Large-scale Indicators for Severe Weather

Transcription

1 Large-scale Indicators for Severe Weather Eric Gilleland Matthew Pocernich Harold E. Brooks Barbara G. Brown Patrick Marsh Abstract Trends in extreme values of a large-scale indicator for severe weather (specifically, convective available potential energy (CAPE) multiplied by 0-6 km wind shear (shear)) are investigated using the generalized extreme value distribution with trends in the location parameter. The study primarily looks at reanalysis observational data for the entire globe, but also performs an initial analysis of a regional climate model (CCSM3) over the United States. Results for global trends from the reanalysis data set are similar to those found previously for the frequency of high values of this indicator. Comparison of the reanalysis data over the United States with the CCSM3 output show numerous discrepancies, some of which are known problems with both the reanalysis and climate model output for precipitation. Key Words: Extreme values, GEV, severe weather, climate models, reanalysis data 1. Introduction Severe weather typically occurs on fine scales that cannot currently be resolved by the largescale climate models. Past studies on climate change have been focused primarily on average weather conditions such as mean temperatures, but more recently concern has arisen regarding the impact of climate change on more severe weather phenomena (e.g., tornados, hurricanes, hail storms, strong winds, etc.) as these types of phenomena can have huge impacts on society in terms of both lives and economic impacts. In order to glean information about extremes under a changing climate, one approach is to investigate large-scale indicators of severe weather. That is, are there variables that can be resolved by existing climate models that can be used to make inferences about the intensity and/or frequency of severe weather for different climate scenarios? It is known that concurrently high values of convective available potential energy (CAPE, J/kg) and 0-6 km wind shear (m/s, henceforth shear) weakly discriminate between types of storms (e.g., Brooks et al., 2003; Rasmussen and Blanchard, 1998) indicated in table 1. Figure 1 shows discrimination plots for different categories of severe storms (table 1) for the product of CAPE and shear. 1 Figure 2 shows density scatter plots for shear against CAPE, which makes it clear that high values of these two variables rarely occur simultaneously. Exploration of the trends in the frequency of threshold exceedances for the product of CAPE and shear as well as the intensity of this variable have been explored using a global reanalysis Research Applications Laboratory, National Center for Atmospheric Research, 3450 Mitchell LN, Boulder, CO, 80301, EricG@ucar.edu Research Applications Laboratory, National Center for Atmospheric Research, 3450 Mitchell LN, Boulder, CO, National Severe Storms Laboratory, Norman, OK Research Applications Laboratory, National Center for Atmospheric Research, 3450 Mitchell LN, Boulder, CO, University of Oklahoma, Norman, OK 1 The product of CAPE and shear has statistical advantages as well as easier discriminatory properties over looking at the two bivariately.

2 Figure 1: Discrimination plots for categories of severe storms as described in table 1. Probability density function (pdf) graphs are shown in the top panel, and cumulative distributijon function (cdf) graphs in the bottom panel; both stratified by storm severity category. Table 1: Definitons for categories of severe storms as shown in figure 1. Non-severe hail < 1.9 cm. (3/4 in.) diameter winds < 55 kts. no tornado Severe Hail 1.9 cm. diameter winds 55 kts. and < 65 kts. or tornado Significant Hail 5.07 cm. (2 in.) diameter Non-tornadic Winds 65 kts. Significant Same as sig. tornadic with F2 (or greater) tornado. Tornadic

3 Figure 2: Density scatter plots of shear vs. CAPE.

4 dataset with this variable derived from the existing data set (Pocernich et al., 2008). Here, the focus rests on issues pertaining to the analysis of the intensities of concurrently large values of these variables. We describe the global reanalysis data and climate model output used here in section 2, followed by an overview of the statistical methods in section 3. Section 4 gives the initial findings from evaluations on the global reanalysis data, and section 5 gives the results for the climate model output. Finally, discussion of future work is given in section Measurements Attention is given to an observational data set consisting of global reanalyses of radio soundings as well as climate model output over the the Continental United States from a global climate model. These sets of measurements are described in the following two subsections. 2.1 Global Reanalysis Observations The reanalysis data are on a o o lon-lat grid with over 17 thousand points, and temporal spacing every 6 hours for 42 years ( ). Further details about the reanalysis data can be found in Brooks et al. (2003). 2.2 Global Climate Model Output Initial exploration of these variables from the current climate as output from the CCSM3 model is also underway. Initially, the output is for 756 grid points at 1.4 o 1.4 o resolution over the United States. 3. Statistical Methods Because the focus of the present study is on the behavior of large values of a process, it is of interest to investigate using extreme value analysis (EVA). We describe the general models applied here in the next subsection, and discuss estimation subsequently. 3.1 Extreme Value Models Similar to the central limit theorem for sums, the maxima for a sample of independent and identically distributed random variables follow asymptotically one of three types of distributions. Provided, of course, that the limiting distribution is non-degenerate. These three types can be written as a single family of extreme value distributions, known as the generalized extreme value (GEV) distribution, and given by { G(z) = exp (1 + ξ } (z µ)) ξ +, (1) σ where µ, ξ (, ), σ > 0 are parameters, and y + = 0 if y 0. The shape parameter, ξ, determines the type of the distribution where ξ = 0 is the light-tailed Gumbel distribution defined by continuity, ξ < 0 gives the Weibull distribution with bounded upper tail at µ σ/ξ, and ξ > 0 yields the heavy-tailed Fréchet distribution with bounded lower tail at µ σ/ξ. Similar results hold for exceedances over thresholds, but such models are left here for future work.

5 When interest is in modeling the GEV distribution (1), one is most often interested in the extreme quantiles, referred to as return levels in this context. Because (1) is invertible, the 1 p quantiles, z p, are easily obtained as z p = { [ ] µ σ ξ 1 log(1 p) ξ, ξ 0 µ σ log (log(1 p)), ξ = 0 (2) In order to analyze trends, or incorporate covariate information, it is natural to model them within the parameters themselves. Typically, models of the following form are considered. n µ µ(t) = µ i f i (t) i=0 σ(t) = nσ σ j g j (t) ξ(t) = j=0 n ξ ξ k h k (t), k=0 where f, g, h are functions (e.g., sine and cosine, identity, exponential, etc.), t are covariates or trend variables. Care must be taken when incorporating covariates into the scale parameter, σ, in order to ensure that it is positive everywhere. Usually, an exponential link function is used so that the model is of the form ln(σ(t)) = nσ σ j g j (t). 3.2 Estimation j=0 Distribution (1) leads to the following log-likelihood equation. (1 1/ξ) n i=1 log L(θ; z) = n log σ [ log 1 + ξ z ] i µ σ n i=1 [ log 1 + ξ z ] 1/ξ i µ, (3) σ subject to the constraint that 1 + ξ(z i µ)/σ > 0. For the Gumbel case, the likelihood simplifies to n [ ] zi µ n [ n log σ log exp z ] i µ. (4) σ σ i=1 There is no analytical solution to the optimization over the parameters for (3) and (4). Therefore, numerical optimization routines are required to find the maximum likelihood estimates (MLE s) for the thre parameters. For small data sets, it is usual to estimate the parameters using L-moments (e.g., Hosking and Wallis, 1997), or the generalized MLE (GMLE) method of Martins and Stedinger (2000), as more stable solutions can be found. However, it is not possible to incorporate trends into the parameter estimates using the L-moments approach. The GMLE approach requires some prior input, which we do not have here. Bayesian estimation (e.g., Coles and Tawn, 1996) is, of course, also possible, and future work will investigate such methods. The likelihoods (3) and (4) can also be written with the incorporation of covariates in the parameters, and iterative likelihood ratio tests can be used to test for significant improvements in the model fits. AIC and BIC approaches are also possible, but are not used here. i=1

6 Figure 3: 20-year return levels for annual maximum CAPE shear (csmax)estimated by the GEV (left) and the 95% quantile of the reanalysis (right). No spatial correlation is accounted for in either graph. 4. Initial results for global reanalysis data Initial investigations have centered on fitting the generalized extreme value (GEV) distribution to annual maxima of the product of CAPE and shear (henceforth, csmax). Figure 3 shows the GEV-estimated 20-year return levels from having fit the GEV individually at each grid point (i.e., no spatial correlation taken into account) as well as the empirical 20-year return levels obtained from the reanalysis csmax (i.e., the 95% quantile taken at each grid point). Of course, estimating a high quantile from such a short record of data is questionable so that the graph on the right is not a very accurate assessment of the true" 20-year return level. Nevertheless, it is clear that although the GEV estimates seem to reproduce the correct spatial structure, they are everywhere too small. This may be a result of large uncertainty in the estimates (not shown), and may be overcome by employing Bayesian estimation (Richard L. Smith, personal communication; see also Coles and Pericchi, 2003). To obtain information about trends in csmax over the 42 years of global reanalysis data, temporal covariates are investigated in the parameters of the GEV. Iteratively more complicated models are tried and tested for significance using the likelihood-ratio test. Where any trends are detected for these data, the only significant ones are linear in the location parameters. That is, µ(year) = µ 0 + µ 1 year Some significant trends in the scale parameter are found, but these occur at grid points where the reanalysis data is less believable such as the polar regions. Therefore, these models are not used. Figure 4 shows the results from fitting a linear trend in the location parameter of the GEV. Checking point-wise significance for these trends is performed. The spatial pattern of the intercept (or constant term) recovers the general pattern of high values of csmax, and the slope terms show a similar pattern as those found from the frequency analysis (Pocernich et al., 2008, not shown). Four regions of interest are inspected more closely. Figure 5 shows these regions without

7 Figure 4: Intercept (left) and slope (right) terms from fitting a linear trend in the shape parameter of the GEV. No significance test is performed in this graph. testing for significance, and figure 6 shows them with point-wise significance. It is important to account for both spatial correlation and multiple testing, however, especially when analyzing over so many points. Therefore, figure 7 shows the results from applying the false discovery rate (fdr) test proposed by Ventura et al. (2004). Significant positive trends (i.e., increasing csmax intensities) are found off the eastern coasts of Asia even after accounting for spatial correlation and multiple testing issues. Some significant decreases in extreme csmax intensities after applying the fdr to the point-wise significance tests are also observed for southern South America, whereas no significant trends remain over the United States. Very little trend activity is detected over Europe, but there exist a few locations of increasing (northern Germany, southern Scandinavia, south-eastern Europe) and decreasing trends (northern Sweden). 5. Results for current climate ( ) as output by CCSM3 over the United States Figure 8 shows the median annual maximum (AM) csmax over from CCSM3 (left) and the reanalysis data (right). While the spatial patterns are similar, there are noticeable differences. Furthermore, there are substantial discrepancies in intensities (in both directions) as can be more easily seen in figure 9, which shows their differences (CCSM3 median AM csmax reanalysis median AM csmax). Performing traditional verification (i.e., point-to-point), which does not account for small spatial discrepancies, shows that the CCSM3 does only slightly better than a completely random model (skill score (SS) of only about 0.5), and not as well as simply using the previous year s reanalysis data (table 2). However, it should be noted that the reanalysis data is not necessarily the truth" as it, for example, shows the higher values of csmax on the lee side of the Rockies as being a bit too far to the east, whereas the CCSM3 captures this spatial feature better than the reanalysis (Harold E. Brooks, personal communication). Similar results are obtained for other aggregations (apart from the median) of csmax, and indeed other series besides AM (e.g., Marsh et al., 2007). Because of the lack of agreement between the reanalysis and CCSM3 output and the reanalysis, and the lack of confidence in what the truth" really is, one must be careful in making strong assertions about csmax under a changing climate. However, it is important to realize that the

8 Figure 5: Trends in location parameter (i.e., the slope term) for csmax over four regions of interest. No significance test is performed in these graphs.

9 Figure 6: Trends in location parameter (i.e., the slope term) for csmax over four regions of interest. Point-wise significance test is performed in these graphs.

10 Figure 7: Trends in location parameter (i.e., the slope term) for csmax over four regions of interest. False discovery rate (fdr) applied to significance tests performed in these graphs.

11 Figure 8: Median annual maximum (AM) series of csmax from for CCSM3 (left) and reanalysis (right). Median AM cape*shear CCSM3 ( ) Median AM cape*shear reanalysis ( ) Table 2: Traditional verification results from comparing the CCSM3 Median AM csmax (forecast) against the reanalysis median AM csmax ( ). MAE 10,660 ME 4,835 MSE MSE - baseline MSE - persistence SS - baseline 0.488

12 Figure 9: Difference between median AM csmax from CCSM3 output minus reanalysis. Median AM CCSM3 Reanalysis ( )

13 climate is essentially the distribution (not necessarily the mean) from which weather is derived. Therefore, an ensemble of climate models should be examined besides just a single realization of the distribution. H.E. Brooks (personal communication) recently found that using the following derived variable in place of CAPE better discriminates severe storms (figure 10). Wmax = 2 CAPE (5) Another advantage of using Wmax in Eq. (5) above over CAPE is that the units are now in m/s; the same as shear. Figure 11 shows the median AM shear Wmax (henceforth swmax) for both reanalysis (left) and CCSM3 output (right). As expected, there are still large differences in intensities, which can be more easily seen in figure 12, which shows the differences. The model shows lower values than the reanalysis over North Dakota and S. Minnesota, as well as Southern Texas, the Caribbean, Florida, and off the southern east coast. The model generally projects higher values of swmax over the Rockies and Appalachians, with extremely higher values over southern Arizona and into Mexico. 6. Initial Conclusions, Future and Ongoing Work The initial analysis of extreme intensities of csmax complements the work of Pocernich et al. (2008) where the frequencies of threshold exceedances are studied. The next logical step would be to model the two aspects simultaneously using a peaks over threshold (POT) extreme value approach. Further, recent findings by H.E. Brooks (personal communication) show that Wmax (5) instead of CAPE may be more useful in identifying likely severe weather scenarios from largescale phenomenon. Additionally, it has been recommended that use of Bayesian estimation may help to reduce uncertainty in GEV-estimated return levels, as well as more accurate estimates consistent with the data. Other possible future directions include using other climate model output in addition to the CCSM3 runs used here. One source of output that will be studied are cases from the North American Regional Climate Change Assessment Program (NARCCAP, for example. Another possibiliy is to use global climate model output to initialize such regional climate models in order to more directly investigate severe weather distributions under a changing climate. Acknowledgments This work is supported by the Weather and Climate Impacts Assessment Science Program ( which is funded by the National Science Foundation (NSF). The authors thank Harold Brooks and Patrick Marsh for providing us with the global reanalysis data and climate model output, as well as for their consultations. We also thank Richard L. Smith for insightful suggestions of future directions. References Brooks, H., J. Lee, and J. Craven, 2003: The spatial distribution of severe thunderstorm and tornado environments from global reanalysis data. Atmos. Res., 67 68, Coles, S. and L. Pericchi, 2003: Anticipating catastrophes through extreme value modelling. Appl. Statist., 52,

14 Figure 10: (0-6km) Shear against Wmax with red contours showing the conditional probabilities of significant severe storms over the United States. Figure courtesy of H.E. Brooks

15 Figure 11: Median ( ) AM shear Wmax for reanalysis (left) and CCSM3 output (right) Coles, S. and J. Tawn, 1996: A bayesian analysis of extreme rainfall data. Appl. Statist., 45, Hosking, J. and J. Wallis, 1997: Regional frequency analysis: An approach based on L-moments. Cambridge University Press, Cambridge, UK, 240 pp. Marsh, P., H. Brooks, and D. Karoly, 2007: Assesment of the severe weather environment in north america simulated by a global climate model. Atmospheric Science Letters, 1 7, doi: /asl.159. Martins, E. and J. Stedinger, 2000: Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resources Res., 36, Pocernich, M., E. Gilleland, H. Brooks, B. Brown, and P. Marsh, 2008: Analysis of atmospheric conditions conducive to small scale extreme events from larger scale global reanalysis data. Manuscript in Preparation. Rasmussen, E. and D. Blanchard, 1998: A baseline climatology of sounding-derived supercell and tornado forecast parameters. Wea. Forecasting, 13, Ventura, V., C. Paciorek, and J. Risbey, 2004: Controlling the proportion of falsely rejected hypotheses when conducting multiple tests with climatological data. J. Climate, 17,

16 Figure 12: Median AM shear Wmax for CCSM3 minus reanalysis ( )