Severity scale for tornadoes

Abstract

Tornadoes are the most destructive winds created by nature. Sometimes tornadoes are strong enough to destroy most things in their path. These types of tornadoes are few compared to all tornadoes confirmed/reported. Currently, existing scales describe severity levels in terms of intensity/magnitude, and they are not sufficient to clearly distinguish the severity level. Several discrepancies between various sources of information complicate the interpretation of trends in tornado data. As a solution to these inconsistencies, a technique is required to compare the severity level of tornadoes. Impact factors, such as the number of fatalities, number of injuries, number of homeless, affected population, affected area, and cost of damage, can be considered to evaluate the severity levels. Prior experience, preparedness, awareness, evolving technology, mitigation methods, and early warning systems may minimize the number of fatalities and injuries. Models are used to identify which of the above factors should be considered in a severity scale to indicate the seriousness of tornadoes. However, the lack of data prevents an in-depth analysis of tornado severity. Extreme value theory is used to study potential severity levels of tornadoes. This paper attempts to develop an initial severity scale for tornadoes, which is a primary stage to develop a multi-dimensional severity scale. This common scale provides the criteria to rank tornadoes and allows the impact of one tornado to be compared to the impact of another tornado. Further, the scale allows the impact of a tornado to be compared to any type of natural disaster that occurs.

Appendix 1: Extreme value distribution and three models of extremes

EVDs are the limiting distributions for the largest or the smallest value of a large collection of random observations from the same arbitrary distribution. Three different methods, block maxima, rth-order statistics, and peak over threshold, can be used to determine the extreme values from a given data set. Annual maxima are the most commonly used values in block maxima, where blocks are defined as a particular year. However, blocks can vary according to the nature of a specific data set in both block maxima and rth-order models. Each method has advantages and disadvantages. Selecting the order is a tedious task in rth-order method in addition to selecting proper blocks. Selecting a threshold value is difficult in the threshold method. In the peak-over-threshold model, all extreme values are included, and they do not need any artificial blocking. In contrast, the peak-over-threshold model is bounded below because the minimum extreme is higher than the threshold selected; consequently, it is only concerned with high extreme values. In the block maxima and rth-order statistic models, a variation of extremes are included; therefore, extreme values, which are not necessarily bounded, can consist of extremes with a full range of data and do not need a minimum threshold value. In contrast, some extremes are neglected in these two methods. Different data sets are selected as extremes according to different methods, as demonstrated by the comparison of Figs. 14, 15 and 16; hence, the results vary. Figure 14 shows five blocks and selected extreme x _i values in the block maxima method; Fig. 15 shows selected extreme x _i values in the second-order statistic (i.e., r = 2) model for the same blocks; and Fig. 16 shows that the selected extreme x _i values exceed the threshold value, u, in the peak-over-threshold method according to the same example data set.

Fig. 14

Extreme values of the block maxima model

Fig. 15

Extreme values of the second-order statistic model

Fig. 16

Extreme values of the peak-over-threshold model

1.1 Appendix 1.1: Methods applied to select the probability distribution functions of block maxima

According to this approach, there were 87 tornado outbreaks identified in North America from 1996 to 2013; the tornado with the maximum fatalities recorded in each outbreak is considered in the extreme value analysis to evaluate the extreme value distribution of tornado fatalities. Maximum likelihood estimators (MLE) are used to determine the distribution parameters using the Xtremes 4.1 software (Reiss and Thomas 2007). A visual comparison between PP plot, QQ plot, and return level plot is used to determine the best-fitted distribution for the data using the estimated parameters of the distribution. The mathematical equations of Gumbel, Frechet, and Weibull distributions (Kotz and Nadarajah 2000) are shown in Eqs. (6), (7), and (8). The best-fitted distribution for fatalities is Frechet (α = 1.14213, µ = 0, σ = 2.01878). Figure 17 demonstrates the return level plot for the fitted distribution. It can be estimated that a tornado that causes 100 fatalities has a return number of 85 outbreaks, and a tornado that causes 150 fatalities has a return number of 135 outbreaks. In this study, the rth-order statistic model is not used, as there was enough data (87 tornado outbreaks) for a block maxima model.

$$ F\left( {x_{i} } \right) = \exp \left( { - {\text{e}}^{{ - \left( {\frac{{x_{i} - \mu }}{\sigma }} \right)}} } \right)\,\,{\text{for}}\,\, - \infty \,< \, x_{i} < \infty \,\, {\text {and}} \,\, \sigma > 0;$$

(6)

$$ F\left( {x_{i} } \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {\,\,{\text{for}}\,\,x_{i} < \mu } \hfill \\ {\exp \left( { - \left( {\frac{{x_{i} - \mu }}{\sigma }} \right)^{ - \alpha } } \right)} \hfill & {\,\,{\text{for}}\,\,x_{i} \ge \mu } \hfill \\ \end{array} } \right.\,\, {\text{and}}\,\, \alpha ,\sigma > 0; $$

(7)

$$ F\left( {x_{i} } \right) = \left\{ {\begin{array}{*{20}l} {\exp \left( { - \left( {\frac{{x_{i} - \mu }}{\sigma }} \right)^{\alpha } } \right)} \hfill & {\,\,\,{\text{for}}\,\,x < \mu } \hfill \\ 1 \hfill & {\,\,\,{\text{for}}\,\,x \ge \mu } \hfill \\ \end{array} } \right.\,\,\, {\text{and}}\,\,\, \alpha ,\sigma > 0; $$

(8)

Fig. 17

Return level plot

1.2 Appendix 1.2: Different methods applied to select the threshold value u ₀ in the peak-over-threshold model

A mean residual plot aids to estimate the threshold, u ₀. For u > u ₀, E (X − u/X > u) is a linear function of u and E (X − u/X > u) is the mean of the values that exceed the threshold, u, for which the sample mean of the threshold values above u provides an empirical estimate. These estimates are expected to change the linearity of E (X − u/X > u) at some value of u along the u-axis. The value of u at which linearity changes is the suitable threshold value for which the generalized Pareto model is appropriate (Coles 2001). The mean residual plot equals $\left\{ {\left[ {u,\frac{1}{{n_{u} }}\sum\nolimits_{i = 1}^{{n_{u} }} {\left( {x_{\left( i \right)} - u} \right)} } \right]:u < x_{\hbox{max} } } \right\}$, where x (1), …, x (n _u ) consists of the n _u observations that exceed u. Figure 18 shows the mean residual plot for the number of fatalities. The graph is approximately linear from u = 0 to u = 18; it is also approximately linear after u = 18, but with a steep slope. This change in slope suggests that u ₀ = 18 is a threshold; however, only nine values exceed the threshold u = 18, and consequently, there are too few data points to make meaningful inferences.

Fig. 18

Mean residual plot

Another method to select the threshold is to use the ratio of k to n (k/n), where k is the number of tail data and n is the total number of data points; k/n should be equal to 0.02 for 50 < n ≤ 500 and k/n should equal 0.1 for 500 < n < 1000 (Boos 1984). According to this method, k is 10 because n = 500. Consequently, the threshold, u ₀, is 16; however, there are just ten values that exceed the threshold u u ₀ = 16, which is too few to make meaningful inferences. Hasofer (1996) suggests that k can be taken to equal approximately 1.5 $\sqrt n$. In that method, k is equal to 33.54. When the threshold is equal to 7, 38 data points exceed the threshold. However, the threshold should be selected by balancing biasness and variance. Therefore, the finite-sample mean-square error (MSE) method for threshold value selection (Caers and Maes 1998), which is a summation of both square bias and variance [Eq. (9)], was chosen as the best method for assessing the stability of parameter estimates based on fitting the models across a range of different thresholds. Minimizing the MSE would balance the biasness and variance because the MSE is the sum of both the square bias and variance.

$$ \begin{aligned} {\text{MSE}}\left( {\hat{\theta }} \right) & = E\left[ {\left( {\hat{\theta } - \theta } \right)^{2} } \right] = \left( {E\left[ {\hat{\theta }} \right] - \theta } \right)^{2} + E\left[ {\left( {\hat{\theta } - E\left[ {\hat{\theta }} \right]} \right)^{2} } \right] \\ & = {\text{bias}}^{2} \left( {\hat{\theta }} \right) + \text{var} \left( {\hat{\theta }} \right); \\ \end{aligned} $$

(9)

Several different estimation methods have been used to estimate parameters; parameters of the distribution need to be known to estimate MSE. MLEs are used to determine the distribution parameters of the general Pareto distribution (GP); MLE and Hill estimators are used to estimate the Pareto distribution (GP1) using the Xtremes 4.1 software. In addition, EasyFit 5.0 Professional software is also used to estimate the GP parameters. Figure 19 shows that the sample MSE for different threshold values, which are calculated using the inverse GP and inverse GP1 equations and Eq. (10), is used to calculate the MSE values. Inverse GP and GP1 equations are shown in Eqs. (11) and (12), respectively (Reiss and Thomas 2007). Figure 19 indicates that the calculated MSE values for GP and GP1, using MLE, are identical; hence, the data follows the Pareto distribution. Figure 19 demonstrates that the minimum MSE is obtained when the threshold equals one for all estimators. Consequently, if more than one fatality occurred during a tornado outbreak, that tornado is an extreme event. Although EasyFit 5.0 professional software gives the minimum MSE most of the time, Table 10 shows that Xtreme 4.1 calculates the MLE, which yields a minimum MSE of the threshold equal to one. Therefore, 172 tornadoes exceed one fatality and follow the Pareto distribution with a shape parameter (α) = 0.878, scale parameter (σ) = 0.308, and location parameter (µ) = 0.963. Tornadoes with more than one fatality, which is 172 tornadoes, are considered to estimate the parameters of threshold u ₀ = 1.

$$ {\text{MSE}} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {\hat{Y}_{i} - Y_{i} } \right)^{2} ; $$

(10)

$$ F^{ - 1} \left( q \right) = \left[ {1 - \left( {\frac{x - \mu }{\sigma }} \right)} \right]^{{ - \frac{1}{\alpha }}} ,\,\,\,\,\,\,\alpha > 0; $$

(11)

$$ F^{ - 1} \left( q \right) = \frac{{\left\{ {\left[ {1 - \left( {\frac{x - \mu }{\sigma }} \right)} \right]^{ - \gamma } - 1} \right\}}}{\gamma } , \,\,\,\,\,\gamma \ne 0. $$

(12)

Fig. 19

Mean square errors for different threshold values

Table 10 MSE values for thresholds equal to one