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  • View in gallery

    (left) Mean annual CG lightning flash density (number of flashes km−2 yr−1), based on data from 1999–2009 in a 25 × 25 km2 grid. (right) Areal interpolated population density (persons km−2) from Canadian census 2001 subdivision boundary in 25 × 25 km2 grid.

  • View in gallery

    (left) Gridded observed tornado counts standardized to 10 000 km−2 yr−1. (center) Posterior mean of tornado occurrence standardized to 10 000 km−2 yr−1. (right) Posterior standard deviation of tornado occurrence standardized to 10 000 km−2 yr−1. (top)–(bottom) The 25 × 25, 50 × 50, and 100 × 100 km2 models, respectively.

  • View in gallery

    Posterior mean of the conditional autocorrelation coefficient ai on a 25 × 25 km2 model grid.

  • View in gallery

    (left) Probability of tornado detection estimated from the model as a function of population density. (right) Percentile of the gridded population density. Models on a 25 × 25, 50 × 50, and 100 × 100 km2 grid are shown in dashed, gray, and black solid line, respectively. The three inset panels show an expanded y axis for percentiles from 0% to 90%.

  • View in gallery

    Posterior mean of Tlatenti on 25 × 25, 50 × 50, and 100 × 100 km2 model grids overlaid on the original census subdivision boundary layer with the population density (person km−2) of each census subdivision shown with gray scales. The gray scale for Tlatenti is the same as those in Fig. 2.

  • View in gallery

    Predicted tornado occurrence (10 000 km−2 yr−1), tornado probability (yr−1), and the return period (yr), based on the posterior mean values of Tlatenti of the 25 × 25 km2 model. Asterisk implies that the median tornado-affected area (F0–F5) of 0.10 km2 from the Schaefer et al. (1986) records is used. An inset map of southwestern and south-central Ontario is shown in the upper right corner.

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Probability of Tornado Occurrence across Canada

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  • 1 Ecological Modelling Laboratory, Department of Physical and Environmental Sciences, and Climate Laboratory, Department of Physical and Environmental Sciences, University of Toronto, and Adaptation and Impacts Research Section, Atmospheric Science and Technology Directorate, Science and Technology Branch, Environment Canada, Toronto, Ontario, Canada
  • | 2 Ecological Modelling Laboratory, Department of Physical and Environmental Sciences, University of Toronto, Toronto, Ontario, Canada
  • | 3 Cloud Physics and Severe Weather Research Section, Atmospheric Science and Technology Directorate, Science and Technology Branch, Environment Canada, Toronto, Ontario, Canada
  • | 4 Adaptation and Impacts Research Section, Atmospheric Science and Technology Directorate, Science and Technology Branch, Environment Canada, Toronto, Ontario, Canada
  • | 5 ** Climate Laboratory, Department of Physical and Environmental Sciences, University of Toronto, Toronto, Ontario, Canada
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Abstract

The number of tornado observations in Canada is believed to be significantly lower than the actual occurrences. To account for this bias, the authors propose a Bayesian modeling approach founded upon the explicit consideration of the population sampling bias in tornado observations and the predictive relationship between cloud-to-ground (CG) lightning flash climatology and tornado occurrence. The latter variable was used as an indicator for quantifying convective storm activity, which is generally a precursor to tornado occurrence. The CG lightning data were generated from an 11-yr lightning climatology survey (1999–2009) from the Canadian Lightning Detection Network. The results suggest that the predictions of tornado occurrence in populated areas are fairly reliable with no profound underestimation bias. In sparsely populated areas, the analysis shows that the probability of tornado occurrence is significantly higher than what is represented in the 30-yr data record. Areas with low population density but high lightning flash density demonstrate the greatest discrepancy between predicted and observed tornado occurrence. A sensitivity analysis with various grid sizes was also conducted. It was found that the predictive statements supported by the model are fairly robust to the grid configuration, but the population density per grid cell is more representative to the actual population density at smaller resolution and therefore more accurately depicts the probability of tornado occurrence. Finally, a tornado probability map is calculated for Canada based on the frequency of tornado occurrence derived from the model and the estimated damage area of individual tornado events.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-13-00093.s1.

Corresponding author address: Vincent Y. S. Cheng, Ecological Modelling Laboratory, University of Toronto, 1265 Military Trail, Scarborough ON M1C 1A4, Canada. E-mail: chengv@geog.utoronto.ca

Abstract

The number of tornado observations in Canada is believed to be significantly lower than the actual occurrences. To account for this bias, the authors propose a Bayesian modeling approach founded upon the explicit consideration of the population sampling bias in tornado observations and the predictive relationship between cloud-to-ground (CG) lightning flash climatology and tornado occurrence. The latter variable was used as an indicator for quantifying convective storm activity, which is generally a precursor to tornado occurrence. The CG lightning data were generated from an 11-yr lightning climatology survey (1999–2009) from the Canadian Lightning Detection Network. The results suggest that the predictions of tornado occurrence in populated areas are fairly reliable with no profound underestimation bias. In sparsely populated areas, the analysis shows that the probability of tornado occurrence is significantly higher than what is represented in the 30-yr data record. Areas with low population density but high lightning flash density demonstrate the greatest discrepancy between predicted and observed tornado occurrence. A sensitivity analysis with various grid sizes was also conducted. It was found that the predictive statements supported by the model are fairly robust to the grid configuration, but the population density per grid cell is more representative to the actual population density at smaller resolution and therefore more accurately depicts the probability of tornado occurrence. Finally, a tornado probability map is calculated for Canada based on the frequency of tornado occurrence derived from the model and the estimated damage area of individual tornado events.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-13-00093.s1.

Corresponding author address: Vincent Y. S. Cheng, Ecological Modelling Laboratory, University of Toronto, 1265 Military Trail, Scarborough ON M1C 1A4, Canada. E-mail: chengv@geog.utoronto.ca

1. Introduction

Tornadoes are one of nature's most hazardous phenomena, capable of causing significant property damage and economic disruption as well as human injuries and fatalities. The tornadic events in Barrie 1985 (Etkin et al. 2001), Edmonton 1987 (Charlton et al. 1995), and southern Ontario 2009 (Ashton et al. 2010a,b) are amongst the most significant and costly tornado events in Canadian history. Multidisciplinary forensic analysis of a number of tornado-damaged areas in eastern Canada revealed that buildings in which more than 90% of the occupants were killed or seriously injured did not have anchorage of house floors into the foundation or anchorage of the roof to the walls (Allen 1992, 1986, 1984; Carter et al. 1989). As a result, the 1995 National Building Code of Canada (NBCC 2005) was updated to include provisions that ensure basic structural resilience under low-end tornadic loads in areas of Canada defined as “tornado prone.” However, the definition of tornado-prone regions requires a rigorous assessment of the spatial frequency of tornado occurrence, and to date these areas have not been clearly identified in Canada and thus these life-saving structural measures are not necessarily being implemented in areas at risk. In light of this need, Environment Canada has recently compiled an updated 30-yr national tornado database, founded upon an expert meteorological assessment of all existing tornado records. This database may serve as the basis for the implementation of tornado resiliency measures in the National Building Code of Canada (Sills et al. 2012).

Despite the notable efforts to consolidate a national tornado database, it is impossible to document all tornado occurrences in every single area, as tornadic events could easily be missed due to their relatively small spatial extent and time duration, as well as due to the absence of observers, structures, or daylight. In fact, tornado observations in any tornado dataset are often biased and underreported (Doswell and Burgess 1988), and the number of observations depends not only on the meteorological factors related to tornado occurrence (e.g., King et al. 2003) but also on nonmeteorological factors such as the monitoring network, proximity to populated areas, and/or radar locations, landscapes, and topography (Schaefer and Galway 1982; Grazulis and Abbey 1983; Ray et al. 2003; King 1997). Thus, the development of tornado climatology and the delineation of tornado-prone areas that could conceivably dictate security standards and infrastructural investments are very challenging. Of all the nonmeteorological factors investigated (e.g., obscured vision due to density of trees and hills, absence of roads, and buildings), population density is documented as the key nonmeteorological factor in determining the observation bias of tornadoes and nontornadic severe thunderstorms in many North American studies (e.g., Anderson et al. 2007; Ray et al. 2003; King 1997; Etkin and Leduc 1994; Paruk and Blackwell 1994; Snider 1977; Tescon et al. 1983; Schaefer and Galway 1982). The population sampling bias is typically accounted for with statistical models that correct tornado observations in areas with high sampling bias using nearby sites with low sampling bias and reliable tornado records. These statistical approaches are generally limited to relatively small spatial domains due to their underlying assumption that the tornado climatology is homogeneous or that there is significant areal coverage of reliable sites to derive credible adjustments (e.g., Anderson et al. 2007; Ray et al. 2003; King 1997). However, this assumption is profoundly violated in countries like Canada, where extensive areas with unreliable data exist, and thus the observational uncertainty can significantly compromise our predictive power.

The latter problem can be overcome when simultaneously considering meteorological covariates of the spatial variability of tornado occurrence. Finding potentially meaningful predictive relationships between meteorological factors and tornado occurrence could be a valuable tool for assessing tornado climatology and risk, particularly for large areas with significant observational uncertainty. There are many studies that have investigated possible linkages between meteorological covariates with tornado observations, for example, sounding-derived parameters such as convective available potential energy, vertical wind shear (Brooks et al. 2003b), and lightning flash polarity (Carey et al. 2003), but none of these relationships has been used in conjunction with predictive frameworks that explicitly accommodate the observation error associated with the tornado data. In addition, most studies typically assume that the tornado observations are independent and ignore other possible sources of spatial correlation within the domain modeled (e.g., Anderson et al. 2007; King 1997). Like in any spatially distributed modeling exercise, the explicit consideration of the error covariance in space is essential for drawing correct statistical inference and for identifying factors unaccounted for by the model (Anderson et al. 2007; Wikle and Anderson 2003). In this study, our thesis is that the Bayesian paradigm offers an effective means to accommodate all the aforementioned error sources and thus impartially communicate the total uncertainty associated with the forecasting of tornado occurrence (Arhonditsis et al. 2007, 2008a,b). Further, the use of hierarchical Bayes offers a conceptually plausible way for addressing the complexity pervading natural systems (Clark 2005; Cheng et al. 2010). In particular, the Bayesian hierarchical modeling can be an indispensable methodological framework to disentangle complex environmental patterns, to exploit disparate sources of information, to accommodate tightly intertwined processes operating at different spatiotemporal scales, and to explicitly consider the variability pertaining to latent variables or other inherently unmeasurable quantities (Clark and Gelfand 2006; Clark 2005; Zhang and Arhonditsis 2009).

The objective of this paper is to predict tornado occurrence across Canada using a Bayesian modeling approach and to shed light on the Canadian tornado climatology. The problem of tornado occurrence assessment is dissected into a two-pronged strategy in which we first consider the covariance between lightning flash density climatology and tornado occurrence, and then we postulate that the likelihood to observe a tornado is closely related to the population density. Our approach also explicitly accommodates the fact that the model residual variability is likely to be characterized by distinct spatial patterns, arising from the model structural uncertainty. This paper is organized as follows: section 2 describes the sources and treatment of the data prior to the modeling analysis. Section 3 describes the rationale and basic features of our Bayesian modeling approach. Section 4 presents the results and discussion, including a sensitivity analysis of the model predictions to the grid resolution. Section 5 presents the conclusions and future perspectives of the present modeling framework.

2. Data sources and processing

a. Tornado data

Our analysis is based on the updated national tornado database for the period 1980–2009 (Sills et al. 2012). This database includes an intensity F-scale rating (Fujita 1981) and a confidence rating of “confirmed” (direct evidence of a tornado, e.g., visual evidence), “probable” (all available evidences point to the likelihood of a tornado but without direct evidence), and “possible” (ambiguous or unreliable tornado evidence). We included only the confirmed and probable events for all F-scale ratings (F0–F5) to ensure that our analysis is not based on nontornadic events that most likely are included in the possible category. The 30-yr study considered herein can also be compared with the last national tornado study based on the previous 30-yr period (1950–79; Newark 1984). To achieve a more accurate representation of the tornado occurrence for the United States–Canada border regions, U.S. tornado data for the same period were also obtained from the U.S. National Weather Service Storm Prediction Center (SPC).

The tornado data were classified into grids as follows: tornado reports of all F scales (F0–F5) were aggregated for the 30-yr period and were plotted as points (tornado touchdown location) or paths (track of the tornado on the ground), if the path information was available. Using the 1-km polygon vertices previously created by Burrows and Kochtubajda (2010) for their lightning study, three separate grids with cell sizes 25 × 25, 50 × 50, and 100 × 100 km2 were overlaid onto Canada to cover areas from the southern Canada–United States border to 70°N. The grid cells that were completely over water bodies (e.g., at the center of lakes and oceans) were removed. We tallied the tornado points and paths that occurred within or passed through each cell i, resulting in a tornado count for each of the cells. In the case of tornado paths, a tornado is counted more than once when it crosses more than one cell, but it cannot be counted multiple times within the same cell. The tally was done separately for the three grids examined. Tornado densities were then calculated by dividing the tornado counts per cell by the cell area. Finally, it should be noted that as the gridcell size increased, the cells at the United States–Canada border extended farther south and thus more U.S. area and SPC data were included in the analysis.

b. Lightning data

There is a body of literature relating the polarity of lightning flashes and, in particular, the predominately positive cloud-to-ground (CG) lightning flashes to tornadic activity (e.g., Reap and MacGorman 1989; Branick and Doswell 1992; Knapp 1994; MacGorman and Burgess 1994). However, recent empirical evidence suggests that a large majority of tornadic storms during the warm season (April–September) can also be associated with predominately negative cloud-to-ground lightning flashes in the contiguous United States (Carey et al. 2003). A similar linkage is found between predominately negative cloud-to-ground lightning and violent tornadic (F4 and F5) storms (Perez et al. 1997). The relationship between lightning polarity and severe storms appears to demonstrate significant regional variability (Carey and Rutledge 2003; Carey et al. 2003), and therefore our understanding is still considered to be at a developing stage (Carey et al. 2003). Given this knowledge gap, we did not attempt to use lightning polarity to model tornado occurrences. Rather, we used the mean annual cloud-to-ground flash density, a typical proxy for the frequency of thunderstorm occurrence and duration (Huffines and Orville 1999; Burrows and Kochtubajda 2010). This implies that we may be introducing an additional source of uncertainty in regards to our capacity to discern the actual tornado occurrence per event recorded, as our predictor is not directly related to the response variable targeted. Nevertheless, our aim is to use the climatology of CG lightning to identify areas in Canada that are prone to thunderstorms, which are typically required for the generation of tornadoes. Mean annual lightning flash density has been shown to be an ideal surrogate variable for thunderstorms (Huffines and Orville 1999), and the lightning network data are spatially richer and more accurate than human-based thunderstorm observations (Huffines and Orville 1999; Changnon 1993; Reap and Orville 1990; Changnon 1988a,b, 1989). We obtained the Canadian Lightning Detection Network (CLDN) CG lightning flash density data for the 1999–2009 period. The 11-yr CG flash density data were originally sorted into 1 × 1 km2 cells (Burrows and Kochtubajda 2010; Shephard et al. 2013). Using the gridded data, mean annual CG flash densities were calculated in three separate grids of 25 × 25 km2 (625 km2), 50 × 50 km2 (2500 km2), and 100 × 100 km2 (10 000 km2) (see also Fig. S1 in the supplementary material).

c. Population density

Initial reporting of tornado observations is typically an interactive process between the general public and the meteorological service personnel, which underscores the presence of a strong positive relationship between tornado observations and population density. However, there are a few variations of the most sensible population density expression, such as (i) the total county population density, where all of the population within a county is included, and (ii) the rural population density, where the densely populated areas (e.g., cities or towns) within a county are excluded. The rationale for using rural population density is that by excluding cities or towns that are densely populated but small in areal extent, the calculated population density provides a more accurate portrayal of the real areal coverage of a study site (e.g., King 1997; Paruk and Blackwell 1994). On the other hand, the rural population density approach would not work in cases where a substantial areal extent of the county is populated (e.g., large metropolitan areas; Anderson et al. 2007). To address this issue, one can use a dataset that has a higher spatial resolution than the county boundary units. For this reason, we obtained Canada's census subdivision boundary units (Statistics Canada 2001), which have more refined spatial resolution with over 40 political administrative boundary unit types, such as cities, towns, villages, Indian reserves, Indian settlements, and unorganized territories. They are all defined in separate entities, such that most small population centers are separated from larger rural communities. Further, since the model grid cells cover both countries at the United States–Canada border, we also obtained the U.S. 2000 census county population density. Population densities in the original boundary units were recalculated into the model grids of 25 × 25 km2 (Fig. 1, right), 50 × 50 km2, and 100 × 100 km2 (see Fig. S2 in the supplementary material) based on the areas of overlap between census subdivision units and model grid cells, assuming that the population density from the original boundary units of the census subdivision is spatially homogeneous (Goodchild et al. 1993). The spatially detailed population density from the census subdivision would only be retained in the smaller model grids, whereas when the grid size increases (e.g., 100 × 100 km2 grid), more population clusters would be averaged within one cell, having an effect similar to that of small densely populated areas for the calculation of county population density.

Fig. 1.
Fig. 1.

(left) Mean annual CG lightning flash density (number of flashes km−2 yr−1), based on data from 1999–2009 in a 25 × 25 km2 grid. (right) Areal interpolated population density (persons km−2) from Canadian census 2001 subdivision boundary in 25 × 25 km2 grid.

Citation: Journal of Climate 26, 23; 10.1175/JCLI-D-13-00093.1

3. Methodology

The tornado cell data are discrete counts that predominantly contain zeros, high observational uncertainty, and intricate spatial covariance driven by nonmeteorological and meteorological factors. These issues have made classical (frequentist) regression methods intractable. In contrast, the use of Bayesian inference techniques offers a flexible means to decompose the problem of tornado occurrence into a series of conditional models coherently linked together via Bayes' rule (Anderson et al. 2007; Wikle and Anderson 2003; Wikle 2003):
e1
where the posterior model [1] reflects our beliefs related to the relative importance of the explanatory (or predictive) factors examined and the underlying parameter values after the consideration of the available data. This in turn can be thought of as the product of the data model [2], specifying the dependence of the observed data on the processes of interest and parameters, with the explanatory model [3] describing the nature of the relationship between response variable and explanatory factors, and the parameter model [4] quantifying the uncertainty in parameter values. This formula will serve as the basis for estimating the probability of tornado occurrence.

a. Data model

Our model postulates that the population density is the primary factor that determines the accuracy of tornado observations in our dataset. In particular, we assume that there is a threshold population density above which all tornadoes are expected to be observed, whereas below this threshold level the probability of tornado detection is proportional to the population density (Newark 1983; King 1997; Anderson et al. 2007). Similar to the Anderson et al. (2007) study, we specify a binomial model in which the observed tornado counts Tobsi for each grid cell i are conditioned upon the actual (but unobserved) tornado occurrences Tlatenti and the probability of detection pi(β):
e2
The probability of detection pi(β) represents the likelihood to observe a tornado and is associated with the population density yi by the following exponential expression:
e3
where β is the population effect parameter and exp(yi) is the exponential transformation of the original population density data. The actual occurrence of tornadoes Tlatenti in the model domain is specified as a Poisson process, conditional on the average or expected tornado occurrence rate per grid cell λi, provided by the predictive model
e4

b. Explanatory model

The explanatory (or predictive) model expresses the logarithm of the expected tornado rate λi in given grid cell i as a linear function of the corresponding mean annual CG flash density xi:
e5
where ai is a gridcell-specific random effect, capturing the residual variability of the tornado frequency in grid cell i, stemming from other explanatory factors/processes unaccounted for by the model. For example, Brooks et al. (2003b) found that certain thresholds of convective available potential energy and vertical wind shear at the 0–6-km layer show a strong correlation with the severity of thunderstorms, and this pattern cannot be explained by the mean annual lightning flash density alone. Moreover, the inclusion of ai aims to address the possibility that the signature of the lightning flash density may not be consistently evident in the tornado frequency records throughout the model domain. That is, the globally common parameterization of our predictive model may be violated in smaller geographic areas due to, for example, orographic-influenced thunderstorms (Taylor et al. 2011) or lake-breeze convergence-influenced convective processes (King et al. 2003). For the same reason, it is reasonable to assume that the random effects of the unaccounted for factors have a regionalized/localized character and thus are spatially correlated.
The characterization of the spatially correlated random terms ai was based on the Bayesian conditional autoregressive (CAR) model (Besag et al. 1991). The random error terms are jointly distributed as a multivariate normal distribution with mean 0 and an unknown covariance matrix (Besag and Kooperberg 1995). In particular, the model postulates that the spatial random effect in cell i depends only on the neighboring cells of i (Ni) and that all of the neighbors have equal influence (weight of 1) on i. The term ai is defined by the conditional normal distribution ai ~ N (μi, σ2/ni), where
e6
and ni is the number of adjacent grid cells. Because we used a first-order neighborhood approach and squared cells, Ni represents the eight immediately adjacent cells.

c. Parameter model

In Bayesian inference, model parameters are treated as random variables rather than fixed quantities. As such, prior distributions can be formulated to depict our knowledge on the relative plausibility of their values before the consideration of the observed data (Gelman et al. 2004). Here, we opted for “noninformative” or “flat” priors reflecting no prior knowledge of the model parameters. In particular, the prior distributions of β, α0, α1, and σ were specified as follows:
e7
e8
e9
e10
where N and IG denote the normal and inverse gamma distributions for the regression coefficients and conditional variance among the spatial random error terms σ2, respectively.

A sequence of realizations from the model posterior were obtained using Markov chain Monte Carlo (MCMC) simulations. Specifically, we used the general normal proposal Metropolis algorithm as implemented in the WinBUGS software (Lunn et al. 2000); this algorithm is based on a symmetric normal proposal distribution, whose standard deviation is adjusted over the first 4000 iterations such that the acceptance rate ranges between 20% and 40%. We collected 20 000 samples from two independent chain runs for each model configuration. The posterior statistics were calculated using a thin of 10, which yielded a sample size of 4000 for all the model configurations considered. The entire modeling process is undertaken in three grid sizes (25-, 50-, and 100-km grid squares) separately in order to assess model sensitivity to the selection of the grid size.

4. Results and discussion

Posterior means and standard deviations of the model parameters for all three model grids are shown in Table 1. The discussion of our results refers to the 25-km model, while the comparison of the inference drawn among the different grid configurations is presented in a following section.

Table 1.

Posterior means and standard deviations (SD) of the parameters and model predictions.

Table 1.

a. Comparison between predicted tornado occurrence and tornado observations

The total predicted tornado occurrence (calculated as the total sum of the posterior means of Tlatenti) for the entire 30-yr period of our study for all three model grids are at least twice the number of recorded tornadoes, based on the counts of tornado points and paths by grid cells (Table 1; Fig. 2). Tornado observations were on average close to 70 yr−1, whereas the model-predicted tornado occurrence was close to 150 yr−1 (Table 1). Some areas with no tornado observations in the 30-yr study period, for example, northern parts of Alberta and Manitoba and northwestern Ontario, are predicted to have a low (~0.10 tornadoes per 10 000 km2) probability of annual tornado occurrence with the 25-km grid model. Similarly, the southern border of Alberta and Saskatchewan are predicted to experience even higher probabilities of tornado occurrence [~(0.5–1.5) tornadoes per 10 000 km2]. Considering the sparse population density and the lack of radar coverage in some of those areas, the discrepancy between observed and predicted tornado occurrence is not surprising. Generally, we note that extensive areas across the prairies are corrected by our model, whereas the more heavily populated areas in the Edmonton–Calgary corridor, Regina, Saskatoon, and Winnipeg are subjected to a lower observation error (Fig. 2). Adjustments also continue eastward to northwestern Ontario and to a lesser extent in the area between northeastern Ontario and Quebec (Fig. 2). There are no major adjustments in southern Ontario or southern Quebec because of the more reliable tornado detection assumed by the much denser population.

Fig. 2.
Fig. 2.

(left) Gridded observed tornado counts standardized to 10 000 km−2 yr−1. (center) Posterior mean of tornado occurrence standardized to 10 000 km−2 yr−1. (right) Posterior standard deviation of tornado occurrence standardized to 10 000 km−2 yr−1. (top)–(bottom) The 25 × 25, 50 × 50, and 100 × 100 km2 models, respectively.

Citation: Journal of Climate 26, 23; 10.1175/JCLI-D-13-00093.1

In a similar manner, the prairies were characterized by the highest standard deviation values of the true tornado occurrences Tlatenti (Fig. 2, right panels). The values decrease proportionally as we move northward to areas with lower predicted tornado occurrences. Further, our analysis suggests that the standard deviations slightly increase as the grid size decreases, which is an expected result as the posterior means also increased slightly with the decrease of the gridcell size. Conversely, in the most populated areas, most notably from southern Ontario to southern Quebec, the standard deviations are zero. That is, the predicted tornado occurrences are simply equal to Tobsi, given that our model postulated high tornado detection efficiency in densely populated areas.

b. Importance of the lightning covariate, population density, and spatial random effects

The posterior estimate for the parameter α1 is 1.53 ± 0.19, indicative of a distinct relationship between the CG flash density and the underlying expected tornado rate (Table 1). The random effects term ai, representing explanatory factors–processes unaccounted for by the model, demonstrates strong spatial covariance with the Tlatenti posterior estimates, in that positive ai values typically coincide with higher Tlatenti and vice versa (Fig. 3). The main exception to this spatial pattern was in southwestern Ontario, where despite the higher tornado occurrence rates relative to the central prairies, the relative ai values are distinctly lower, especially when a larger gridcell size is used (see Figs. S3,S4 in the supplementary material). It is also worth noting that southwestern Ontario experiences the most intense CG lightning flash densities in Canada (Fig. 1, right). We believe that the lower ai values for southwestern Ontario could be a reflection of the nature of the two structural components of our model. First, the lower ai values could stem from the underlying assumption that the lightning–tornado relationship is identical throughout the study domain. Namely, the assumption of a globally common α1 value may oversimplify the regional variability in the nature of the explanatory linkage between CG lightning densities and tornado frequency, which in turn it is depicted by the systematic trends of the ai values in the central prairies and southern Ontario. The problems arising from a geographically constant lightning–tornado relationship can be exemplified by one facet of the spatial variability of the lightning densities: namely, the length of lightning season. Different regions may experience similar mean annual lightning densities, but the length of the lightning season can be significantly different, reflecting the potential differences in the underlying thermodynamic processes (Burrows and Kochtubajda 2010). For example, the Pacific coastal region and southern Nova Scotia appear to have higher lightning flash densities due to a virtual year-round lightning season, as winter lightning commonly occurs when Arctic air masses pass over much warmer water (Burrows and Kochtubajda 2010). It is significantly different than in the Edmonton area, where over 95% of the annual lightning occurred in May–August (Burrows and Kochtubajda 2010). Hence, a different lightning–tornado relationship may be expected between these regions.

Fig. 3.
Fig. 3.

Posterior mean of the conditional autocorrelation coefficient ai on a 25 × 25 km2 model grid.

Citation: Journal of Climate 26, 23; 10.1175/JCLI-D-13-00093.1

Second, the spatial patterns of the ai posteriors could similarly be a reflection of the assumption of globally common population effects on the likelihood to observe a tornado pi(β). Possible regional population bias may exist because of differences in the quality of the monitoring networks, number of trained weather spotters, training of meteorological service staff, and general public awareness to tornadoes, but these factors are not explicitly considered in the model (Anderson et al. 2007). The omission of this potential bias may influence the probability of tornado detection and subsequently the uncertainty estimates of Tlatenti. In such a case, it is conceivable that the estimation of the random terms ai may offset this systematic error, regardless of the CG lightning flash densities.

An appealing approach to investigate the effects of potential region-specific relationships is a Bayesian hierarchical configuration of the present model. Under this framework, the assumption of globally common relationships is relaxed, and the model is dissected into levels (hierarchies) that explicitly account for the role of significant sources of spatial variability (e.g., geographical locations, climatic regimes, particular features of regional monitoring networks, road density, or landscape), thereby allowing for site-specific parameter estimates (Gelman and Hill 2007; Cheng et al. 2010). Finally, the posterior estimate of the model intercept α0 is −3.536 ± 0.201 (or 0.030 ± 0.006 in the original scale), suggesting that the baseline expected tornado rate across the study area, when we account for the flash density and the spatial random effects, is very low (Table 1).

The posterior estimate of the population effect parameter β is 3.129 ± 0.143 (Table 1). Substituting the mean value back to Eq. (2), we infer that the probability of tornado detection pi(β) is approximately equal to 1 when the population density reaches the level of 6 individuals km−2 (Fig. 4, left). In effect, the model allows identifying a population threshold when all tornadoes can be observed, and thus the variables Tlatenti and Tobsi are identical. Interestingly, our analysis suggests a population threshold similar to the value reported by King (1997) using data from southern Ontario. When the population density is below 6 individuals km−2, the tornado observational uncertainty is partly reflected in the uncertainty estimates of Tlatenti. As the Tlatenti posteriors are also dependent on the meteorological submodel, it is worth noting that the standard deviations of Tlatenti stem from the relationship between lightning and tornado frequency as well as the population density effects on the probability of tornado detection (Fig. 2).

Fig. 4.
Fig. 4.

(left) Probability of tornado detection estimated from the model as a function of population density. (right) Percentile of the gridded population density. Models on a 25 × 25, 50 × 50, and 100 × 100 km2 grid are shown in dashed, gray, and black solid line, respectively. The three inset panels show an expanded y axis for percentiles from 0% to 90%.

Citation: Journal of Climate 26, 23; 10.1175/JCLI-D-13-00093.1

c. Model sensitivity to gridcell size

Posterior means of the population effect parameter β (Table 1) are quite consistent across the three model gridcell sizes used, although a slight increase for the posterior standard deviation is observed with the coarser spatial resolution. This is not surprising as the population densities calculated for larger grid cells are less accurate compared to the smaller grid cells. The posterior means of the parameters α0 and α1 are larger as the gridcell size increases (Table 1), which simply stems from the scale-dependent units of Tobsi (counts of tornadoes per cell) relative to the normalized units of the mean annual CG lightning flash density (flash km−2). The conditional variance σ of the random terms αi is similar for the 25 × 25 and 50 × 50 km2 grids but increases slightly with the 100 × 100 km2 grid. Some notable differences among the three grid configurations are as follows: (i) as the grid size increases, the proportion of cells with zero tornado observations is lower (i.e., 93%, 87%, and 80% of cells for the 25 × 25, 50 × 50, and 100 × 100 km2 grids, respectively); (ii) the population density is somewhat higher with the coarser resolution (e.g., the proportion of cells with population density greater than 6 individuals km−2 is 6%, 7%, and 9% for the 25 × 25, 50 × 50, and 100 × 100 km2 grid cells, respectively; Fig. 4); and (iii) each population center is associated with Tlatenti projections that are spread over a larger areal extent with the coarser resolution relative to the finer grid, and thus the predicted tornado occurrence rates are higher with the smaller grid size (Fig. 5). Overall, our analysis shows that the predictive statements supported by the model are robust and do not depend strongly on the grid sizes, although the smaller gridcell resolution more accurately portrays the actual population densities and therefore better represent the probability of tornado occurrence.

Fig. 5.
Fig. 5.

Posterior mean of Tlatenti on 25 × 25, 50 × 50, and 100 × 100 km2 model grids overlaid on the original census subdivision boundary layer with the population density (person km−2) of each census subdivision shown with gray scales. The gray scale for Tlatenti is the same as those in Fig. 2.

Citation: Journal of Climate 26, 23; 10.1175/JCLI-D-13-00093.1

d. Extent of tornado occurrence probabilities

Based on the Tlatenti predictions with the 25 × 25 km2 grid configuration (Fig. 2), we used the ordinary kriging method (see, e.g., Ray et al. 2003) to create a map that depicts a smoothed pattern of the regions with different tornado occurrence rate (Fig. 6). Based on the full range of tornado occurrence rate and with reference to other Canadian studies (Newark 1984; Hage 2003), we identified four levels [≥0.05, ≥0.10, ≥1.0, and ≥2.0 (per 10 000 km2 yr)] to classify regions of very low, low, medium, and high probability of tornado occurrence, respectively. The very low probability areas begin from central British Columbia in the west and move northward to northern Alberta, before dipping southward in Saskatchewan and extend northward again to the northeastern tip of Manitoba. The same very low probability area extends to northern Ontario, southern Quebec, eastern New Brunswick, and Prince Edward Island. The low probability areas follow a smaller in extent but similar pattern, except for a discontinuity north of Lake Superior. The medium probability areas have a distinctly different pattern. The area starts near the foothills of the Alberta Rocky Mountains and extends eastward before dipping south to central Saskatchewan and ending near southern Manitoba. For the eastern part of Canada, the area of medium probability of tornado occurrence begins between Georgian Bay and Lake Ontario in southern Ontario and extends across southwestern Ontario, where there are also some regions with higher probabilities. Some of the significant tornado events that have occurred in the medium probability areas are listed in Table 2. One of the high probability areas of tornado occurrence resides in the central prairies, with a broad clockwise rotated “J-shaped” region over south-central Saskatchewan and southern Manitoba. The highest tornado occurrence predictions within this region are found in the center of southern Saskatchewan, southeastern Saskatchewan, and southwestern and southeastern Manitoba, with the latter three lying just north of the broad “C-shaped” tornado alley in the central plains of the United States (Brooks et al. 2003a). Because of the absence of Canadian data along with the smoothing effect of the Brooks et al. (2003a)'s study, the C-shaped tornado alley did not extend to the northern plains/Canada–United States border (e.g., Dakotas and Montana). Our analysis suggests that the broad C-shaped tornado alley likely extends north into Manitoba and Saskatchewan. Other high probability areas are located within southern Ontario. One area can be found in the southernmost part of Canada near Windsor, Ontario, and appears to be a northeastern extension of the aforementioned tornado alley identified by Brooks et al. (2003a). The other area is located in the inland area of southern Ontario between Lakes Erie, Huron, and Ontario. This region is largely influenced by the lake-breeze circulations generated by these lakes; the tornado activity tends to be suppressed in regions near the lakes regularly visited by lake-modified air, while it is enhanced at inland locations along lake-breeze fronts as well as at sites where lake-breeze fronts interact (King et al. 2003). All the high probability areas have experienced deadly tornadoes in the past (Table 2).

Fig. 6.
Fig. 6.

Predicted tornado occurrence (10 000 km−2 yr−1), tornado probability (yr−1), and the return period (yr), based on the posterior mean values of Tlatenti of the 25 × 25 km2 model. Asterisk implies that the median tornado-affected area (F0–F5) of 0.10 km2 from the Schaefer et al. (1986) records is used. An inset map of southwestern and south-central Ontario is shown in the upper right corner.

Citation: Journal of Climate 26, 23; 10.1175/JCLI-D-13-00093.1

Table 2.

Regions of ≥1 and ≥2 tornado occurrences per 10 000 km2 × yr−1 and historically significant tornado events occurred within each region.

Table 2.

When tornado-affected areas are known, tornado occurrence in area and time (10 000 km−2 yr−1) can be expressed as tornado probabilities PT in unit per time (yr−1). Tornado probabilities are typically calculated as follows:
e11
where l is the length of the tornado t, w is the width of the tornado t, A is the area of the grid, Y is the number of years, and T is the number of tornadoes in area A. However, for most of the observations in this dataset, the tornado-affected areas are unknown, and thus it would be impossible to use Eq. (11). In an attempt to predict tornado probabilities, we assume our sampled tornado-affected areas are similar to the large dataset of Schaefer et al. (1986). We applied their sampled median affected areas of all tornadoes (F0–F5) of 0.10 km2 along with our Tlatenti to predict tornado probabilities, which are calculated as
e12
Based on this assumption, the tornado probabilities and the return periods are directly proportional to the predicted tornado occurrences (Fig. 6): for example, areas where the tornado occurrences are predicted to be 1 tornado 10 000 km−2 yr−1 have a tornado probability of 10−5 yr−1, which is equivalent to a return period of 105 yr. Our results in tornado probabilities can be compared with the F0–F5 projections reported by Schaefer et al. (1986). In Schaefer et al. (1986), the areas with tornado probabilities greater than 10−6 yr−1 covered most of the United States east of the Rocky Mountains. Based on our results, the same frequency areas should extend well northward to the Canadian plains, east of the Alberta Rocky Mountains, and also to northwestern Ontario. The same frequency also characterizes areas that extend from southwestern Ontario northward to the east of Lake Superior and eastward to southwestern Quebec and New Brunswick. Regarding the tornado probabilities of greater than 10−5 yr−1, Schaefer et al. (1986)'s projections lie closer to southwestern Ontario than to the Canadian Prairies. However, our result suggests that a significant portion of the Canadian Prairies close to the U.S. northern prairies regions should also be included, as well as a much larger portion of southwestern Ontario (i.e., from Windsor, Ontario, extending to the east of Georgian Bay).

5. Conclusions and future perspectives

Newark (1984) provided a first national view of the tornado climatology of Canada. In that study, the tornado observations were assumed to be reliable in areas with population density ≥1 person km−2, while areas with lower population density were extrapolated based on subjective meteorological knowledge. In this study, we developed a Bayesian modeling approach founded upon the explicit consideration of the population sampling bias in tornado observations and the predictive relationship between cloud-to-ground lightning flash climatology and tornado occurrence, in order to predict the probability of tornado occurrence in Canada. The key findings of our analysis are as follows:

  • There is a population density threshold of 6 individuals km−2, below which there is uncertainty in the detection of tornadoes. Nonetheless, regional variability may be associated with differences in the quality of the monitoring networks, number of trained weather spotters, training of meteorological service staff, and general public awareness.

  • Mean annual CG lightning density is an important meteorological covariate for partially explaining tornado occurrence, although substantial spatial variability exists in regards to the strength of this predictive relationship.

  • In sparsely populated areas, our analysis shows that the probability of tornado occurrence is significantly higher than what is represented in the 30-yr data record. Areas with low population density but high lightning flash density demonstrate the greatest discrepancy between predicted and observed tornado occurrence.

  • The total predicted tornado occurrences for the entire 30-yr period of our study were at least twice the number of recorded tornadoes, based on the counts of tornado points and paths by grid cells. Tornado observations were on average close to 70 yr−1, whereas the model predicted tornado occurrence close to 150 yr−1.

The sensitivity analysis shows that the predictive statements supported by the model are fairly robust to the grid configuration; hence, the modeling framework should be a sound foundation for objectively defining tornado-prone areas for the National Building Code of Canada. The ability of the model to delineate the importance of meteorological factors and the observational error associated with the population density lays the groundwork for a more detailed investigation of these relationships. Future research goals would be to examine additional lightning flash properties (e.g., lightning polarity, total lightning density, multiplicity, and first stroke peak current) and other atmospheric meteorological parameters (e.g., convective potential available energy and deep-level wind shear) that could potentially be relevant to tornado occurrence and their intensity (Brooks et al. 2003b). Moreover, future investigation of the lightning–tornado relationship via a Bayesian hierarchical framework will be valuable in elucidating the underlying drivers of spatial variability. Similarly, this modeling framework would be helpful in identifying potential regional population bias associated with the likelihood of tornado detection. Such a hierarchical configuration can conceivably augment the capacity of tornado occurrence models to effectively support hazard assessment (Banik et al. 2007), as the realistic representation of spatial heterogeneity is critical in developing tornado hazard maps (Tan and Hong 2010). Further, manufactured structures, such as mobile homes and school portables are vulnerable even to weaker (F0 and F1) tornadoes (Ashley 2007; Sutter and Simmons 2010), whereas critical and hazardous facilities, such as hospitals and nuclear plants, may be subjected to a greater risk with more intense (F2 and higher) tornadoes. It is thus critical to develop models for various tornado intensities, so that the associated risks with different types of infrastructure can be evaluated. Such a modeling exercise will be presented in a future study.

From a socioeconomic perspective, one of most direct impacts of tornadoes is their financial cost on individuals and infrastructure. The large financial losses suffered from recent tornadic events, such as the Goderich 2011 (CAD 75 million) and the Leamington 2010 (CAD 120 million) Ontario tornadoes (Insurance Bureau of Canada 2012), have shown that the destruction of infrastructure and losses to individual owners and insurers can be overwhelming. These events have highlighted the importance of developing tornado forecasting tools and risk information systems for engineering and financial loss models that will improve our capacity to cope with damage control and minimize the societal risk of tornadoes. Canada is undergoing significant population growth and expansion, centering mainly in Alberta and southern Ontario, and thus it is expected that societal impacts of tornadoes will increase. It is critical that the tornado risk to Canada's society is properly assessed, for example, through tornado casualty models, and that adequate tornado vigilance from the Meteorological Service of Canada, emergency groups, and by the general public are in place. Our predictions of tornado occurrence should be essential in integrating tornado risk with future socioeconomic studies.

Acknowledgments

Vaisala Inc. of Tucson, Arizona, processed the Canadian Lightning Detection Network data and provided telecommunication services to Environment Canada. We thank Bill Burrows (Environment Canada) for processing and providing the 1 × 1 km2 gridded lightning data.

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