lightning climatologies have been developed using both satellite and sensor network systems to support studies seeking to understand earth system processes such as the atmospheric interaction between the global electrical circuit and climatic cycles. Cecil et al. (2014) combined satellite-detected lightning flashes from the Optical Transient Detector and the Lightning Imaging Sensor, with approximately 10 km and 5 km resolution respectively, to produce a 0.5 degree resolution global lightning climatology (Fig. 1). Virts et al. (2013) used data from the World Wide Lightning Location Network, which detects lightning with approximately 5 km accuracy using sensors that monitor very low-frequency radio waves, to produce a 0.25 degree resolution global lightning climatology. While global lightning climatologies can enable studies about lightning’s interaction with atmospheric processes (Beirle et al., 2014), they are not well suited for examining questions about biosphere-lightning interactions. For example, the interaction between lightning and wildfires is cross-scalar and links broader scale atmospheric processes to finer scale terrestrial processes and patterns (Macias Fauria et al., 2011; Whitlock et al., 2010). Most lightning caused wildfires are less than one hectare in size (Dowdy and Mills, 2012; Müller et al.,
2013), and hence predicting their occurrence requires lightning climatologies at a resolution that is much higher than that offered by a global LLS.
Many countries have a national LLS capable of detecting lightning locations with subkilometre accuracy (Burrows et al., 2002; Chronis, 2012; Cummins and Murphy, 2009; Gijben, 2012; Liu et al., 2014; Schulz et al., 2005), but the method used to analyse the lightning data affects the resolution of any climatology derived from it. The traditional approach to produce a lightning climatology is to define a grid of square cells over a study region, and then count the number of strikes per cell using the point location estimate of each lightning strike. However, this approach completely ignores the spatial uncertainty associated with each point location estimate of a lightning strike. Schultz et al. (2005) used simulation modelling to demonstrate that even when accuracy is ±500 m, a spatial resolution of 40 × 40 km is required to ensure that 95 % of strike locations are assigned to the correct grid cell. This result clearly demonstrates that limiting uncertainty in grid cell-counts requires sacrificing spatial resolution. In addition, the cell-count method is sensitive to cell size, as at higher resolutions cells contain fewer lightning strikes, resulting in an inflated level of apparent spatial stochasticity (Bourscheidt et al., 2014; Shephard et al., 2013).
Motivated by the issues in applying the cell-count approach to developing sub-kilometre resolution lightning climatologies suitable for studying biosphere-lightning interactions, we present a novel geocomputational approach for analysing lightning data based on probabilistic computational methods. Our computational method uses an open-source Python approach founded on the NumPy and SciPy packages (Oliphant, 2007), and enables the development of sub-kilometre resolution lightning climatologies. We demonstrate our methods by applying them to the mid-latitude oceanic landmass of New Zealand where levels of lightning activity are much lower than in continental areas (Figure 1). As most other national lightning climatologies come from continental or tropical areas where lightning activity is relatively high (Burrows et al., 2002; Chronis, 2012; Gijben, 2012; Hidayat and Ishii, 1998; Kuleshov et al., 2006; Liu et al., 2014; Orville et al., 2011; Pinto et al., 2009; Schulz et al., 2005), our analyses extend knowledge about spatial and temporal lightning activity patterns into an area with geographic conditions underrepresented in existing lightning climatologies.
2. Methods
2.1. The New Zealand Lightning Locating System The New Zealand LLS has been operating since 22nd August 2000 and consists of ten Vaisala IMPACT ESP2 lightning detection sensors deployed to detect cloud-to-ground lightning strokes, rather than flashes, with an expected detection efficiency > 90 % and location accuracy < 1 km across the majority of New Zealand (Rodger et al., 2006). The New Zealand LLS is a proprietary system operated by the Meteorological Service of New Zealand Limited who provide lightning data for purchase from September 2000 (http://about.metservice.com/weather-for-business/custom-forecasts/#Lightning). During this period there have only been very minor changes in two of the sensor locations and minor upgrades in LLS software and hardware, neither of which are expected to have had any notable effect on either detection efficiency or location accuracy (personal communication, Meteorological Service of New Zealand Limited, 2014).
The New Zealand LLS is well suited to developing a sub-kilometre resolution climatology to inform the study of atmosphere-biosphere interactions, as the LLS is deployed to optimise detection of cloud-to-ground and has a fairly low cloud-to-cloud detection efficiency (Rodger et al., 2006). However, experiments with similar LLS technology have shown that strokes with small positive polarities are very likely to be cloud-to-cloud discharge (Cummins and Murphy, 2009). Therefore, although we purchased data classified by the New Zealand LLS as positively and negatively charged cloud-to-ground strikes, to ensure that our analysis minimised contamination of cloud-to-cloud strikes and was consistent with other recent lightning climatologies (Orville et al., 2011; Shephard et al., 2013), we followed the advice of Cummins and Murphy (2009) and excluded any positive polarity strokes < 15 kA. The final set of data we analysed covered the period from September 2000 to December 2014, and comprised 2,512,574 cloud-to-ground lightning strikes.
2.2. Application of the cell-count method We began our analyses by applying the traditional cell-count method to the New Zealand LLS data to provide a comparison for our new approach. We did this at a resolution of 1 km, which is the highest resolution at which similar LLS data have been analysed using the cellcount method (Chronis, 2012; Schulz et al., 2005; Shephard et al., 2013), and also at a resolution of 100 m to meet our requirement of a sub-kilometre resolution lightning climatology.
We then applied the methodology of Schultz et al. (2005) to examine how the accuracy of the cell-count method would vary with cell resolution as a function of the spatial uncertainty in the strike locations detected by the New Zealand LLS. Schultz et al. (2005) considered strike locations to possess a circle that described the locational uncertainty of a strike, and that a strike could be confidently attributed to a cell if it was greater than its radius of uncertainty from the edge of a cell it was located within.
The spatial accuracy of the estimated strike locations recorded by the New Zealand LLS is a function of the number and geometrical positioning of lightning sensors detecting each strike, and is expressed as a probability ellipse describing the 50 % probability of occurrence that is centred on the estimated strike location (Stansfield, 1947). As Schultz et al. (2005) consider uncertainty of strike locations as a circle, we used the mean of the semi-major and semiminor axis lengths as a radius of uncertainty for each strike location that had a point estimate occurring on land (Figure 2a). We then defined grids of varying cell resolutions (1 km, 5 km, 10 km, 17.5 km, 25 km, 37.5 km, 50 km, 100 km) across New Zealand, and calculated the percentage of strike locations in each cell that were further than their uncertainty radius from a cell edge, and hence can be confidently assigned to the cell to which they are most likely to belong (Figure 2b).
2.3. Probabilistic lightning density across space To map lightning density we used a probabilistic approach (Bourscheidt et al., 2014) that calculates strike density by representing each strike as a bivariate normal probability distribution:
( Where:
(1)
)
[ (
)
(
)(
)
(
) ]
The New Zealand LLS provides all the unknowns for Eq. 1 via the orientation (θ) and lengths of the semi-major and semi-minor axes of the 50 % probability ellipses that are converted to lengths of one standard deviation (σx, σy) by down-scaling by 1.177 (Bourscheidt et al., 2014). Using Eq. 1 means that rather than considering each lightning strike to be at a point location, they can be represented as a continuous probability density that acknowledges the inherent uncertainty of each individual lightning strike (Figure 3a). A cumulative lightning strike density map is then developed as the sum of the densities across all strikes (Figure 3b). We mapped lightning density across New Zealand using a grid with 100 × 100 m cells to match the precision of the lengths of the 50 % probability ellipse semi-major and semi-minor axes recorded by the New Zealand LLS.
Creating a reliable high-resolution lightning strike density map from data collected over a short time-frame is challenging as there will be localised natural variation in lightning density that may disappear over longer time-frames (Shephard et al., 2013). Therefore, we applied spatial smoothing to reduce the effect of localized and rare lightning events. Bourscheidt et al. (2014) applied a constant multiplicative smoothing factor to the bivariate normal distribution describing each strike’s location. However, spatial variations in lightning activity mean that no single spatially invariant smoothing factor will be at the appropriate spatial scale for the entire map. For example, in some locations there may be a finer scale pattern of localised lightning activity, whereas in others there may be a broader scale pattern of more random lightning activity. If a single smoothing factor is applied at a scale that is somewhere between the finer localised and broader random patterns, then finer scale localised patterns may be smoothed away and lost, while broader random patterns may not be smoothed enough. A much better solution would be to minimise smoothing where the probability of clear localised patterns is high, and maximise smoothing where the probability of clear localised patterns is low. To achieve this spatially varying smoothing an adaptive probabilistic spatial smoothing methodology is required.
The standardised Getis-Ord spatial autocorrelation statistic (zGi*) assesses, for a given probability level, if localised values are different from those expected if the global values were randomly distributed (Getis and Ord, 1992; Ord and Getis, 1995). The zGi* was applied by Shephard et al. (2013) to identify, for a given probability, the distance at which
patterns of lightning strike densities became non-random for individual scenes of a lightning strike density map. Each scene was therefore smoothed by a distance related to the underlying randomness of lightning measured in the scene.
We developed an approach based on zGi* that is designed for the finer spatial resolution required for a sub-kilometre climatology. With reference to the global mean (̅) and variance (
) of lightning density values (x) within some area of interest consisting of n cells (Figure
4a), the zGi* statistic is calculated for a focal grid cell (i) using a circular neighbourhood of radius d that determines a set of neighbouring cells (j) of number nd that includes i itself. As we weighted each neighbourhood cell equally, zGi* can be calculated via: ∑
( )
(
(
̅ )
)
Where: ̅ ∑(
̅)
∑ (2)
As long as nd is sufficiently large to meet assumptions of normality, then the resulting zGi* can be interpreted as a z-score from a normal distribution and used to identify the probability of the values in the neighbourhood around the focal cell to have occurred by chance alone. By varying the size of the neighbourhood windows for each cell in turn, the smallest neighbourhood window that produces a zGi* significantly different from random can be identified for each cell (Figure 4b). The smoothed lightning strike density value for each cell is the mean of the strike density values within this smallest significantly non-random circular neighbourhood window (Figure 4c).
We applied this adaptive probabilistic approach to our total strike density map, using the mean and variance of the lightning strike density values across the landmass of New Zealand to assess the level of randomness of the local values. We examined circular windows ranging in radius from five cells (500 m), which was the smallest window that ensured the zGi* statistic conformed to normality, to 50 cells (5 km). The window size chosen for each cell was the smallest window size that produced a significant zGi* value in excess of three
standard deviations. If there were no significant window sizes the largest window size (5 km) was used.
2.4. Probabilistic lightning frequency across time Calculating temporal variation in lightning frequency relies on there being accurate counts of lightning strikes in a specific area of interest. When islands are being considered it is important to determine whether lightning strikes occurred on land or on the sea as temporal patterns of lightning vary between the two (Chronis, 2012; Hidayat and Ishii, 1998). This division is particularly important given that we are primarily interested in developing a subkilometre lightning climatology to study biosphere-lightning interactions such as wildfire, which means that for temporal patterns to be relevant they must be based on strikes occurring on land.
Using a probabilistic approach enables the probability of strikes being inside or outside an area of interest, such as the land surface, to be estimated (Figure 3). Therefore, we calculated the probability of each lightning strike occurring on land to produce probabilistic hourly, monthly, and annual counts of terrestrial lightning strikes that were the sum of all probability values. As the peak of lightning activity in the Southern Hemisphere is in the Austral summer (Kuleshov et al., 2006), the annual counts of lightning strikes were based on an annual period from July to June.
The El Niño Southern Oscillation (ENSO) has been shown to be related to inter-annual variation in lightning activity (Chronis et al., 2008; Sátori et al., 2009). This association has important implications for understanding risks associated with biosphere-lightning interactions such as wildfire. However, associations between ENSO and lightning activity have been confirmed only in lower latitude areas with much higher levels of lightning activity than New Zealand (Figure 1) raising the question of whether the inter-annual variation in lightning in New Zealand has as strong El Niño association. To assess if terrestrial lightning activity in New Zealand is associated with inter-annual climatic variation we compared the annual lightning strike counts with mean annual values of the multivariate ENSO index. The multivariate ENSO index provides a measure of the strength of El Niño and La Niña events with positive and negative values respectively (Wolter and Timlin, 1998). The multivariate ENSO index data are available from NOAA’s Earth System Research Laboratory (http://www.esrl.noaa.gov/psd/enso/mei/table.html).
3. Results The median location uncertainty, defined as the mean of the 50 % probability ellipse semimajor and semi-minor axes lengths (Figure 2a), was variable across the landmass of New Zealand. Location uncertainty was highest at the extremities of the network (Figure 5a) where 50 % probability ellipses generally became larger and more elongated. This spatial variation in location uncertainty means that the proportion of strikes that can be confidently located using the cell-count method also varies across New Zealand. For example, at a cell resolution of 10 km, only 41 % of strikes across New Zealand as a whole can be confidently located, but this value varies from around 20 % to 60 % depending on the precise location being considered (Figure 5b). Such scale-dependent uncertainty means that no single resolution is appropriate for the whole of New Zealand, unless a cell resolution towards 100 km is used at which point the number of confidently located strikes at locations of varying uncertainty begins to converge (Figure 5b).
When applied at a 100 m resolution the cell-count method did not produce meaningful results over either national (Figure 6a) or local (Figure 6d) extents. At a 100 m cell resolution, the vast majority of the cells contained no strikes, and hence a strike density of zero, while a minority of cells had estimated strikes densities of up to 76.74 km-2 yr-1, a figure which significantly exceeds the expected level of lightning activity for New Zealand (Figure 1). That applying the cell-count method at very high resolutions inflates the level of apparent spatial stochasticity is consistent with observations from other studies (Bourscheidt et al., 2014; Shephard et al., 2013).
At a national extent, the cell-count method applied at 1 km resolution (Figure 6b) and the probabilistic method applied at 100 m resolution (Figure 6c) exhibit similar spatial patterns of lightning activity. In all cases lightning activity is concentrated in the mountainous western edge of New Zealand’s South Island, and both maps had levels of lightning activity comparable to those from previous analyses of data from the New Zealand LLS (Rodger et al., 2006), the World Wide Lightning Location Network (Virts et al., 2013) and those detected by satellite imagery (Cecil et al., 2014 - as shown in Figure 1).
At a local extent, the cell-count method applied at 1 km resolution (Figure 6e) and the probabilistic method applied at 100 m resolution (Figure 6f) exhibit broadly similar spatial
patterns of lightning activity with finer scale patterns of lightning distribution on mountain peaks and ridges. However, the cell-count method produced a more ‘noisy’ result, with unrealistically abrupt breaks in lightning activity occurring for neighbouring cells.
We observed clear daily temporal patterns, with peaks of lightning activity during the midafternoon local time (Figure 7a). While there is an increased level of lightning activity during the Austral summer months (Figure 7b), in any given year the peak of monthly activity was highly variable (Figure 7c). There was considerable inter-annual variability in lightning activity (Figure 7c), but there was a positive association with ENSO activity as a third of the variation in annual lightning activity is explained by the multivariate ENSO index (r2 = 0.31, p = 0.05).
4. Discussion Applying the cell-count method at a 100 m resolution (Figure 6a) clearly demonstrates that the cell-count approach cannot produce a meaningful sub-kilometre lightning climatology suitable for examining cross-scalar processes such as wildfires that link the atmosphere and biosphere. In contrast, the 1 km resolution cell-count climatology (Figure 6b) did produce results that were broadly consistent with the 100 m resolution probabilistic climatology (Figure 6c). However, given that at a resolution of 1 km < 1 % of strike locations can be confidently attributed to the cell to which they most likely belong (Figure 5b), should we be surprised that the 1 km resolution cell-count approach (Figure 6b) and 100 m resolution probabilistic approach (Figure 6c) produced such similar results? Perhaps not. Although we cannot be confident that individual strikes are allocated to the correct cell, so long as the number of strikes is large, and there is no systematic bias, then any misclassification of a strike into a neighbouring cell should be balanced by a reciprocal misclassification. In other words, although individual strikes may be allocated incorrectly, the total counts for both cells will be correct. Therefore, the cell-count approach is a form of heuristic algorithm, as while it is not necessarily strictly correct, for lightning data of similar spatial accuracy to ours it does provide a meaningful solution in a computationally efficient manner at or above 1 km resolution when there are sufficiently many lightning strikes to ensure that any misclassifications even out.
However, when a sub-kilometre lightning climatology is required then we recommend the use of a probabilistic geocomputation approach as it provides information at a resolution
more relevant to those questions. Identifying sub-kilometre patterns using the cell-count method is not possible due to the inflated level of apparent spatial stochasticity that occurs when the cell-count method is applied at such high resolutions (Bourscheidt et al., 2014; Shephard et al., 2013). It is important to note, however, that both the cell-count approach and our probabilistic approach are limited by the underlying accuracy of the strike location data. Therefore, the application of any lightning climatology should consider the distribution of location accuracy (Figure 5a), and less confidence should be placed on the lightning climatology in areas with lower accuracy levels.
We also believe that a probabilistic approach to lightning climatologies is well suited to developing sub-kilometre lightning climatologies for New Zealand in particular. Unlike the traditional cell-count method, the probabilistic approach is independent of the cell size of the map grid (see Figure 9 in Bourscheidt et al., 2014). This independence is particularly useful for application in New Zealand, as the uncertainty of strike data varies spatially (Figure 5a) making it impossible to select a single optimal cell size for the whole country. In addition, as at higher resolutions the cell-count method relies on reciprocal misclassification to produce meaningful results, the cell-count approach will be more suitable for areas of higher lightning activity. However, the low level of lightning activity in New Zealand in comparison to much of the rest of the world (Figure 1), results in a higher risk of cell-counting generating a misleading pattern.
Our lightning climatology has unparalleled spatial resolution and confirms that relative to global levels, lightning activity in New Zealand is exceptionally low. However, the effect of topography on lightning occurrence, and the daily and monthly patterns of lightning activity are consistent with lightning climatologies from continental and tropical areas where lightning activity is high (Burrows et al., 2002; Chronis, 2012; Gijben, 2012; Hidayat and Ishii, 1998; Kuleshov et al., 2006; Liu et al., 2014; Orville et al., 2011; Pinto et al., 2009; Schulz et al., 2005). The positive association between the annual level of lightning activity and the ENSO cycle is also consistent with tropical regions where lightning activity is higher during El Niño events compared to La Niña events (Sátori et al., 2009). Therefore, we conclude that while lightning occurs at lower levels relative to global patterns, the processes governing lightning activity in New Zealand appear to be similar to those elsewhere.
Given the potential of a probabilistic approach to developing lightning climatologies, to demonstrate and encourage the use and development of the method, we provide Python scripts containing the computer code that generate Figure 3 and Figure 4. Example data are generated using the NLMpy package (Etherington et al., 2015), data computations are done using the NumPy and SciPy packages (Oliphant, 2007), and plotting is done using the Matplotlib package (Hunter, 2007). All these packages are open-source and freely available.
Acknowledgments This work was funded by The University of Auckland Faculty Research Development Fund 3702237.
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Figure 1. The global distribution of lightning flashes detected by satellites. The flash counts are determined from a combination of counts from the Optical Transient Detector and the Lightning Imaging Sensor (Cecil et al., 2014). The underlying data are available from the Global Hydrology Resource Center, dataset: LISOTD_HRFC_V2.3.2013.hdf (http://thunder.nsstc.nasa.gov/data/data_lis-otd-climatology.html). Figure 2. An illustration of the process of defining a strike uncertainty circle and evaluating whether a strike can be confidently assigned to a given grid cell. (a) The radius of an uncertainty circle is defined as the mean of the semi-major and semi-minor axis lengths associated with the 50 % probability ellipse. (b) Strikes such as A whose uncertainty circle overlap the edge of a grid cell cannot be confidently allocated to a grid cell, while strikes B and C can be confidently allocated. Figure 3. Mapping lightning density across space using a probabilistic approach. (a) The strike density for each strike is calculated from a bivariate normal probability distribution. (b) By summing the densities across multiple strikes a total strike density surface is produced. The probability of individual strikes occurring in an area of interest can also be calculated. A strike with an estimated point location in an area of interest may actually be more likely to have occurred outside the area of interest – see the strike at the top left hand corner of the area of interest in (b). Probabilistic lightning strike counts for an area of interest are calculated as the sum of probabilities (∑p) across all lightning strikes. The grid extent for these figures is 10 × 10 km and resolution is 100 × 100 m. Figure 4. An example of a spatially adaptive smoothing process based on spatial autocorrelation. (a) With reference to the mean and variance of lightning strike values within an area of interest, (b) the standardized Getis-Ord spatial autocorrelation statistic identifies the smallest circular neighbourhood window distance that contains a significantly nonrandom pattern. (c) Identifying this neighbourhood distance enables strike densities to be smoothed based on the scale of the spatial structure surrounding each individual cell. The grid extent for these figures is 50 × 50 km and resolution is 100 × 100 m. Figure 5. Assessment of the changes in location uncertainty as a function of cell resolution using the cell-count method. (a) The median strike uncertainty of strikes located in 10 × 10
km cells across New Zealand. (b) The percentage of strikes that can be confidently located within grid cells defined with different cell resolutions as a function of each strike’s accuracy, which was defined as the mean of the 50 % probability ellipse semi-major and semi-minor axis lengths. The percentage of confidently located strikes are shown for New Zealand as a whole, and cells at locations A, B, and C that occur in areas of differing location uncertainty Figure 6. The spatial pattern of cloud-to-ground lightning strike density in New Zealand using a cell-count approach at a cell resolution of (a) 100 × 100 m, and (b) 1 × 1 km, and (c) the probabilistic approach at a cell resolution of 100 × 100 m. Landscape perspective views in a 50 × 50 km section of mountainous terrain for the cell-count approaches at (d) 100 × 100 m, and (e) 1 × 1 km, and (f) the probabilistic approach at a cell resolution of 100 × 100 m. Note the maximum lightning strike density value varies for each method. Figure 7. The temporal pattern of terrestrial cloud-to-ground lightning strikes in New Zealand. (a) Distribution of strikes by local hour of the day. (b) Distribution of strikes by month. (c) Stacked bar chart showing the total number of annual strikes (from July-June) broken down into counts for each month, with horizontal lines showing the mean (± 1 standard deviation) annual strike rate of 37,928 (±12,895). Also plotted is the multivariate El Niño Southern Oscillation (ENSO) index that is positively correlated with annual variation in lightning activity (r2 = 0.31).
Figure 1. The global distribution of lightning flashes detected by satellites. The flash counts are determined from a combination of counts from the Optical Transient Detector and the Lightning Imaging Sensor (Cecil et al., 2014). The underlying data are available from the Global Hydrology Resource Center, dataset: LISOTD_HRFC_V2.3.2013.hdf (http://thunder.nsstc.nasa.gov/data/data_lis-otd-climatology.html).
Figure 2. An illustration of the process of defining a strike uncertainty circle and evaluating whether a strike can be confidently assigned to a given grid cell. (a) The radius of an uncertainty circle is defined as the mean of the semi-major and semi-minor axis lengths associated with the 50 % probability ellipse. (b) Strikes such as A whose uncertainty circle overlap the edge of a grid cell cannot be confidently allocated to a grid cell, while strikes B and C can be confidently allocated.
Figure 3. Mapping lightning density across space using a probabilistic approach. (a) The strike density for each strike is calculated from a bivariate normal probability distribution. (b) By summing the densities across multiple strikes a total strike density surface is produced. The probability of individual strikes occurring in an area of interest can also be calculated. A strike with an estimated point location in an area of interest may actually be more likely to have occurred outside the area of interest – see the strike at the top left hand corner of the area of interest in (b). Probabilistic lightning strike counts for an area of interest are calculated as the sum of probabilities (∑p) across all lightning strikes. The grid extent for these figures is 10 × 10 km and resolution is 100 × 100 m.
Figure 4. An example of a spatially adaptive smoothing process based on spatial autocorrelation. (a) With reference to the mean and variance of lightning strike values within an area of interest, (b) the standardized Getis-Ord spatial autocorrelation statistic identifies the smallest circular neighbourhood window distance that contains a significantly nonrandom pattern. (c) Identifying this neighbourhood distance enables strike densities to be smoothed based on the scale of the spatial structure surrounding each individual cell. The grid extent for these figures is 50 × 50 km and resolution is 100 × 100 m.
Figure 5. Assessment of the changes in location uncertainty as a function of cell resolution using the cell-count method. (a) The median strike uncertainty of strikes located in 10 × 10 km cells across New Zealand. (b) The percentage of strikes that can be confidently located within grid cells defined with different cell resolutions as a function of each strike’s accuracy, which was defined as the mean of the 50 % probability ellipse semi-major and semi-minor axis lengths. The percentage of confidently located strikes are shown for New Zealand as a whole, and cells at locations A, B, and C that occur in areas of differing location uncertainty.
Figure 6. The spatial pattern of cloud-to-ground lightning strike density in New Zealand using a cell-count approach at a cell resolution of (a) 100 × 100 m, and (b) 1 × 1 km, and (c) the probabilistic approach at a cell resolution of 100 × 100 m. Landscape perspective views in a 50 × 50 km section of mountainous terrain for the cell-count approaches at (d) 100 × 100 m, and (e) 1 × 1 km, and (f) the probabilistic approach at a cell resolution of 100 × 100 m. Note the maximum lightning strike density value varies for each method.
Figure 7. The temporal pattern of terrestrial cloud-to-ground lightning strikes in New Zealand. (a) Distribution of strikes by local hour of the day. (b) Distribution of strikes by month. (c) Stacked bar chart showing the total number of annual strikes (from July-June) broken down into counts for each month, with horizontal lines showing the mean (± 1 standard deviation) annual strike rate of 37,928 (±12,895). Also plotted is the multivariate El Niño Southern Oscillation (ENSO) index that is positively correlated with annual variation in lightning activity (r2 = 0.31).
Highlights
A sub-kilometre resolution lightning climatology is produced
Spatially adaptive probabilistic geocomputation methods are developed
Geocomputation workflow provided using an open-source Python approach
New Zealand has exceptionally low levels of lightning activity
PII: DOI: Reference:
S0098-3004(16)30444-7 http://dx.doi.org/10.1016/j.cageo.2016.09.010 CAGEO3841
To appear in: Computers and Geosciences Received date: 15 April 2016 Revised date: 9 August 2016 Accepted date: 25 September 2016 Cite this article as: Thomas R. Etherington and George L.W. Perry, Spatially adaptive probabilistic computation of a sub-kilometre resolution lightning climatology for New Zealand, Computers and Geosciences, http://dx.doi.org/10.1016/j.cageo.2016.09.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Spatially adaptive probabilistic computation of a sub-kilometre resolution lightning climatology for New Zealand Thomas R. Etheringtona,b,c*, George L.W. Perryc a
Royal Botanic Gardens Kew, Wakehurst Place, West Sussex, UK
b c
Department of Zoology, University of Oxford, Oxford, UK
School of Environment, The University of Auckland, Auckland, New Zealand
*
Corresponding author. [email protected]
Abstract Lightning is a key component of the Earth’s atmosphere and climate systems, and there is a potential positive feedback between a warming climate and increased lightning activity. In the biosphere, lightning is important as the main natural ignition source for wildfires and because of its contribution to the nitrogen cycle. Therefore, it is important to develop lightning climatologies to characterise and monitor lightning activity. While traditional methods for constructing lightning climatologies are suitable for examining lightning’s influence on atmospheric processes, they are less well suited for examining questions about biosphere-lightning interactions. For example, examining the interaction between lightning and wildfires requires linking atmospheric processes to finer scale terrestrial processes and patterns. Most wildfires ignited by lightning are less than one hectare in size, and so require lightning climatologies at a comparable spatial resolution. However, such high resolution lightning climatologies cannot be derived using the traditional cell-count methodology. Here we present a novel geocomputational approach for analysing lightning data at high spatial resolutions. Our approach is based on probabilistic computational methods and is capable of producing a sub-kilometre lightning climatology that honours the spatial accuracy of the strike locations and is adaptive to underlying spatial patterns. We demonstrate our methods by applying them to the mid-latitude oceanic landmass of New Zealand, an area with geographic conditions that are under-represented in existing lightning climatologies. Our resulting lightning climatology has unparalleled spatial resolution, and the spatial and temporal patterns we observe in it are consistent with other continental and tropical lightning climatologies. To encourage further use and development of our probabilistic approach, we provide Python scripts that demonstrate the method alongside our resulting New Zealand lightning climatology. Keywords: Getis-Ord; GIS; Python; spatial autocorrelation; smoothing; wildfire
1. Introduction Lightning is intrinsically linked with Earth’s climate system (Williams, 2005), and global warming and drying are predicted to result in an increase in lightning activity (Price, 2009; Romps et al., 2014). Increased lightning activity may further warm the climate as lightning increases atmospheric nitrogen oxide that affects production of ozone and hence the Earth’s radiation balance (Schumann and Huntrieser, 2007). In the biosphere increased lightning activity may also affect climate indirectly by increasing levels of atmospheric carbon dioxide and aerosols by igniting wildfires (Bowman et al., 2009), which may also become more common as the warmer and drier global climate increase fuel-bed flammability (Macias Fauria et al., 2011). Given the potential for positive feedbacks between a changing climate and increased lightning activity in both the atmosphere and biosphere, there is a pressing need for lightning climatologies that quantify and monitor variations in lightning density across space and frequency across time.
Lightning climatologies are developed from data collected by a lightning location system (LLS) that can detect lightning optically via satellites or by a network of sensors that detect lightning on the basis of the electromagnetic energy lightning produces (Cummins and Murphy, 2009). Global coverage lightning climatologies have been developed using both satellite and sensor network systems to support studies seeking to understand earth system processes such as the atmospheric interaction between the global electrical circuit and climatic cycles. Cecil et al. (2014) combined satellite-detected lightning flashes from the Optical Transient Detector and the Lightning Imaging Sensor, with approximately 10 km and 5 km resolution respectively, to produce a 0.5 degree resolution global lightning climatology (Fig. 1). Virts et al. (2013) used data from the World Wide Lightning Location Network, which detects lightning with approximately 5 km accuracy using sensors that monitor very low-frequency radio waves, to produce a 0.25 degree resolution global lightning climatology. While global lightning climatologies can enable studies about lightning’s interaction with atmospheric processes (Beirle et al., 2014), they are not well suited for examining questions about biosphere-lightning interactions. For example, the interaction between lightning and wildfires is cross-scalar and links broader scale atmospheric processes to finer scale terrestrial processes and patterns (Macias Fauria et al., 2011; Whitlock et al., 2010). Most lightning caused wildfires are less than one hectare in size (Dowdy and Mills, 2012; Müller et al.,
2013), and hence predicting their occurrence requires lightning climatologies at a resolution that is much higher than that offered by a global LLS.
Many countries have a national LLS capable of detecting lightning locations with subkilometre accuracy (Burrows et al., 2002; Chronis, 2012; Cummins and Murphy, 2009; Gijben, 2012; Liu et al., 2014; Schulz et al., 2005), but the method used to analyse the lightning data affects the resolution of any climatology derived from it. The traditional approach to produce a lightning climatology is to define a grid of square cells over a study region, and then count the number of strikes per cell using the point location estimate of each lightning strike. However, this approach completely ignores the spatial uncertainty associated with each point location estimate of a lightning strike. Schultz et al. (2005) used simulation modelling to demonstrate that even when accuracy is ±500 m, a spatial resolution of 40 × 40 km is required to ensure that 95 % of strike locations are assigned to the correct grid cell. This result clearly demonstrates that limiting uncertainty in grid cell-counts requires sacrificing spatial resolution. In addition, the cell-count method is sensitive to cell size, as at higher resolutions cells contain fewer lightning strikes, resulting in an inflated level of apparent spatial stochasticity (Bourscheidt et al., 2014; Shephard et al., 2013).
Motivated by the issues in applying the cell-count approach to developing sub-kilometre resolution lightning climatologies suitable for studying biosphere-lightning interactions, we present a novel geocomputational approach for analysing lightning data based on probabilistic computational methods. Our computational method uses an open-source Python approach founded on the NumPy and SciPy packages (Oliphant, 2007), and enables the development of sub-kilometre resolution lightning climatologies. We demonstrate our methods by applying them to the mid-latitude oceanic landmass of New Zealand where levels of lightning activity are much lower than in continental areas (Figure 1). As most other national lightning climatologies come from continental or tropical areas where lightning activity is relatively high (Burrows et al., 2002; Chronis, 2012; Gijben, 2012; Hidayat and Ishii, 1998; Kuleshov et al., 2006; Liu et al., 2014; Orville et al., 2011; Pinto et al., 2009; Schulz et al., 2005), our analyses extend knowledge about spatial and temporal lightning activity patterns into an area with geographic conditions underrepresented in existing lightning climatologies.
2. Methods
2.1. The New Zealand Lightning Locating System The New Zealand LLS has been operating since 22nd August 2000 and consists of ten Vaisala IMPACT ESP2 lightning detection sensors deployed to detect cloud-to-ground lightning strokes, rather than flashes, with an expected detection efficiency > 90 % and location accuracy < 1 km across the majority of New Zealand (Rodger et al., 2006). The New Zealand LLS is a proprietary system operated by the Meteorological Service of New Zealand Limited who provide lightning data for purchase from September 2000 (http://about.metservice.com/weather-for-business/custom-forecasts/#Lightning). During this period there have only been very minor changes in two of the sensor locations and minor upgrades in LLS software and hardware, neither of which are expected to have had any notable effect on either detection efficiency or location accuracy (personal communication, Meteorological Service of New Zealand Limited, 2014).
The New Zealand LLS is well suited to developing a sub-kilometre resolution climatology to inform the study of atmosphere-biosphere interactions, as the LLS is deployed to optimise detection of cloud-to-ground and has a fairly low cloud-to-cloud detection efficiency (Rodger et al., 2006). However, experiments with similar LLS technology have shown that strokes with small positive polarities are very likely to be cloud-to-cloud discharge (Cummins and Murphy, 2009). Therefore, although we purchased data classified by the New Zealand LLS as positively and negatively charged cloud-to-ground strikes, to ensure that our analysis minimised contamination of cloud-to-cloud strikes and was consistent with other recent lightning climatologies (Orville et al., 2011; Shephard et al., 2013), we followed the advice of Cummins and Murphy (2009) and excluded any positive polarity strokes < 15 kA. The final set of data we analysed covered the period from September 2000 to December 2014, and comprised 2,512,574 cloud-to-ground lightning strikes.
2.2. Application of the cell-count method We began our analyses by applying the traditional cell-count method to the New Zealand LLS data to provide a comparison for our new approach. We did this at a resolution of 1 km, which is the highest resolution at which similar LLS data have been analysed using the cellcount method (Chronis, 2012; Schulz et al., 2005; Shephard et al., 2013), and also at a resolution of 100 m to meet our requirement of a sub-kilometre resolution lightning climatology.
We then applied the methodology of Schultz et al. (2005) to examine how the accuracy of the cell-count method would vary with cell resolution as a function of the spatial uncertainty in the strike locations detected by the New Zealand LLS. Schultz et al. (2005) considered strike locations to possess a circle that described the locational uncertainty of a strike, and that a strike could be confidently attributed to a cell if it was greater than its radius of uncertainty from the edge of a cell it was located within.
The spatial accuracy of the estimated strike locations recorded by the New Zealand LLS is a function of the number and geometrical positioning of lightning sensors detecting each strike, and is expressed as a probability ellipse describing the 50 % probability of occurrence that is centred on the estimated strike location (Stansfield, 1947). As Schultz et al. (2005) consider uncertainty of strike locations as a circle, we used the mean of the semi-major and semiminor axis lengths as a radius of uncertainty for each strike location that had a point estimate occurring on land (Figure 2a). We then defined grids of varying cell resolutions (1 km, 5 km, 10 km, 17.5 km, 25 km, 37.5 km, 50 km, 100 km) across New Zealand, and calculated the percentage of strike locations in each cell that were further than their uncertainty radius from a cell edge, and hence can be confidently assigned to the cell to which they are most likely to belong (Figure 2b).
2.3. Probabilistic lightning density across space To map lightning density we used a probabilistic approach (Bourscheidt et al., 2014) that calculates strike density by representing each strike as a bivariate normal probability distribution:
( Where:
(1)
)
[ (
)
(
)(
)
(
) ]
The New Zealand LLS provides all the unknowns for Eq. 1 via the orientation (θ) and lengths of the semi-major and semi-minor axes of the 50 % probability ellipses that are converted to lengths of one standard deviation (σx, σy) by down-scaling by 1.177 (Bourscheidt et al., 2014). Using Eq. 1 means that rather than considering each lightning strike to be at a point location, they can be represented as a continuous probability density that acknowledges the inherent uncertainty of each individual lightning strike (Figure 3a). A cumulative lightning strike density map is then developed as the sum of the densities across all strikes (Figure 3b). We mapped lightning density across New Zealand using a grid with 100 × 100 m cells to match the precision of the lengths of the 50 % probability ellipse semi-major and semi-minor axes recorded by the New Zealand LLS.
Creating a reliable high-resolution lightning strike density map from data collected over a short time-frame is challenging as there will be localised natural variation in lightning density that may disappear over longer time-frames (Shephard et al., 2013). Therefore, we applied spatial smoothing to reduce the effect of localized and rare lightning events. Bourscheidt et al. (2014) applied a constant multiplicative smoothing factor to the bivariate normal distribution describing each strike’s location. However, spatial variations in lightning activity mean that no single spatially invariant smoothing factor will be at the appropriate spatial scale for the entire map. For example, in some locations there may be a finer scale pattern of localised lightning activity, whereas in others there may be a broader scale pattern of more random lightning activity. If a single smoothing factor is applied at a scale that is somewhere between the finer localised and broader random patterns, then finer scale localised patterns may be smoothed away and lost, while broader random patterns may not be smoothed enough. A much better solution would be to minimise smoothing where the probability of clear localised patterns is high, and maximise smoothing where the probability of clear localised patterns is low. To achieve this spatially varying smoothing an adaptive probabilistic spatial smoothing methodology is required.
The standardised Getis-Ord spatial autocorrelation statistic (zGi*) assesses, for a given probability level, if localised values are different from those expected if the global values were randomly distributed (Getis and Ord, 1992; Ord and Getis, 1995). The zGi* was applied by Shephard et al. (2013) to identify, for a given probability, the distance at which
patterns of lightning strike densities became non-random for individual scenes of a lightning strike density map. Each scene was therefore smoothed by a distance related to the underlying randomness of lightning measured in the scene.
We developed an approach based on zGi* that is designed for the finer spatial resolution required for a sub-kilometre climatology. With reference to the global mean (̅) and variance (
) of lightning density values (x) within some area of interest consisting of n cells (Figure
4a), the zGi* statistic is calculated for a focal grid cell (i) using a circular neighbourhood of radius d that determines a set of neighbouring cells (j) of number nd that includes i itself. As we weighted each neighbourhood cell equally, zGi* can be calculated via: ∑
( )
(
(
̅ )
)
Where: ̅ ∑(
̅)
∑ (2)
As long as nd is sufficiently large to meet assumptions of normality, then the resulting zGi* can be interpreted as a z-score from a normal distribution and used to identify the probability of the values in the neighbourhood around the focal cell to have occurred by chance alone. By varying the size of the neighbourhood windows for each cell in turn, the smallest neighbourhood window that produces a zGi* significantly different from random can be identified for each cell (Figure 4b). The smoothed lightning strike density value for each cell is the mean of the strike density values within this smallest significantly non-random circular neighbourhood window (Figure 4c).
We applied this adaptive probabilistic approach to our total strike density map, using the mean and variance of the lightning strike density values across the landmass of New Zealand to assess the level of randomness of the local values. We examined circular windows ranging in radius from five cells (500 m), which was the smallest window that ensured the zGi* statistic conformed to normality, to 50 cells (5 km). The window size chosen for each cell was the smallest window size that produced a significant zGi* value in excess of three
standard deviations. If there were no significant window sizes the largest window size (5 km) was used.
2.4. Probabilistic lightning frequency across time Calculating temporal variation in lightning frequency relies on there being accurate counts of lightning strikes in a specific area of interest. When islands are being considered it is important to determine whether lightning strikes occurred on land or on the sea as temporal patterns of lightning vary between the two (Chronis, 2012; Hidayat and Ishii, 1998). This division is particularly important given that we are primarily interested in developing a subkilometre lightning climatology to study biosphere-lightning interactions such as wildfire, which means that for temporal patterns to be relevant they must be based on strikes occurring on land.
Using a probabilistic approach enables the probability of strikes being inside or outside an area of interest, such as the land surface, to be estimated (Figure 3). Therefore, we calculated the probability of each lightning strike occurring on land to produce probabilistic hourly, monthly, and annual counts of terrestrial lightning strikes that were the sum of all probability values. As the peak of lightning activity in the Southern Hemisphere is in the Austral summer (Kuleshov et al., 2006), the annual counts of lightning strikes were based on an annual period from July to June.
The El Niño Southern Oscillation (ENSO) has been shown to be related to inter-annual variation in lightning activity (Chronis et al., 2008; Sátori et al., 2009). This association has important implications for understanding risks associated with biosphere-lightning interactions such as wildfire. However, associations between ENSO and lightning activity have been confirmed only in lower latitude areas with much higher levels of lightning activity than New Zealand (Figure 1) raising the question of whether the inter-annual variation in lightning in New Zealand has as strong El Niño association. To assess if terrestrial lightning activity in New Zealand is associated with inter-annual climatic variation we compared the annual lightning strike counts with mean annual values of the multivariate ENSO index. The multivariate ENSO index provides a measure of the strength of El Niño and La Niña events with positive and negative values respectively (Wolter and Timlin, 1998). The multivariate ENSO index data are available from NOAA’s Earth System Research Laboratory (http://www.esrl.noaa.gov/psd/enso/mei/table.html).
3. Results The median location uncertainty, defined as the mean of the 50 % probability ellipse semimajor and semi-minor axes lengths (Figure 2a), was variable across the landmass of New Zealand. Location uncertainty was highest at the extremities of the network (Figure 5a) where 50 % probability ellipses generally became larger and more elongated. This spatial variation in location uncertainty means that the proportion of strikes that can be confidently located using the cell-count method also varies across New Zealand. For example, at a cell resolution of 10 km, only 41 % of strikes across New Zealand as a whole can be confidently located, but this value varies from around 20 % to 60 % depending on the precise location being considered (Figure 5b). Such scale-dependent uncertainty means that no single resolution is appropriate for the whole of New Zealand, unless a cell resolution towards 100 km is used at which point the number of confidently located strikes at locations of varying uncertainty begins to converge (Figure 5b).
When applied at a 100 m resolution the cell-count method did not produce meaningful results over either national (Figure 6a) or local (Figure 6d) extents. At a 100 m cell resolution, the vast majority of the cells contained no strikes, and hence a strike density of zero, while a minority of cells had estimated strikes densities of up to 76.74 km-2 yr-1, a figure which significantly exceeds the expected level of lightning activity for New Zealand (Figure 1). That applying the cell-count method at very high resolutions inflates the level of apparent spatial stochasticity is consistent with observations from other studies (Bourscheidt et al., 2014; Shephard et al., 2013).
At a national extent, the cell-count method applied at 1 km resolution (Figure 6b) and the probabilistic method applied at 100 m resolution (Figure 6c) exhibit similar spatial patterns of lightning activity. In all cases lightning activity is concentrated in the mountainous western edge of New Zealand’s South Island, and both maps had levels of lightning activity comparable to those from previous analyses of data from the New Zealand LLS (Rodger et al., 2006), the World Wide Lightning Location Network (Virts et al., 2013) and those detected by satellite imagery (Cecil et al., 2014 - as shown in Figure 1).
At a local extent, the cell-count method applied at 1 km resolution (Figure 6e) and the probabilistic method applied at 100 m resolution (Figure 6f) exhibit broadly similar spatial
patterns of lightning activity with finer scale patterns of lightning distribution on mountain peaks and ridges. However, the cell-count method produced a more ‘noisy’ result, with unrealistically abrupt breaks in lightning activity occurring for neighbouring cells.
We observed clear daily temporal patterns, with peaks of lightning activity during the midafternoon local time (Figure 7a). While there is an increased level of lightning activity during the Austral summer months (Figure 7b), in any given year the peak of monthly activity was highly variable (Figure 7c). There was considerable inter-annual variability in lightning activity (Figure 7c), but there was a positive association with ENSO activity as a third of the variation in annual lightning activity is explained by the multivariate ENSO index (r2 = 0.31, p = 0.05).
4. Discussion Applying the cell-count method at a 100 m resolution (Figure 6a) clearly demonstrates that the cell-count approach cannot produce a meaningful sub-kilometre lightning climatology suitable for examining cross-scalar processes such as wildfires that link the atmosphere and biosphere. In contrast, the 1 km resolution cell-count climatology (Figure 6b) did produce results that were broadly consistent with the 100 m resolution probabilistic climatology (Figure 6c). However, given that at a resolution of 1 km < 1 % of strike locations can be confidently attributed to the cell to which they most likely belong (Figure 5b), should we be surprised that the 1 km resolution cell-count approach (Figure 6b) and 100 m resolution probabilistic approach (Figure 6c) produced such similar results? Perhaps not. Although we cannot be confident that individual strikes are allocated to the correct cell, so long as the number of strikes is large, and there is no systematic bias, then any misclassification of a strike into a neighbouring cell should be balanced by a reciprocal misclassification. In other words, although individual strikes may be allocated incorrectly, the total counts for both cells will be correct. Therefore, the cell-count approach is a form of heuristic algorithm, as while it is not necessarily strictly correct, for lightning data of similar spatial accuracy to ours it does provide a meaningful solution in a computationally efficient manner at or above 1 km resolution when there are sufficiently many lightning strikes to ensure that any misclassifications even out.
However, when a sub-kilometre lightning climatology is required then we recommend the use of a probabilistic geocomputation approach as it provides information at a resolution
more relevant to those questions. Identifying sub-kilometre patterns using the cell-count method is not possible due to the inflated level of apparent spatial stochasticity that occurs when the cell-count method is applied at such high resolutions (Bourscheidt et al., 2014; Shephard et al., 2013). It is important to note, however, that both the cell-count approach and our probabilistic approach are limited by the underlying accuracy of the strike location data. Therefore, the application of any lightning climatology should consider the distribution of location accuracy (Figure 5a), and less confidence should be placed on the lightning climatology in areas with lower accuracy levels.
We also believe that a probabilistic approach to lightning climatologies is well suited to developing sub-kilometre lightning climatologies for New Zealand in particular. Unlike the traditional cell-count method, the probabilistic approach is independent of the cell size of the map grid (see Figure 9 in Bourscheidt et al., 2014). This independence is particularly useful for application in New Zealand, as the uncertainty of strike data varies spatially (Figure 5a) making it impossible to select a single optimal cell size for the whole country. In addition, as at higher resolutions the cell-count method relies on reciprocal misclassification to produce meaningful results, the cell-count approach will be more suitable for areas of higher lightning activity. However, the low level of lightning activity in New Zealand in comparison to much of the rest of the world (Figure 1), results in a higher risk of cell-counting generating a misleading pattern.
Our lightning climatology has unparalleled spatial resolution and confirms that relative to global levels, lightning activity in New Zealand is exceptionally low. However, the effect of topography on lightning occurrence, and the daily and monthly patterns of lightning activity are consistent with lightning climatologies from continental and tropical areas where lightning activity is high (Burrows et al., 2002; Chronis, 2012; Gijben, 2012; Hidayat and Ishii, 1998; Kuleshov et al., 2006; Liu et al., 2014; Orville et al., 2011; Pinto et al., 2009; Schulz et al., 2005). The positive association between the annual level of lightning activity and the ENSO cycle is also consistent with tropical regions where lightning activity is higher during El Niño events compared to La Niña events (Sátori et al., 2009). Therefore, we conclude that while lightning occurs at lower levels relative to global patterns, the processes governing lightning activity in New Zealand appear to be similar to those elsewhere.
Given the potential of a probabilistic approach to developing lightning climatologies, to demonstrate and encourage the use and development of the method, we provide Python scripts containing the computer code that generate Figure 3 and Figure 4. Example data are generated using the NLMpy package (Etherington et al., 2015), data computations are done using the NumPy and SciPy packages (Oliphant, 2007), and plotting is done using the Matplotlib package (Hunter, 2007). All these packages are open-source and freely available.
Acknowledgments This work was funded by The University of Auckland Faculty Research Development Fund 3702237.
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Figure 1. The global distribution of lightning flashes detected by satellites. The flash counts are determined from a combination of counts from the Optical Transient Detector and the Lightning Imaging Sensor (Cecil et al., 2014). The underlying data are available from the Global Hydrology Resource Center, dataset: LISOTD_HRFC_V2.3.2013.hdf (http://thunder.nsstc.nasa.gov/data/data_lis-otd-climatology.html). Figure 2. An illustration of the process of defining a strike uncertainty circle and evaluating whether a strike can be confidently assigned to a given grid cell. (a) The radius of an uncertainty circle is defined as the mean of the semi-major and semi-minor axis lengths associated with the 50 % probability ellipse. (b) Strikes such as A whose uncertainty circle overlap the edge of a grid cell cannot be confidently allocated to a grid cell, while strikes B and C can be confidently allocated. Figure 3. Mapping lightning density across space using a probabilistic approach. (a) The strike density for each strike is calculated from a bivariate normal probability distribution. (b) By summing the densities across multiple strikes a total strike density surface is produced. The probability of individual strikes occurring in an area of interest can also be calculated. A strike with an estimated point location in an area of interest may actually be more likely to have occurred outside the area of interest – see the strike at the top left hand corner of the area of interest in (b). Probabilistic lightning strike counts for an area of interest are calculated as the sum of probabilities (∑p) across all lightning strikes. The grid extent for these figures is 10 × 10 km and resolution is 100 × 100 m. Figure 4. An example of a spatially adaptive smoothing process based on spatial autocorrelation. (a) With reference to the mean and variance of lightning strike values within an area of interest, (b) the standardized Getis-Ord spatial autocorrelation statistic identifies the smallest circular neighbourhood window distance that contains a significantly nonrandom pattern. (c) Identifying this neighbourhood distance enables strike densities to be smoothed based on the scale of the spatial structure surrounding each individual cell. The grid extent for these figures is 50 × 50 km and resolution is 100 × 100 m. Figure 5. Assessment of the changes in location uncertainty as a function of cell resolution using the cell-count method. (a) The median strike uncertainty of strikes located in 10 × 10
km cells across New Zealand. (b) The percentage of strikes that can be confidently located within grid cells defined with different cell resolutions as a function of each strike’s accuracy, which was defined as the mean of the 50 % probability ellipse semi-major and semi-minor axis lengths. The percentage of confidently located strikes are shown for New Zealand as a whole, and cells at locations A, B, and C that occur in areas of differing location uncertainty Figure 6. The spatial pattern of cloud-to-ground lightning strike density in New Zealand using a cell-count approach at a cell resolution of (a) 100 × 100 m, and (b) 1 × 1 km, and (c) the probabilistic approach at a cell resolution of 100 × 100 m. Landscape perspective views in a 50 × 50 km section of mountainous terrain for the cell-count approaches at (d) 100 × 100 m, and (e) 1 × 1 km, and (f) the probabilistic approach at a cell resolution of 100 × 100 m. Note the maximum lightning strike density value varies for each method. Figure 7. The temporal pattern of terrestrial cloud-to-ground lightning strikes in New Zealand. (a) Distribution of strikes by local hour of the day. (b) Distribution of strikes by month. (c) Stacked bar chart showing the total number of annual strikes (from July-June) broken down into counts for each month, with horizontal lines showing the mean (± 1 standard deviation) annual strike rate of 37,928 (±12,895). Also plotted is the multivariate El Niño Southern Oscillation (ENSO) index that is positively correlated with annual variation in lightning activity (r2 = 0.31).
Figure 1. The global distribution of lightning flashes detected by satellites. The flash counts are determined from a combination of counts from the Optical Transient Detector and the Lightning Imaging Sensor (Cecil et al., 2014). The underlying data are available from the Global Hydrology Resource Center, dataset: LISOTD_HRFC_V2.3.2013.hdf (http://thunder.nsstc.nasa.gov/data/data_lis-otd-climatology.html).
Figure 2. An illustration of the process of defining a strike uncertainty circle and evaluating whether a strike can be confidently assigned to a given grid cell. (a) The radius of an uncertainty circle is defined as the mean of the semi-major and semi-minor axis lengths associated with the 50 % probability ellipse. (b) Strikes such as A whose uncertainty circle overlap the edge of a grid cell cannot be confidently allocated to a grid cell, while strikes B and C can be confidently allocated.
Figure 3. Mapping lightning density across space using a probabilistic approach. (a) The strike density for each strike is calculated from a bivariate normal probability distribution. (b) By summing the densities across multiple strikes a total strike density surface is produced. The probability of individual strikes occurring in an area of interest can also be calculated. A strike with an estimated point location in an area of interest may actually be more likely to have occurred outside the area of interest – see the strike at the top left hand corner of the area of interest in (b). Probabilistic lightning strike counts for an area of interest are calculated as the sum of probabilities (∑p) across all lightning strikes. The grid extent for these figures is 10 × 10 km and resolution is 100 × 100 m.
Figure 4. An example of a spatially adaptive smoothing process based on spatial autocorrelation. (a) With reference to the mean and variance of lightning strike values within an area of interest, (b) the standardized Getis-Ord spatial autocorrelation statistic identifies the smallest circular neighbourhood window distance that contains a significantly nonrandom pattern. (c) Identifying this neighbourhood distance enables strike densities to be smoothed based on the scale of the spatial structure surrounding each individual cell. The grid extent for these figures is 50 × 50 km and resolution is 100 × 100 m.
Figure 5. Assessment of the changes in location uncertainty as a function of cell resolution using the cell-count method. (a) The median strike uncertainty of strikes located in 10 × 10 km cells across New Zealand. (b) The percentage of strikes that can be confidently located within grid cells defined with different cell resolutions as a function of each strike’s accuracy, which was defined as the mean of the 50 % probability ellipse semi-major and semi-minor axis lengths. The percentage of confidently located strikes are shown for New Zealand as a whole, and cells at locations A, B, and C that occur in areas of differing location uncertainty.
Figure 6. The spatial pattern of cloud-to-ground lightning strike density in New Zealand using a cell-count approach at a cell resolution of (a) 100 × 100 m, and (b) 1 × 1 km, and (c) the probabilistic approach at a cell resolution of 100 × 100 m. Landscape perspective views in a 50 × 50 km section of mountainous terrain for the cell-count approaches at (d) 100 × 100 m, and (e) 1 × 1 km, and (f) the probabilistic approach at a cell resolution of 100 × 100 m. Note the maximum lightning strike density value varies for each method.
Figure 7. The temporal pattern of terrestrial cloud-to-ground lightning strikes in New Zealand. (a) Distribution of strikes by local hour of the day. (b) Distribution of strikes by month. (c) Stacked bar chart showing the total number of annual strikes (from July-June) broken down into counts for each month, with horizontal lines showing the mean (± 1 standard deviation) annual strike rate of 37,928 (±12,895). Also plotted is the multivariate El Niño Southern Oscillation (ENSO) index that is positively correlated with annual variation in lightning activity (r2 = 0.31).
Highlights
A sub-kilometre resolution lightning climatology is produced
Spatially adaptive probabilistic geocomputation methods are developed
Geocomputation workflow provided using an open-source Python approach
New Zealand has exceptionally low levels of lightning activity