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  • View in gallery

    Intensity forecasts (instantaneous 10-m wind) for the four initializations in the three models. The color scheme—red for COTC, orange for GFDL, blue for HWRF—is maintained for all following comparison images. Black is for the observations [taken from the hurricane database (HURDAT)].

  • View in gallery

    Storm frequency binned by intensity (10-m maximum instantaneous wind speed at a model grid point).

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    Average spread of all centers, calculated using Eq. (1), over all times, storms, and models. Black is for all models.

  • View in gallery

    Center locations at 500 hPa, forecast hour 12, for TC Emilia in COTC. Abbreviations are explained in Table 2. Winds are shaded; heights are contoured.

  • View in gallery

    As in Fig. 4, but for TC Debby in HWRF at forecast hour 24.

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    Scatterplot of 500-hPa spread (km) vs 900-hPa maximum mean tangential wind (m s−1) for all models.

  • View in gallery

    As in Fig. 6, but for 300 hPa and for a magnitude change along the ordinate.

  • View in gallery

    As in Fig. 4, but for TC Daniel in GFDL at 300-hPa height and forecast hour 30.

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    The 950-hPa radial and tangential wind differences for TC Emilia, forecast hour 6 for a 1200 UTC 10 Jun initialization, between PVC060060 and MAVWBS in COTC.

  • View in gallery

    As in Fig. 9, but for TC Daniel at forecast hour 12 and for difference between MAVWBS and PVC300300.

  • View in gallery

    Radial and tangential wind means and standard deviations at 950 hPa for TC Emilia (0000 UTC 12 Jul) at forecast hour 12 in GFDL.

  • View in gallery

    The 900–700-hPa average tilt magnitude for all methods and for only those storms of hurricane strength or greater.

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    Significance matrix of 900–700-hPa tilt differences between all methods across all models and for only those storms of hurricane strength; the significance is multiplied by the sign of the t score. Green (blue) colors indicate where the method along the abscissa is less (greater) than the method along the ordinate.

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    As in Fig. 12, but for 700–500 hPa.

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    As in Fig. 12, but for 500–300 hPa.

  • View in gallery

    As in Fig. 13, but for 500–300 hPa.

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    As in Fig. 15, but for the entire dataset and for an increased scale.

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    Tilt of TC Daniel in GFDL, forecast hour 24, for a select number of methods. Larger markings indicate lower-level centers, decreasing in size with height at 900, 700, 500, and 300 hPa.

  • View in gallery

    As in Fig. 18, but for TC Emilia (1200 UTC 10 Jul) in HWRF, forecast hour 36.

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An Investigation of Center-Finding Techniques for Tropical Cyclones in Mesoscale Models

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  • 1 Fleet Numerical Meteorology and Oceanography Center, Monterey, California
  • 2 Department of Earth, Ocean, and Atmospheric Sciences, Florida State University, Tallahassee, Florida
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Abstract

A variety of tropical-cyclone (TC) center-finding methods aggregated from previous works of mesoscale modeling and operational analysis are compared. The previous methods used can be divided into three classes: local extreme, weighted grid point, and minimization of azimuthal variance. To analyze these methods, four representative separate TC forecasts from three operational models—the Coupled Ocean–Atmosphere Mesoscale Prediction System Tropical Cyclone version, a Geophysical Fluid Dynamics Laboratory model, and the Hurricane Weather Research and Forecasting Model—are examined. It is found that for this dataset the spread of the derived TC centers is fairly small between 1000 and 600 hPa but begins to increase rapidly at higher levels. All models exhibit increased center spread at upper levels when the TCs’ strengths fall below approximately hurricane strength. On a given pressure level, tangential wind differences calculated from different centers are generally small and localized, whereas radial wind differences are often much larger in both space and relative magnitude. Center-finding techniques that use mass fields to calculate centers exhibit the smallest vertical tilts for hurricane-strength TCs. Conversely, potential vorticity centroids with large weighting areas produce the largest tilts. Given the potential sensitivity of center determination and implied tilt for various other measures of TC structure (radius of maximum winds), these results may have large repercussions on both past and future analyses.

Corresponding author address: David R. Ryglicki, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Rm. 254, Bldg. 704, Monterey, CA 93940. E-mail: david.ryglicki.ctr@nrlmry.navy.mil

Abstract

A variety of tropical-cyclone (TC) center-finding methods aggregated from previous works of mesoscale modeling and operational analysis are compared. The previous methods used can be divided into three classes: local extreme, weighted grid point, and minimization of azimuthal variance. To analyze these methods, four representative separate TC forecasts from three operational models—the Coupled Ocean–Atmosphere Mesoscale Prediction System Tropical Cyclone version, a Geophysical Fluid Dynamics Laboratory model, and the Hurricane Weather Research and Forecasting Model—are examined. It is found that for this dataset the spread of the derived TC centers is fairly small between 1000 and 600 hPa but begins to increase rapidly at higher levels. All models exhibit increased center spread at upper levels when the TCs’ strengths fall below approximately hurricane strength. On a given pressure level, tangential wind differences calculated from different centers are generally small and localized, whereas radial wind differences are often much larger in both space and relative magnitude. Center-finding techniques that use mass fields to calculate centers exhibit the smallest vertical tilts for hurricane-strength TCs. Conversely, potential vorticity centroids with large weighting areas produce the largest tilts. Given the potential sensitivity of center determination and implied tilt for various other measures of TC structure (radius of maximum winds), these results may have large repercussions on both past and future analyses.

Corresponding author address: David R. Ryglicki, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Rm. 254, Bldg. 704, Monterey, CA 93940. E-mail: david.ryglicki.ctr@nrlmry.navy.mil

1. Introduction

As computing power has increased over the past 50 years, so too has our understanding of tropical cyclones (TC). What were once anomalously high magnitude and unrealistically large, in a spatial sense, vorticity and moisture signatures in models with very coarse resolution (Manabe et al. 1970) have become more defined, articulate, and realistic reproductions of actual TCs (e.g., Braun 2006; Davis et al. 2008; Rotunno et al. 2009). Models that were once run with a grid spacing of hundreds of kilometers can now be run on the order of a kilometer or even tens of meters. Of course, these extremely high resolution simulations come at the expense of massive computing costs, as time steps in Eulerian models shrink with grid size nonlinearly. Simulations run at large-eddy resolution are still not viable for operational use (Rotunno et al. 2009), but operationally viable simulations of TCs have recently reached 3 km [Hurricane Weather Research and Forecasting Model (HWRF; Gopalakrishnan et al. 2012)]. As resolution increases, inner-core processes of TCs become better resolved. What these high-resolution simulations show is that smooth and relatively homogeneous fields—such as wind, pressure, or temperature—may be appropriate for two-dimensional, axisymmetric maintenance theories such as the wind-induced surface heat exchange theory (Emanuel 1986) but that their existence is perhaps not the case in three-dimensional reality. Boundary layer parameterizations tend to smooth turbulent eddies in the eyewall of TCs, thus reducing the radial entropy gradient. As simulations begin to resolve turbulent eddies explicitly, the entropy gradients in models across the eyewall (Bryan and Rotunno 2009) begin to approach observational values (Bell and Montgomery 2008).

Even though high-resolution simulations provide more information about inner-core processes, they present unique challenges when compared with global models, not the least of which is determining the center of the models’ TCs. In the past in global models, TCs have usually been detected by a combination of a low-level pressure (or height) minimum, a low-level vorticity maximum, a low-level wind speed maximum, a temperature anomaly aloft, and a duration threshold. In addition, some previous works attach a geographical restriction to limit false alarms at higher latitudes (Tsutsui 2002). There have been a plethora of previous works that use these basic parameters as guides for locating TCs in global models, the specifics of which having been concisely summarized in Walsh et al. (2007). Additional previous works have added motion criteria (Hart 2003; Halperin et al. 2013). In operational settings, the Marchok (2002) approach has been widely accepted because it uses a Barnes analysis on several low-level fields in an attempt to produce the best location for a low-level center. This technique is currently used by the Environmental Modeling Center and the Fleet Numerical Meteorology and Oceanography Center as well as in the Automated Tropical Cyclone Forecasting system that is used by the National Hurricane Center and the Joint Typhoon Warning Center to track TCs.

The techniques that are primarily used in global models are convenient for tracking the position of a storm. What they are unable to do is to present any information on the finer details of the vertical structure of the storm. Part of the reason for this fact is simply because, at grid spacings on the order of degrees, finer details of the vertical structure cannot be adequately resolved. A step forward in the analysis of vertical structure of a storm was the cyclone phase space (Hart 2003), a method that allowed for a distinct quantification of the vertical and horizontal structure of a cyclone by using thermal wind and storm-motion-relative thickness asymmetries. With this tool, one can determine whether a cyclone is warm core (tropical) or cold core (extratropical) on the basis of large-scale features of the cyclone. It has also been evaluated for its use in diagnosing structural transitions (Hart 2003; Evans and Hart 2003; Kofron et al. 2010a,b). Drawbacks of this technique are that it integrates the vertical structure details into two sublayers and also that it integrates the horizontal structure into two hemispheres rather than presenting those details explicitly; therefore, information on the finer vortex scale remains hidden.

At the mesoscale-model level (around 10 km), the vortex scale can begin to be resolved. A multitude of papers have attempted to discern the center of TCs at these resolutions. It is at this point where center-finding techniques begin to diverge. Some previous works have used domain extremes, such as pressure minima, to prescribe the center (Jones 1995; Frank and Ritchie 1999). Other previous works have used potential vorticity (PV) centroids, but the size of subdomain used to calculate the centroid is not consistent (Reasor and Montgomery 2001; Jones 2004; Riemer et al. 2010). As an alternative, one could use a mass field—pressure or geopotential—and perform a similar calculation (Ryglicki and Hart 2012; Nguyen et al. 2014). Yet another option could be to favor the location at which the radius of maximum winds (RMW) is at its greatest magnitude (Marks et al. 1992; Lee and Marks 2000), to find the point at which the radial pressure variance is at its smallest (Cram et al. 2007), or to locate the position where gradient wind balance is most closely satisfied (Xu et al. 2015).

Other trackers do exist for observational data, but because the trackers themselves are limited in scope in that they are usually applicable to a singular point or level, they will not be analyzed further. Wood (1994) used an idealized Rankine vortex fitted with Doppler velocity data. Wong et al. (2004, 2008) presented an algorithm to calculate the center on the basis of rainband spiral patterns in radar reflectivity. Following that, Wong et al. (2007) and Wong and Yip (2009) developed an algorithm that used feature tracking in sequential radar reflectivity images to calculate the TC rotational center. Chang et al. (2009) developed an algorithm that also uses sequential images, but this method determines the low-reflectivity region of the cyclone’s inner core. As opposed to using Doppler radar data, a center-tracking method has been developed by Wimmers and Velden (2010) that uses passive microwave satellite imagery and is called automated rotational center hurricane eye retrieval. It uses two components: a spiral-centering algorithm finds the optimal center of a spiral pattern in the image, and a ring-fitting algorithm finds the edges that are most likely associated with the inner eyewall. For in situ flight-level data, Willoughby and Chelmow (1982) attempted to find a streamfunction minimum. In other cases, finding the center of a TC is at the discretion of the observer, such as marking the center at which axes of an elliptical eyewall intersect (J. Franklin 2012, personal communication).

It is fairly easy to see that, because there are so many methods to define the center of a TC, selecting the “best” metric is not straightforward. It is also dependent on the goals of the user. After all, an attempt to locate the center of a TC, whether in core-resolving simulations or in observations, is an attempt to quantify an abstract quality. Various previous works allude to these quandaries. Cram et al. (2007) developed a hybrid technique that sought to maximize the radial gradient of pressure but noted that “[t]he alternative method of using the location of minimum surface pressure was found to track centers of mesovortices rather than the storm-scale center of pressure.” In addition, Reasor et al. (2013, p. 2966) noted, when discussing the lack of a strong relationship between shear magnitude and vortex tilt,

[t]he tilt metric used here emphasizes the core, which is typically more upright than the broader-scale vortex (Jones 1995; Reasor et al. 2004). Comparison of the perturbation vorticity field (relative to the low-level center) for low- and high-shear stratifications indeed reveals a more significant upper-level wavenumber-1 asymmetry within the eyewall at higher shear (not shown). Thus, center estimates based upon the vorticity centroid […] would potentially reveal a stronger correlation between vortex tilt and shear strength.

It is precisely questions such as these that this work will address. All center-finding metrics have certain advantages while also possessing certain limitations. This paper will begin to explore these advantages and limitations, but it will only focus on the techniques used on gridded model data, as opposed to irregularly spaced dropsonde data, for example. The goal is not to find an absolute correct answer for TC center; it is assumed that there is no ubiquitous technique that will work for all situations. The goals are to survey previous techniques used for finding model TC centers, to quantify differences among the methods, and to highlight potential consequences of these differences. Determining which specific method is best for a specific case is not an explicit goal of this particular paper, although that goal is a natural follow-up to the results presented herein.

A discussion of the datasets used and the methods themselves will be presented in section 2. The analysis results of the techniques on the individual model storms will be presented in section 3. Summary, conclusions, and recommendations for use will be presented in section 4.

2. Method

a. Dataset

To assess the viability and utility of methods used for determining the center of model TCs, a select dataset was chosen to illustrate differences in center-finding methods. The three operational models chosen are a Geophysical Fluid Dynamics Laboratory model (GFDL; Bender et al. 2007), the Coupled Ocean–Atmosphere Mesoscale Prediction System Tropical Cyclone version, or COAMPS-TC (herein also called COTC for brevity) (Doyle et al. 2012), and the Hurricane Weather Research and Forecasting Model (HWRF; Gopalakrishnan et al. 2012). At their finest nests, these are run operationally at 8-, 5-, and 3-km resolutions, respectively. The resolution, initialization, and physics differences among the three models should give a good representation of how center-finding methods behave with different models in this resolution range. Four deterministic TC forecasts were chosen from the 2012 hurricane season: North Atlantic (NATL) Debby [initialization 0000 UTC 25 June, intensity of 50 kt (1 kt ≈ 0.5 m s−1), and minimum surface pressure of 990 hPa], eastern Pacific (EPAC) Daniel (1200 UTCJuly 8, 95 kt, and 965 hPa), and EPAC Emilia twice (1200 UTC July 10, 115 kt, and 948 hPa; 0000 UTC July 12, 105 kt, and 958 hPa). Emilia was chosen twice because it possessed two relative maxima of intensity and enlarged its eye during an annular episode (Cangialosi 2012). Daniel was chosen because it was a small hurricane (Avila and Hogsett 2012). Debby was chosen because it was a weak storm with a broad circulation and ill-defined features (Kimberlain 2012). GFDL and HWRF each possesses 22 six-hourly output time steps per simulation, whereas COTC only possesses 21 six-hourly output time steps per simulation. The exception to these forecast-duration parameters will be Debby. Forty-eight hours into the simulation of Debby, it dissipated in COTC and was absorbed by a midlatitude front; HWRF and GFDL followed this behavior shortly thereafter. As a result, Tropical Storm Debby’s presence in the dataset is subjectively limited to 42 h (eight time steps). Note that Emilia and Daniel, owing to their tracks in the eastern Pacific Ocean, both migrated out to open ocean and never encountered any land, whereas Debby dissipated east of Florida. Each dataset is also reduced in the vertical direction for ease of analysis. Nineteen levels are coincident among the three models from 1000 to 100 hPa in increments of 50 hPa.

Since this work is not a verification analysis, the specific forecast performance of each model with regard to observations is irrelevant to this study. For the sake of completeness, the forecast performance of each model for each initialization is presented in Fig. 1. Equally unimportant is the ability of the models, at these resolutions, to reproduce all of the minute details of inner-core hurricane structure, since many previous studies indicate that vortex resiliency is related to the larger-scale vortex structure as a whole (e.g., Jones 1995, 2004; Reasor et al. 2004); therefore, what is relevant for this discussion is the binned intensity distribution. Figure 2 shows the binned total count of intensities—as based on maximum instantaneous 10-m wind speed at a given data point in the domain—broken down by model. While the dataset is heavily skewed toward tropical storms and category-1 hurricanes, it is important to keep in mind that this work is not merely meant to address how center-finding techniques behave for hurricanes but also for the times at which center-finding methods begin to diverge. As will be shown in section 3, above a certain wind speed threshold, specific intensities are not important with relation to the spread of the centers, and therefore this work will naturally extrapolate to major hurricanes.

Fig. 1.
Fig. 1.

Intensity forecasts (instantaneous 10-m wind) for the four initializations in the three models. The color scheme—red for COTC, orange for GFDL, blue for HWRF—is maintained for all following comparison images. Black is for the observations [taken from the hurricane database (HURDAT)].

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 2.
Fig. 2.

Storm frequency binned by intensity (10-m maximum instantaneous wind speed at a model grid point).

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

b. Center-finding methods

Center-finding techniques will be separated into three distinct classes: local extreme (LE), weighted grid points (WGP), and the minimization of azimuthal variance (MAV). The first class, LE, is straightforward and is generally the simplest. The LE techniques merely locate the maximum or minimum value of a given field in a given search area and are, as such, limited to gridpoint locations in the domain chosen. A mass minimum (pressure or geopotential height) has historically been used most often (e.g., Jones 1995). The PV maximum can also be used for the LE class (Frank and Ritchie 1999). The second class is the WGP class. The PV centroid (PVC) and the mass-ring centroid fall into this class. The WGP methods function by weighting each grid point that falls within bounds, whether it is over the domain, over a subdomain, or within specific contours, by the value of the selected variable. For this research, those weights are either PV or geopotential height. The third class that will be discussed is the MAV. These techniques, as their names suggest, seek to minimize the variance—or maximize the mean—of a given quantity inside a circular shape, whether it is a complete circle or an annulus. This class assumes a generally circular storm shape, where the symmetric mode of a given variable—wind or pressure, for example—dominates any asymmetric modes that develop.

An aggregation of previous studies that use various center-finding techniques can be found in Table 1. It is clear that there is much variety in the methods to determine the center of a TC. Jones (1995) used both a height minimum and a PV maximum. Frank and Ritchie (1999), for example, used a pressure minimum in the vertical direction, arguing that at 5-km resolution the PV maximum’s location was too unstable. Stern and Zhang (2013) smoothed the pressure field before choosing the minimum. In the applications of PV centroid, there is disagreement on the weighting area. Reasor and Montgomery (2001), Jones (2004), Davis et al. (2008), and Riemer et al. (2010) all used different sizes for their weighting areas. The mass-ring centroid used by Ryglicki and Hart (2012), instead of using a weighting box that depends on a fixed grid size, only incorporates grid points that fall within a specified range of geopotential height values. This allows local pressure minima from near-core eddies to be eliminated from the calculation. Nguyen et al. (2014), after examining a few choices on a case study, also decided on a pressure centroid. For the MAV techniques, much like the PVC WGP techniques, there is some disagreement on the details of implementation. Braun (2002) and the first-generation Pennsylvania State University–National Center for Atmospheric Research Cloud Model (CM1; G. H. Bryan 2012, personal communication) both use what can be deemed the “brute-force method.” This method iterates over grid points in a subdomain, looking for the grid point at which either the standard deviation of pressure in the azimuth is a minimum or at which the mean tangential wind is at its maximum using the pressure minimum as a first guess.

Table 1.

Center-finding methods employed by previous studies. For Cram et al. (2007), it is not quite a strict MAV technique, because it is an MR(adial)V technique, but the concept is similar.

Table 1.

Marks et al. (1992) employed an iterative “simplex” method (Nelder and Mead 1965), which is categorized as an MAV class, using the domain center of their radar data as a first guess to ascertain the true center. For mathematical completeness, a simplex may be defined as the smallest convex set (i.e., a set is convex if it contains all of the line segments connecting all pairs of its points) containing any number of given vertices. The relevance to this problem is a 2-simplex—a triangle. The method is iterative, in which a first guess is supplied and a triangle is drawn in a circle surrounding the initial guess. Given a first-guess radius of maximum winds, the wind is checked at each vertex inside the prescribed circle [in all cases, this is 2 times the horizontal grid spacing, as per Reasor et al. (2009, 2013)]. The triangle is made smaller in the direction of the vertex with the highest wind value. The triangle is then made iteratively smaller until meeting an iteration criterion or a convergence criterion using the tangential wind speed calculated in the prescribed annulus. The appeals of this technique are computational speed, because the RMW annulus is only found once, and flexibility in freely positioning the center between grid points. The brute-force methods that iterate over grid points must find the RMW and the tangential wind maximum at each radius at each point, thus limiting the potential solutions to the grid points themselves, much like the LE class.

The simplex technique has been used primarily for observational radar data, whether those data originate from dual-Doppler flight data (Marks et al. 1992; Reasor et al. 2000, 2009) or from ground-based velocity-track single-Doppler data (Lee and Marks 2000; Murillo et al. 2011). Many studies have attempted to expand on the original simplex application by using a variational statistical approach on their datasets to determine the best center, altering the starting location and/or the annulus width and processing the data through the simplex process several times to improve the robustness of the results (Lee and Marks 2000; Reasor et al. 2009; Murillo et al. 2011; Bell and Lee 2012). The reason it appears that so many resources have been spent on refining the simplex, apart from the rest, is that computing a vorticity-based or mass-based centroid from an observational radar dataset is very challenging (Reasor et al. 2013). Despite the fact that the simplex has been used primarily for observational data, it is unique from the satellite-derived techniques described previously in that it is both vertical-level and data-source agnostic, because it can be used either on 500-hPa model data or on 5-km observational radar data. It is important to keep in mind that, even though we can compare the simplex technique with, perhaps, a PV centroid on model data, it is impossible to perform this comparison on a real storm using radar data unless the storm is excessively sampled such that copious information about the thermal field and the inner-core winds is provided.

The various simplex techniques are fundamentally similar in a physical sense, but the same cannot be stated for the WGP methods. The physical interpretation of the “center” that these techniques find is dependent upon the weighting area, as was discussed in Reasor and Eastin (2012). Davis et al. (2008) used PV centroids with weighting boxes ranging from 120 to 200 km on a side to determine the tilt of storms undergoing extratropical transition, whereas Jones (2004) used a 300-km circle for a weighting area. The larger weighting area presents more information on the larger system and would most likely yield information about the large-scale tilt (Jones 1995; Reasor et al. 2004). The limit at which the weighting boxes reduce to zero size yields a representative LE class—in the case of PV, a local PV maximum. In general, the smaller weighting areas will tend to be sensitive to the inner-core vortex scale, whereas the larger weighting areas will reveal more information about the full scale of the system. If a storm possesses a large quantity of PV at larger radii in the outer regions of the vortex (as will be shown in an example in section 3), then a PV centroid with a large weighting area will be dependent on the evolution of that region. Conversely, PV centroids with smaller boxes will be more resistant to that evolution. The opposite is also true: PV centroids with small weighting areas will be more sensitive to dynamic rearrangement in the core at low levels or perhaps convective plumes aloft, whereas larger weighting areas will be more resistant to such features. Similar arguments can be made for pressure centroids.

For this work, 11 variations of the methods plus the mean center will be calculated for a total of 12 center metrics per height per storm per model. Each of the variations was selected to mimic a method found in the literature. Not all previous metrics are represented—the metric of Braun (2002) was deemed sufficiently comparable to the MAV technique that is based on the RMW, for instance. The complete list can be found in Table 2. In total, there are two LE, five WGP, four MAV, and the mean. For the PV centroids, “convergence” means that the nearest grid point is the same after two consecutive cycles through the process. For the three simplex routines (MAVWBS, MAVWZMS, MAVWCS), convergence is when the centers are within 100 m apart, as per previous work (Reasor et al. 2009, 2013).

Table 2.

Center-finding methods used in this work. DOF is degrees of freedom, and abbrev gives the abbreviations and acronyms used in the text and figures to label the methods.

Table 2.

An alternative way to characterize center-finding methods is to document the degrees of freedom of each method as the number of adjustable parameters. For example, the mass minimum, as it is classified as LE, has three degrees of freedom: search parameter (geopotential height), search domain (entire grid), and search function (location of minimum). As an alternative LE method, one could search for PV maximum with its degrees of freedom as search parameter (PV), search domain (entire grid), and search function (location of maximum). The simplex method, in the MAV class, has five degrees of freedom: search parameter (tangential wind speed), first-guess location (height minimum), first-guess RMW, annular width, and convergence/iteration cease condition. For the MAV techniques that involve the simplex method—MAVWBS, MAVWCS, MAVWZMS—the first-guess RMW is not provided explicitly, and therefore the code determines the first-guess RMW as the radius at which the tangential mean wind is largest from the given starting position. The annulus in which the wind is averaged is always 2 times the horizontal grid spacing dx—16-km annulus for GFDL, 10-km annulus for COTC, and 6-km annulus for HWRF. This is to maintain consistency with prior work (Reasor et al. 2009, 2013). Sensitivity tests varying the annulus beyond the 2dx setting were not performed.

Note that in this research there are no stability criteria. In Reasor et al. (2009), if the center at one level is beyond a certain distance from the center at the previous level, it is rejected as bad data. For the purposes of this research, no such criteria were established. This is once again to investigate the conditions under which these center-finding methods begin to diverge. Future work may examine the temporal evolution of centers as determined by the various methods here.

3. Results

There are potentially innumerable ways to quantify differences in the use of center-finding techniques, and this paper is in no way meant to be fully exhaustive. The research presented in this section will demonstrate a few key relevant areas that highlight and quantify the differences in selecting center-finding methods. It is important to remember that, once again, defining the center in a physically meaningful way is somewhat nebulous. At this point, we are not making claims on a true center; therefore, we can merely compute differences between different methods. Three forms of analysis are performed: determining the spread of the centers at each level, computing the differences when decomposing the wind into its cylindrical components, and calculating the vertical tilt magnitude.

It also bears mentioning that in this section the nature of statistically significant difference between analyses is addressed. For the purposes of this research, the dataset, when taken as a collection of 219 individual forecast times, is considered to be independent. Wilks (2011) intimates that in meteorology observations and forecasts over a limited area usually exhibit high degrees of serial correlation. It is fairly obvious that for a given storm in a given model there is most likely a high degree of serial correlation in that specific forecast run. It is not entirely clear that the same storm, modeled in three different systems, is either independent or serially correlated. We assume that different storms are independent from one another, given the same model, and that different storms in different models are also independent. Despite the fact that all three models presumably pull from the same cadre of observations, they do not necessarily use the same boundary conditions, data assimilation systems, physics parameterization options, initialization routines, or dynamical cores. Determining the serial correlation of the same storm simulated in different models is beyond the scope of this paper and will be addressed in a future work.

a. Spread of the centers

One simple and straightforward way to quantify the difference in using different center-finding methods is to calculate the spread among the centers. First, the average center is computed from the 11 methods at each level at each time step for each model. Next, the Euclidean distance is calculated for each method relative to the mean center. The mean spread at each level is defined as
e1
where s is the spread, z is the vertical level, N is the total number of forecast lead times, M is the total number of methods, and Δm is the Euclidean distance between a calculated center and the mean. This calculation is not meant to identify outliers, and therefore determining which center is consistently external of the others is not covered in this work. The average spread at each level is shown in Fig. 3. The spread of the centers is fairly consistent up to 550 hPa. The spread then appears to increase rapidly as pressure decreases. At lower pressure, it is natural to question what exactly the center-finding methods are detecting, as the storm’s structure changes aloft. In climatological terms, the size of the cyclonic circulation shrinks and is replaced radially by a growing anticyclonic circulation (Frank 1982).
Fig. 3.
Fig. 3.

Average spread of all centers, calculated using Eq. (1), over all times, storms, and models. Black is for all models.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

One reason for the distribution of results may be from the choice of dataset. As was shown in Fig. 2, most of the storms in this dataset are of tropical-storm strength. Center distribution of a hurricane-strength storm (Emilia) at 500 hPa in COTC is shown in Fig. 4. All of the centers are, for the most part, tightly packed together. Contrast that with Fig. 5, a tropical-storm-strength vortex (Debby) in HWRF. Accounting for the fact that the domain is larger in Fig. 5, the spread of the centers appears to be much greater and less tightly grouped. To get a more complete view of how intensity and spread are related, Figs. 6 and 7 show scatterplots of spread versus 900-hPa maximum mean tangential wind (taken from the MAVWBS calculation) for 500 and 300 hPa, respectively. What these plots show is that below approximately 30 m s−1 the spread begins to increase. Given that wind speeds at 900 hPa are stronger than at the surface in TCs in this dataset, it would appear that the vertical coherence of a vortex, as interpreted as greater spread in these results, begins to break down slightly below hurricane strength. An example can be shown in Fig. 8, a sub-hurricane-strength GFDL forecast of Daniel, for which the different metrics are not clustered together. This is consistent behavior among all three models, regardless of differing physics and resolutions (not shown).

Fig. 4.
Fig. 4.

Center locations at 500 hPa, forecast hour 12, for TC Emilia in COTC. Abbreviations are explained in Table 2. Winds are shaded; heights are contoured.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 5.
Fig. 5.

As in Fig. 4, but for TC Debby in HWRF at forecast hour 24.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 6.
Fig. 6.

Scatterplot of 500-hPa spread (km) vs 900-hPa maximum mean tangential wind (m s−1) for all models.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for 300 hPa and for a magnitude change along the ordinate.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 8.
Fig. 8.

As in Fig. 4, but for TC Daniel in GFDL at 300-hPa height and forecast hour 30.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

b. Wind field decomposition

One aspect of diagnostic analysis that is affected by center positioning is the conversion of meridional and zonal winds on a Cartesian grid to tangential and radial winds on a polar grid. Previous works have performed eddy energy calculations and azimuthal Fourier transforms to document structural evolution of a vortex (e.g., Reasor et al. 2009; Kwon and Frank 2008; Riemer et al. 2010), but the results could be sensitive to the decomposition of the Cartesian winds to cylindrical winds—which again is dependent on the determination of the center. This phenomenon has been qualitatively assessed as an “alpha gyre,” in which a wavenumber-1 asymmetry is artificially created by aliasing wavenumber 0 because of a misplaced center (Willoughby 1992), but these phenomena have not been structurally quantified. A few select examples are presented to demonstrate these issues.

Figure 9 shows an example from COTC—forecast hour 6 of the first Emilia simulation. This difference is between the MAVWBS and PVC060060 (see Table 2) centers. While tangential wind differences are generally small except for very localized maxima around the core, radial wind differences can be very large, comparatively, in terms of both relative magnitude and areal extent. Radial wind variances usually appear to be on the order of the radial wind itself inside the eye—approximately 10 m s−1. In certain cases, as seen for Daniel in COTC at in Fig. 10, a center-finding method will put the center of the TC closer to the RMW. Because of a broad profile of PV inherent to storms in the model (not shown), the PVC300300 method is pulled closer to the eyewall in COTC here. This results in a large reduction of tangential winds on the order of 30 m s−1 while transferring those speeds to the radial wind component. This behavior is not exclusive to COTC, of course. Figure 11 shows the mean radial wind and mean tangential wind fields of Emilia at 950 hPa at forecast hour 12 in GFDL. The tangential wind variance for this storm is much more localized and focused on specific points, while the radial wind variance once again show a very large wavenumber-1 difference that covers practically the entire storm.

Fig. 9.
Fig. 9.

The 950-hPa radial and tangential wind differences for TC Emilia, forecast hour 6 for a 1200 UTC 10 Jun initialization, between PVC060060 and MAVWBS in COTC.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for TC Daniel at forecast hour 12 and for difference between MAVWBS and PVC300300.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 11.
Fig. 11.

Radial and tangential wind means and standard deviations at 950 hPa for TC Emilia (0000 UTC 12 Jul) at forecast hour 12 in GFDL.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

In qualitative terms, what these images highlight is one of the potential pitfalls of only using one method for center determination. In Fig. 9, it appears that the direction of the eye outflow (warmer colors) is pointing in different radial directions. If one were to use the PVC060060 technique, the primary outflow would be to the southeast, whereas if one used the MAVWBS technique, a more intense outflow would be detected flowing westward from the center. This could potentially lead to challenges and inconsistencies when attempting to perform near-core and inner-core analyses when transposing data from a Cartesian grid to a cylindrical grid.

c. Vortex tilt

Perhaps one of the most important calculations for analyzing vertical structure of a storm is determining the tilt. As was noted in Reasor et al. (2013), there was not a strong correlation between tilt and shear in their study. They propose two reasons: the first is the timing of their observations, and the second is the center-finding metric. Here, we address the second reason. In their work, they argue that their center-finding technique, the equivalent of MAVWCS in this work, is biased toward smaller tilts while a PV-based centroid would provide larger tilt estimates. To address this quandary, this section is divided in the following ways. First, tilt calculations are divided into three layers characterizing low-level (900–700 hPa), midlevel (700–500 hPa), and upper-level (500–300 hPa) tilt. Second, the analysis is carried out twice—once for all storms and then again for only those storms of hurricane strength. The vertical breakdown is to isolate how different center-finding metrics behave as the storms’ gradients in wind, vorticity, and pressure begin to weaken. As seen in Figs. 6 and 7, the spread is generally larger at 300 hPa than at 500 hPa. The tiered breakdown will permit a more specific analysis of how each center metric behaves as the storm gradients become more diffuse.

The focus of this section will primarily be on those storms of hurricane strength. The results can be generalized to all storms, but differences will be noted when significant. Of a possible 219 storm instances (as noted in section 2a) across all three models, 72 were of hurricane strength. It is expected that all tilts for the stronger, hurricane-force storms will be smaller than for weaker storms given previous work (Jones 1995, 2004; Reasor et al. 2004; Hodyss and Nolan 2008; Reasor et al. 2013).

Starting at the bottom of the storm, Fig. 12 shows the average tilt magnitude of each center for each model between 900 and 700 hPa. The two pressure-based metrics—ZMIN and ZRING—give the smallest tilts of ~1–2 km on average. The two PV centroids with smaller weighting areas, PVC060060 and PVC120120, indicate tilts on the order of 3–4 km, whereas all of the MAV techniques, for this layer, indicate tilts of 5–6 km. The largest tilts of 10–20 km in this layer are given by the PV centroids with large weighting areas and by the PV maximum. Figure 13 indicates that most of these results are significant across all hurricanes and models, at least to the 90th percentile and in some cases to the 95th percentile. The differences of centers across the entire dataset are structurally similar and are generally as significant as for the hurricanes (not shown).

Fig. 12.
Fig. 12.

The 900–700-hPa average tilt magnitude for all methods and for only those storms of hurricane strength or greater.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 13.
Fig. 13.

Significance matrix of 900–700-hPa tilt differences between all methods across all models and for only those storms of hurricane strength; the significance is multiplied by the sign of the t score. Green (blue) colors indicate where the method along the abscissa is less (greater) than the method along the ordinate.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

For the 700–500-hPa layer, as seen in Fig. 14, the results are essentially the same as for the 900–700-hPa layer. In terms of significance (not shown), the pattern is similar to that of the lower layer. At this height, however, the differences between the PV centroids with smaller weighting areas are not different in a statistically significant sense from the MAV techniques, excluding MAVWCS, as both methods would result in an average tilt of 5 km for this layer. ZMIN remains the metric that yields the smallest tilt of around 2 km, and the difference is significant against all other metrics.

Fig. 14.
Fig. 14.

As in Fig. 12, but for 700–500 hPa.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

In the highest layer analyzed here, 500–300 hPa, a similar pattern once again emerges, as seen in Fig. 15. The two methods that are based on use of the pressure field, ZMIN and ZRING, are still yielding the smallest tilts, significantly so according to Fig. 16, of ~5 km. Over all models, PVC060060 finally displays a larger tilt (10 km) when compared with the MAV techniques (7–8 km). PVC120120 appears to be slightly smaller than PVC060060 and slightly larger than the MAVs, but it is not statistically significant in either case except for against MAVWZMS. PVCONCE and PVMAX continue to be the two methods that would indicate the largest tilts, anywhere from 15 to 30 km, in this layer. The large anomaly shown by PVCONCE in COTC is a result of too much PV in the outer regions of storms in the model, as was alluded to in section 3b.

Fig. 15.
Fig. 15.

As in Fig. 12, but for 500–300 hPa.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 16.
Fig. 16.

As in Fig. 13, but for 500–300 hPa.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

When the entire dataset is considered, including all of the weaker storms whose vertical coherence is less pronounced, a radically different result is attained. Figure 17 shows the average tilt magnitude of the upper layer over the entire dataset. As expected, the average tilts over all metrics are greater than their hurricane counterparts, agreeing with previous observation and theory (Reasor et al. 2013). MAVWBS and MAVWZMS become the metrics that would indicate the largest tilts, whereas PV centroids with larger weighting areas would give smaller tilts. There are two possible reasons for this difference. First, as seen in a comparison between Figs. 15 and 17, ZMIN goes from the method that yields the smallest result to the one that yields one of the largest results. Once storms weaken, the mass minimum aloft may not always be located in the storm’s cyclonic circulation, because the center of the storm may invert to a mass maximum in the anticyclone or could possibly be located outside the storm altogether if it translates into an area with a strong upper-level mass gradient (such as, hypothetically, near a jet). Since three of the MAV techniques use the pressure minimum as the first guess, they will all be negatively affected by that first guess. PV centroids with large weighting boxes can mitigate this problem. As the storms weaken, the gradients become weaker, and so the PV centroids with larger weighting boxes will potentially give a more accurate representation of the broad circulation and not of any localized features.

Fig. 17.
Fig. 17.

As in Fig. 15, but for the entire dataset and for an increased scale.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Vortex tilt magnitude is only one facet of tilt analysis; the other is tilt direction. Tilt direction analyses are only meaningful if they can be normalized against the shear vector, since storms generally tilt slightly to the left of the shear vector (Reasor et al. 2013). Determining tilt direction and computing correlation with tilt magnitude require an extra calculation of the shear vector over each layer, which is a problem that deserves its own focus (e.g., Hanley et al. 2001; Reasor and Eastin 2012). That attention is currently beyond the scope of this research and is reserved for future work. What can be shown here, before a full statistical analysis can be performed using a multiyear dataset, is a sample of the potential variance in the tilt direction. Figure 18 is a snapshot of a few selected center methods—MAVWBS, ZMIN, ZRING, PVC060060, and PVC300300—for Daniel in GFDL at forecast hour 24. The center methods all provide good agreement on the general southward tilt of Daniel in GFDL at this time, and therefore one could confidently say that Daniel here is tilting to the south, despite the fact that the magnitudes are all different. Conversely, Fig. 19 shows a snapshot of Emilia in HWRF. MAVWBS would show a predominantly westward tilt, whereas ZRING and ZMIN would both indicate a small tilt that is westward at low levels before turning to the north at upper levels. The PV centroids also disagree: PVC060060 indicates a general northward tilt, and PVC300300 indicates a southward tilt at low levels yet a westward tilt at upper levels. As can easily be seen from these figures, determining how the tilts are related to each other, to the model used, to storm strength, to shear direction, and to temporal evolution will be an important investigation for a future study.

Fig. 18.
Fig. 18.

Tilt of TC Daniel in GFDL, forecast hour 24, for a select number of methods. Larger markings indicate lower-level centers, decreasing in size with height at 900, 700, 500, and 300 hPa.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

Fig. 19.
Fig. 19.

As in Fig. 18, but for TC Emilia (1200 UTC 10 Jul) in HWRF, forecast hour 36.

Citation: Journal of Applied Meteorology and Climatology 54, 4; 10.1175/JAMC-D-14-0106.1

4. Summary and discussion

This work has sought to quantify the differences in various methods used to track the center of mesoscale-model TCs through the depth of the troposphere. The forecasts from four separate TCs from 2012 (NATL Debby, EPAC Daniel, and EPAC Emilia twice) from three operational models (GFDL, COAMPS-TC, and HWRF) at three different resolutions (8, 5, and 3 km), respectively, were used to analyze different center-finding methods. The center-finding methods can be categorized into three classes: local extreme, weighted grid points, and minimization of azimuthal variance. This research used two LE, five WGP, and four MAV, plus the mean, for a total of 12 center methods. A variety of analyses were performed.

Spreads of the centers at 500 and 300 hPa were shown to increase greatly in storms whose intensities are below category-1 strength but to remain relatively consistent for hurricanes. The spread for storms of approximately hurricane strength appears to remain near 25 km at both 500 and 300 hPa; below hurricane intensity, however, the spread increases—drastically, in cases of very weak storms. This can be caused by multiple factors. As was shown in Fig. 5, the PVMAX center was located very far away from the rest of the center calculations. For weaker storms that are embedded in strong pressure gradients, the lowest heights may not be located within the storm. Because the height minimum is used as a first guess for many center-finding methods both here and in the literature, this poor first guess acts as an anchor from which some methods simply cannot recover.

The spread of the centers, even when they are near to each other, can manifest itself in differences in cylindrical wind decomposition. When comparing the decomposition relative to different centers, tangential wind differences appear to be localized while radial wind differences appear to cover a much broader area and are on the magnitude of the radial wind itself. What has been shown is that when calculating metrics that use cylindrical decomposition—calculations such as azimuthal Fourier transforms, momentum fluxes, energetics, and divergence—the solutions and results may be sensitive to decomposition of the wind field as based on the location of the computed center. This is an important issue not only when re-examining past work but also, more important, moving forward and will be explicitly quantified in a future work.

A few important results emerged from the tilt analyses. In storms whose strength is of hurricane force or greater, mass-based center-finding methods yield the smallest tilts. At lower levels, PV centroids with small weighting boxes produce the next smallest tilts, followed in order by the MAV techniques, the PV centroid with a larger weighting box, and the PV maximum. Most of these differences are deemed statistically significant at least at the 90% confidence level. As the height in the storm increases, the PV centroid with the smallest weighting area becomes larger than in the MAV techniques, but the mass-based centers always indicate the smallest tilts. When the entire dataset is considered, the mean tilt calculations aloft drastically change. The height minimum suddenly becomes one of the largest tilts aloft. This result could be due to the fact that, as a storm weakens, the height falls aloft of the vortex can be less pronounced than those of the surrounding environment. Because the mass minimum is often used as a first guess, all subsequent center-finding calculations will be affected. In addition, for MAV techniques the wind gradients also begin to break down; therefore, searching for a strong azimuthal wind maximum becomes problematic. Determining which center-finding metric is best for a certain situation, while also keeping in mind its physical meaning, is an important topic for future discussion.

Although a complete statistical analysis was not performed on tilt direction and its correlation to shear because of the additional complexity of calculating vertical wind shear itself, preliminary investigations apparently indicate that merely accepting tilt calculations from a given method can sometimes lead to questionable conclusions. In some cases, methods will agree in the direction but not quite in the magnitude of the tilt (Fig. 18). In others, methods will diverge not only in distance but also in direction (Fig. 19). The primary goal of this research was to begin to answer some questions posed both directly and indirectly by previous works, such as whether one center-finding method will give different answers (and/or physically different interpretations) than those from another method.

It has been shown that PV centroids may not, in fact, give a larger tilt than a MAV technique, depending on the size of the weighting grid box. It is natural, however, to question what a center-finding method is in fact detecting. Perhaps in the future, when discussing tilt, it may be prudent to discuss various aspects of the tilt. For example, the large-scale tilt may be found with a PV centroid that uses a large weighting box, whereas the tilt of the inner-core vortex may require the use of a MAV technique. Perhaps more information is present than merely a bulk tilt from 900 to 300 hPa from any one given method. This work certainly seems to imply so. Visualizations of other storms at other times also apparently support this conclusion. In addition, Reasor et al. (2013) showed that the direction of tilts on average, calculated using the simplex, of observational storms is slightly to the left of the shear vector. When performing this work on HWRF storms, they used a pressure centroid and found that the tilt vectors noticeably shifted to the right of their observational counterparts (Reasor et al. 2014). Reusing the simplex might potentially have yielded results that were more coincident with observations. These results highlight what many future authors may wish to do: calculate center using a variety of methods to test the robustness of results.

With the relationship between shear, storm strength, and tilt still somewhat of an unknown quantity, it is important moving forward to define “tilt” more completely. Observational data support the notion of two distinct centers aloft in some cases, such as different locations for pressure minimum and circulation maximum (Black et al. 1972). In physical terms, each center-finding method represents something different, and differentiating center-finding methods on the basis of physical processes is an important avenue for future work.

This research is not without its deficiencies. This study neglects very-high-resolution model simulations—those on the order of 1-km horizontal grid resolution. At these resolutions, previous works indicate that PV centroids and local extremes begin to behave erratically because of the presence of mesovortices; therefore, new techniques beyond PV centroids and the simple mass minimum had to be used or developed (Braun 2002; Cram et al. 2007; Ryglicki 2015). Some of those methods were not analyzed here. In addition, as was noted in the discussion on the centers themselves, the sensitivity of the MAV techniques to annulus size was not tested. Investigating this aspect of the MAV could add to its usability parameters in future works.

The dataset analyzed is small, and, despite the statistical significance, a larger dataset may make the answers more robust and may define more the universe of model behavior. This study ignores extratropically transitioning systems and storms undergoing trough interactions. Performing these calculations would add much specificity for certain situations and perhaps reveal more information about what center-finding metric should be used and when—an aspect that this work downplays for the sake of generality. Previous work has also established that processing centers in time can yield more robust and smoother results about center location (Bell and Lee 2012), a facet of center finding that this work ignores because of the long time steps between analyses (P. Harr 2014, personal communication; P. Black 2014, personal communication). This will also be an interesting avenue for future research, to see how each of these centers evolve and coevolve in time. It would also be worthwhile to see how forecast skill and tilt relate to each other, along with seeing how the tilts themselves evolve in time.

Prior work has established that shear magnitude, shear direction, storm size, storm strength, and tilt may all be related. Observations and simulations of sheared TCs all indicate that storms tilt slightly to the left of the shear vector, but more recent observations could not draw any firm conclusions on the relationship between tilt magnitude and shear magnitude, and this difficulty perhaps may be due to the method used for determining tilt (Reasor et al. 2013). These both remain open questions and shall be addressed in future work.

Acknowledgments

A portion of this work was performed while the first author held a Science, Mathematics, and Research for Transformation (SMART) fellowship at Florida State University. DRR acknowledges Elizabeth Satterfield (NRL) and Justin McLay (NRL) for guidance on statistical analysis, Raymond Lee (FNMOC) for helpful opinions on visualization, Kevin Viner (NRL), and Benjamin Schenkel (University at Albany) for enlightening discussion on the topic at hand, and Jonathan Hodapp (FNMOC) for editorial assistance. We also acknowledge the helpful suggestions of two anonymous reviewers. Their contributions greatly enhanced this paper’s structure and content. DRR dedicates this work to his friend D. Philip Lane, gone too soon. DRR also thanks Paul Reasor (HRD) for supplying the initial simplex routine and for initial guidance on his Ph.D work.

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