1. Motivation
Identifying and tracking storm cells from remotely sensed imagery such as satellite and radar have long been of great interest in meteorology (Tag et al. 2000; Rinehart and Garvey 1978; Augustine and Howard 1988; Crane 1979; Rosenfeld 1987; Johnson et al. 1998; Yang et al. 2006; Lakshmanan et al. 2003, 2009; Dixon and Wiener 1993; Morel et al. 1997; Han et al. 2009; Roberts et al. 2009; Lakshmanan and Smith 2010; Bedka et al. 2009; Sieglaff et al. 2013) because algorithms that extract properties of storm cells and track these properties over time provide information that is important to forecasters for assessing storm intensity, growth, and decay (Wilson et al. 1998; Lakshmanan and Smith 2009). Because of this paramount need to support forecasters in real time, widely employed algorithms for tracking storms such as the algorithms of Johnson et al. (1998), Dixon and Wiener (1993), and Lakshmanan and Smith (2009) have all focused on devising causal methods (i.e., methods that track storms by using only data that are already known). In other words, storm-tracking algorithms do not “peek” into the future to determine the optimal set of cell associations between time steps. The causal requirement was slightly relaxed by Lakshmanan et al. (2013) and Miller et al. (2013) when they devised a time-lagged quality-control technique for the purpose of cleaning up accumulation products. Nevertheless, the accumulation products were created in real time by applying temporal continuity criteria only to earlier frames of the sequence being accumulated.
Even though storm-tracking methods such as the Storm Cell Identification and Tracking algorithm (SCIT; Johnson et al. 1998); Thunderstorm Identification, Tracking and Nowcasting (TITAN; Dixon and Wiener 1993); and segmentation–motion estimation (w2segmotion; Lakshmanan et al. 2003; Lakshmanan and Smith 2009) are constrained to work in a purely causal fashion, these algorithms have been widely employed by the meteorological research community to do case studies and formulate spatiotemporal relationships (e.g., Antonescu et al. 2013; Rossi et al. 2013; Peleg and Morin 2012; Kohn et al. 2011; Brown et al. 2002; Weusthoff and Hauf 2008; Durrans et al. 2002).
Using a storm-tracking algorithm that is constrained to work in real time to do postevent analysis is suboptimal. There is more information (about which cells persist and the direction in which they move) that is available if the entire set of storm-cell identifications over the complete dataset is used to determine thunderstorm tracks. In this paper, we describe a way of clustering a set of storm-cell identifications over time into trajectories, where a trajectory is the line (or curve) that best fits the position of an individual storm cell over time.
This work was performed to improve spatiotemporal relationships between radar-derived storm characteristics and the subsequent onset of specific weather hazards such as cloud-to-ground lightning, hail, and tornadoes (Kuhlman et al. 2008). Such hazard probabilities can be derived from storm attributes using the method of Lakshmanan and Smith (2009) on a multiyear reanalysis dataset created as described in Cintineo et al. (2012), but the reliability and skill of these probabilities are limited by the quality of the storm tracks used to train the data-mining algorithms.
The rest of this paper is organized as follows. In section 2, the statistical methods that underlie the proposed technique are described. In section 3, the proposed technique for postevent storm-track extraction is described. In section 4, the results of postevent storm-track extraction are compared with the storm tracks extracted in real time. In section 5, we discuss some possible enhancements to the procedure described in this paper.
2. Statistical methods and definitions
Given a set of storm centroids, our approach is to cluster the centroids into individual tracks. We can treat each storm cell as a point in three-dimensional space (two spatial dimensions and one temporal dimension) so that the coordinates of a storm cell are (x, y, t).
a. K-means clustering
The technique of K-means clustering (Gowda 1984) is a popular iterative clustering approach that can be used to extract storm tracks from a set of storm centroids. Naively applying a clustering procedure to the set of 3D points (x, y, t) will not work, however. To see why, consider the storm centroids shown in Fig. 1a.
(a) Storm-cell positions from 17 Jun 2012. (b) The storm cell at point “a” needs to be assigned to one of the two clusters whose centers are shown at c1 and c2. (c) The storm cell at point “a” needs to be assigned to one of the two trajectories c1 or c2.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
The K-means clustering approach consists of four steps: 1) start with K clusters, 2) assign each storm to the cluster to which it is closest, 3) recompute the cluster properties (such as the cluster center), and 4) compute the variance within the cluster and, if it is small enough, stop. Otherwise, return to step 2. There are three components to the K-means clustering method outlined above: 1) a way to choose the initial set of clusters, 2) a way to compute the distance between a storm cell and a cluster, and 3) a way to compute the cluster center. We can initialize the set of clusters by using a traditional real-time storm-tracking algorithm; that is, we can use SCIT, TITAN, or w2segmotion to provide a first-guess estimate of storm tracks. The next two components of the standard clustering algorithm are more problematic.
b. The need for a trajectory
If each storm cell is represented by three coordinates (x, y, t), a typical choice of distance metric is the Euclidean distance. The cluster centroid (the mean of the vector components) is often chosen as the cluster center. With these choices, however, storm cells at the extremities of a track might well be closer to the centers of other tracks. For example, in Fig. 1b, the cell at location “a” is closer to the cluster center c1 than it is to cluster center c2. Therefore, one has to make choices for the distance metric and cluster center that are more appropriate.
Our approach in this paper is to fit the cells within a cluster to a curve (a “trajectory”). The distance between a cell and a cluster is defined as the Euclidean distance between the cell and its projected position within the trajectory. For example, in Fig. 1c, point “a” is closer to the trajectory c2 than to the trajectory c1. When storm cells are added to (or removed from) a cluster, the trajectory corresponding to the cluster can be recomputed by recomputing the best-fit curve.
c. Theil–Sen trajectory
For estimating the slope of the best-fit line through a set of points, the Theil–Sen slope estimator is superior to a least squares regression (LSR) estimator in several ways (Sen 1968). 1) It is nonparametric whereas an LSR estimate assumes that errors are normally distributed. 2) It requires fewer independent observations than the LSR to attain a given sample variance. 3) It is less sensitive to outliers. 4) It is impervious to linear transformations of the input variables—a fact that is important because the track of a storm should not vary depending on the map projection in which the coordinates are represented and because storm tracking can happen at multiple scales. 5) It is more robust to skewed data, as can happen when centroid estimates are skewed because of mergers and splits.
d. Cluster center and distance measure
The definitions of the cluster center and distance measure are critical to K-means clustering. As pointed out in section 2b, a naive definition of these terms will not work for storm tracks.
(a) The cluster center is computed by considering all pairs of points within the cluster. The median slope in the x direction is u. (b) The distance of a candidate storm cell to the cluster is computed by considering its projected location at time t from the trajectory of the cluster.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
3. Extracting postevent storm tracks
The method for extracting postevent storm tracks proceeds as follows:
Find an initial estimate of tracks in the dataset. This can be obtained from any robust storm-tracking algorithm, even a real-time one such as that of Johnson et al. (1998) or Dixon and Wiener (1993). In the results reported in this paper, we used the w2segmotion algorithm described in Lakshmanan and Smith (2009).
While treating each track (set of storm cells with the same identifier) as a cluster, compute the Theil–Sen slope and constants (u, υ, x0, y0, t0) for each cluster.
For every storm cell in the dataset, use Eq. (3) to find the nearest cluster, taking care to compute the distance only for points within the buffered bounding box of the cluster in x, y, t. If the nearest cluster is different from the cluster of which the cell is currently a part and if the distance is less than some reasonable threshold D, move the storm cell to the nearest cluster.
Compute the Theil–Sen fit for each cluster, prune the set of clusters (as explained in the next paragraph), and carry out step 3, repeating steps 3 and 4 until there are no more changes or until the number of iterations reaches some maximum (we used three iterations as this maximum number since we found substantial convergence after three iterations for all of the cases that we considered).
At the end of step 3, there may be several trajectories that are substantially identical. Trajectories whose space–time bounding boxes overlap are considered in pairs. If the two trajectories are within a spatial distance D at all points and if the two trajectories overlap by less than 600 s in time, the two trajectories are combined into a single cluster and the Theil–Sen fit is recomputed. It is also possible that, after storm cells have been moved into their best fit, there are clusters with very few storm cells. Clusters with fewer than three storm cells are pruned, and the cells within them are moved to the closest trajectory (if one exists). It is possible that there are multiple cells at the same time assigned to the same cluster—this situation is normal and is typically indicative of errors in storm identification (e.g., part of the storm cell may have fallen below a threshold at one time step and become identified as two separate cells).
4. Evaluation
a. Track statistics
The w2segmotion algorithm for storm identification and tracking that is described by Lakshmanan et al. (2003, 2009) and Lakshmanan and Smith (2009) allows storm identification to be carried out at multiple scales. We performed the clustering procedure on storms identified on mosaicked radar-reflectivity data collected by the federal WSR-88D network over the coterminous United States (CONUS) and mapped to the height of the −10°C isotherm in real time using the approach described in Lakshmanan et al. (2006) and Lakshmanan and Humphrey (2014). The storms were identified at scales of 200, 600, and 1000 km2. We tried different values of D on storm tracks extracted from the mosaicked radar-reflectivity data at the different scales and, after trial and error, employed a D of 0.1 decimal degrees for the results in this paper.
By following the method of Lakshmanan and Smith (2010), we did a statistical analysis of the set of storm tracks extracted from the radar data (mosaicked over the CONUS) for five days in 2012 (the 17th of the month from April to August). For the data on 17 April 2012, at the most detailed (200 km2) scale, the number of trajectories is cut by approximately one-half as a result of postevent track extraction (see Fig. 3). The error in size fit [computed by fitting the sizes of the storm cells within a trajectory to a “growth and decay” parabola and looking for deviations from that fit; see Lakshmanan and Smith (2010) for details], which is an indicator of how likely it is that two separate tracks are wrongly combined, increases by one-third. The position error, which is an indicator of how likely it is that storm cells are added to tracks of which they are not a part, also increases but remains limited to be below the limit imposed by D. The bottom-right panel in Fig. 3 demonstrates the benefit of postevent track extraction: the mean duration of the tracks nearly doubles, from an average of ~1700 s to an average of ~3300 s.
Effect of postevent extraction of storm tracks at different scales on the set of storm positions extracted for the CONUS on 17 April 2012. The “before” data refer to the tracks as identified by a real-time storm-tracking algorithm (w2segmotionll), and the “after” data refer to the tracks as extracted by the postevent storm-track-extraction method that is described in this paper.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
At the moderate (600 km2) and coarse (1000 km2) scales, the number of trajectories increases in the postevent storm-extraction stage and the errors in size fit and position decrease. This is likely because “jumpy” tracks in the real-time storm-tracking algorithm are getting fixed by the postevent storm-track-extraction method described in this paper. Even with a greater quantity of storm tracks that exhibit less positional error, the mean duration of the tracks increases by a considerable amount at the medium scale and a small amount at the coarse scale. Thus, one gets a significant improvement in the form of longer-lived tracks, and this improvement is associated with only a small risk of potentially wrong associations.
Similar statistics for the four other days that were considered are shown in Fig. 4. In all of these cases, improvements in mean durations are seen at the detailed and moderate scales accompanied by either a small increase (detailed scale) or decrease (medium scale) in positional error. At the detailed scale, one sees an overall decrease in mean duration, but it is accompanied by some improvement in the positional error.
As in Fig. 2, but on four other days in 2012.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
We may conclude that, at the detailed and medium scales, applying a postevent storm-extraction technique is beneficial. At the large scale, the errors associated with extracting centroids of large objects2 probably outweigh any potential improvement that could be delivered by the method described in this paper.
b. Case study
On 23 April 2008, large-scale ascent ahead of an incoming short-wave trough coupled with a moist and unstable air mass supported widespread convective initiation during the midafternoon hours along the Texas and New Mexico border. One cell in particular formed initially out of multicellular convection and persisted for over 210 min across Curry County, New Mexico, and Parmer and Castro Counties in Texas (a total distance of 105 km). This storm produced a long swath of hail with reported sizes up to 4.4 cm in diameter (NCDC 2008). This cell was tracked using the w2segmotion real-time storm-tracking algorithm and the postevent storm-track-extraction method that is described in this paper.
Tracking this cell using w2segmotion from 1859 until 2229 UTC yielded four unique cell identifiers (IDs) for the time period, with each cell ID persisting for a variable amount of time (see Fig. 5). The first identification (storm A) occurred at 1859 UTC and continued for 30 min as the storm developed from an initial multicellular state to a more isolated system. At 1934 UTC, this cell was assigned a new ID (storm B), which persisted for 55 min as the storm strengthened and moved into Parmer County, Texas. The third reclassification (storm C) occurred at 2034 UTC and lasted for 120 min as it moved through Parmer County into Castro County. This time period was not contiguous, however, because at 2159 UTC the cell was reclassified for one volume scan (storm D). During the period between 2034 and 2159 UTC, the storm reached its peak severity, with a descending 70-dBZ core observed during the time of the severe-hail reports. Plotting of the centroids of the four clusters for the single storm cell shows a disjointed track that could easily be confused as individual cells (see Fig. 6a).
Radar imagery from (a) 1859, (b) 1934, (c) 2034, and (d) 2159 UTC 23 Apr 2008 showing (left) w2segmotion output at the four reclassification points during the storm lifetime in comparison with (right) the postevent track-extraction method at the same four time steps. Note that the four different cell IDs on the left (1006, 1157, 1426, and 1886) are combined into the same track (456) on the right.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
Spatiotemporal plots of storm centroids across object lifetime for the 23 Apr 2008 case: (a) w2segmotion real-time storm-tracking algorithm and (b) postevent track-extraction algorithm from this paper. The numbers indicate the storm IDs assigned to an object and are the same as those in Fig. 5. The same ID in successive frames indicates that the storms are part of the same track.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
Tracking this cell with the postevent track-extraction technique described in this paper provided a contiguous track from 1859 to 2229 UTC with the same assigned ID (right-hand panels in Fig. 5). During all periods in which w2segmotion would reclassify the cell, the postevent extracted-storm-track ID persisted through the dataset. Furthermore, tracking the centroids at each time step (Fig. 6b) provided a more uniform trajectory when compared with the originally classified clusters. Overall, where a traditional storm-tracking technique would have identified four storms with the same convective mode consisting of 30-, 55-, 125-, and 5-min durations, the postevent track-extraction technique described in this paper provided a single trajectory object of 210-min duration. Thus, this postprocessing technique can provide a substantial benefit over traditional techniques, especially when evaluating spatiotemporal trends such as analyses of storm lifetime.
One question that arises is whether the clusters as determined by the real-time w2segmotion algorithm hold information of dynamical significance. By merging four separate tracks into one, we posit an objective judgment that these four tracks belong to one storm. The parts may correspond to different stages in the life cycle of the storm, however, or may correspond to storm splits and mergers. Therefore, the information that this track consists of four constituent parts ought to be retained. In “w2besttrack,” which is the implementation of the algorithm described in this paper that is available as part of the Warning Decision Support System (Lakshmanan et al. 2007), the original cell IDs are attached to the information of each track and can be used in later data mining.
c. Usage in a lightning-jump algorithm
Total lightning in thunderstorms has been shown to behave as an indicator of severe weather because of the relationship between lightning activity and the strength of a storm’s updraft (e.g., Saunders et al. 2006; MacGorman et al. 2008; Calhoun et al. 2013). Although there have been some studies that have shown that instantaneous flash rates can discriminate between severe and nonsevere storms (e.g., Williams et al. 1999; Rudlosky and Fuelberg 2013), it is methods that examine the trend in total lightning that have perhaps shown the most promise in delineating storms that produce severe weather (e.g., Goodman et al. 1988; Schultz et al. 2011).
One such method is the “2-sigma” lightning-jump algorithm (Schultz et al. 2009, 2011). This algorithm searches for rapid increases in total lightning in a storm (lightning jumps). To search for jumps, this algorithm requires the recent history of total lightning activity in a storm 1) to determine a baseline of electrical activity in the storm and then 2) to determine whether the most recent observation of total lightning in the storm was significantly larger than the baseline. Once a lightning jump is triggered, a “warning” is issued that can be verified by the production of severe weather. Because of the reliance on the history of the total lightning activity, identifying lightning jumps with a storm-tracking algorithm can be problematic if the tracking fails over the lifetime of the storm.
Take, for example, the case of a storm over central Oklahoma on 19 May 2010 (Fig. 7). To compute the 2-sigma algorithm for this storm, the w2segmotion storm-tracking algorithm was run to obtain the instantaneous flash rates for each minute of the storm (Fig. 8). Manual analysis revealed that this storm was tracked as three separate entities by w2segmotion. The top plot in Fig. 8 shows the times at which w2segmotion misidentified the storm as a new entity. Lightning jumps were computed for each of these identified objects (“clusters”) over their lifetime, and jumps from these clusters are also shown in Fig. 8.
A storm over central Oklahoma on 19 May 2010: (a) merged composite reflectivity and (b) an identifying cluster from w2segmotion and the lightning initiations from that time.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
The total flash rate for a storm over central Oklahoma on 19 May 2010: (top) the time at which the storm was identified by a new cluster(vertical black lines) and the lightning jumps computed for each cluster(dashed blue lines) and (bottom) the lightning jumps computed from the flash rates obtained by merging the three clusters.
Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0132.1
These clusters were then processed by the technique that is described in this paper. It was noticed that the three clusters from w2segmotion were correctly merged into one trajectory. The flash rates and computed lightning jumps over the more complete trajectory are shown in the bottom plot in Fig. 8. Notice that the fourth jump computed when using the improved trajectory did not appear when the lightning jumps were computed with the flash rates from the individual clusters. This mismatch occurs because, shortly before the fourth jump, the storm was misidentified as a new cluster, and not enough time had elapsed for the lightning-jump algorithm to run on the new cluster. Although this storm never produced severe weather, and thus the lightning jump did not “verify,” it does serve as an example of how postevent storm-track extraction can improve upon traditional storm-tracking algorithms.
5. Discussion
a. Why not MHT?
In this paper, we suggest the use of a clustering approach to address the problem of obtaining a set of tracks from a dataset. Another noncausal approach to storm tracking is to use multiple hypothesis tracking (MHT; Reid 1979; Cox and Hingorani 1996). In meteorology, this method has been employed for storm tracking by B. Root et al. (2012, personal communication) and for quality control by Lakshmanan et al. (2013) and Miller et al. (2013). So, a natural question that arises is why we chose to use clustering and not MHT.
First, MHT is an improvement only of assignment errors between storm cells at adjacent frames and does not address problems that result from poor storm identification. It is common, for example, for a storm to be mistakenly delineated as two cells at one time step because of a poor choice of threshold, movement through an area of beam blockage, and so on. By clustering storm cells over all frames in the dataset, one imposes order on this problem by assuming that if there is no other storm trajectory close by then both of the cells belong to the same trajectory. During postprocessing, it is then possible to combine cells at the same time in the same trajectory into a single unit, thus fixing storm-identification errors.
Second, MHT implementations have a “look back” period after which tracks are fixed. This is an ideal approach for real-time tracking in domains such as missile tracking in which there are frequent observations of targets, and in which the number of potential targets is small and the targets are discrete and well defined. The use of a look-back period seriously limits the applicability of MHT on large datasets of amorphous, short-lived entities.
b. More-complex trajectories
Given three points, it is possible to find an arc that connects them, and, given more points, it is possible to find the best-fit arc through those points. The parameters could be estimated by using some nonlinear least squares estimation method. [See Venables and Ripley (2002) for details.] The additional number of parameters and the use of least squares minimization leads us to suspect that the resulting tracks will not be as robust as the simpler Theil–Sen nonparametric fits. Whether this intuition holds true is worth exploring, at least at the largest scale.
In this paper, storm cells were represented solely in terms of their centroids (x, y, t) and these centroids were fit into trajectories. More information about storms is available in the form of a fit of the storm shape to ellipses, and it should be possible to use overlap in addition to centroid position when associating storms. Indeed, previous work has shown that a combination of overlap and centroid position is better at tracking storms than is either one alone (Lakshmanan and Smith 2010).
There are three potential approaches to incorporating the complete shape of a storm when tracking. In particle filtering (Maskell and Gordon 2001), the set of pixels in which an object could possibly be is modeled as a joint probability distribution. Another is to explicitly model the probabilities of new births, deaths, mergers, and splits and to perform a Bayesian MHT (Makris and Prieur 2014). A third approach that we have not observed in the literature but that might be more conducive to the storm-tracking problem might be to combine ideas from the method of this paper: particle filtering and Bayesian MHT. In such a hybrid method, one might fit identified storms to ellipses and selectively sample (as in particle filtering) the centroid as well as the outer boundaries of the ellipse to follow over time as Theil–Sen trajectory fits, but taking into account explicit models of new births, deaths, mergers, and splits.
c. Summary
Many studies of storm life cycles and climatological descriptions have simply utilized a storm-tracking algorithm designed to be used in real time. It was demonstrated that one can take advantage of noncausal information when fitting storm cells to postevent trajectories. This was done by fitting the storm cells found by a real-time storm-tracking algorithm to a best-fit trajectory by using an iterative clustering method in which the cluster parameters are estimated through a Theil–Sen fit of the points in the cluster. It was shown that such clustering usually results in fewer, longer-lived tracks. Bulk analysis of the set of storms over a dataset shows that the improvement in duration is achieved without much degradation of position and size errors. The improvement was also illustrated by means of two case studies, and the impact of such an improvement is shown in its effect on a lightning-jump algorithm.
Acknowledgments
Funding for the authors was provided by the NOAA/Office of Oceanic and Atmospheric Research under NOAA–OU Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce. The technique described in this paper has been implemented within the Warning Decision Support System-Integrated Information (WDSS-II; Lakshmanan et al. 2007) as the tool w2besttrack.
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A Kendall tau value of zero indicates that the two quantities are independent. In this context, this result indicates that the observed storm-cell location is as likely to lie “above” the trajectory as it is lie “below” it.
The centroid of a slowly moving object may change significantly as a result of changes in shape. This problem is more acute the larger the object is because of the increased likelihood of adding or removing data at the fringes of the object.
