## 1. Introduction

Traditional observational analyses of vorticity, divergence, and deformation fields usually rely on interpolating observations to either a Cartesian or spherical grid and then evaluating the appropriate finite-difference equations. While this approach has the benefit of creating a set of gridded data that is easily processed by uniform spacing finite-difference methods, it has been shown that greater accuracy can be obtained by using a line-integration method, which employs Green’s theorem, on triangular regions (Spencer and Doswell 2001).

An early method of calculating vorticity and divergence on a triangular region is the Bellamy method (Bellamy 1949). In this method, a triangle is constructed using three observations and then a single observation is advected by its wind while the other two remain stationary, resulting in a new triangle. The partial divergence associated with the advected observation can then be calculated by subtracting the original area from the area of the new triangle and then dividing by the original area as well as the time over which the advection is applied. Repeating this process for each of the other two observations and finding the sum of the partial divergences produces the total divergence associated with the three observations. By rotating the wind vectors clockwise by 90° and using the same process, the vorticity can be calculated as well. While the Bellamy method was designed for use with triangular regions, the method can also be used on higher-order polygonal regions without modification. Davies-Jones (1993) has shown that the Bellamy method is mathematically identical to methods that employ Green’s theorem (e.g., Spencer and Doswell 2001).

*u*and

*υ*are the zonal and meridional wind components, respectively, and

*L*is the wavelength. Using these prescribed wind fields allows for the easy calculation of an analytical vorticity against which the results of the line-integration and Barnes interpolation methods can be compared. The results of Spencer and Doswell (2001) indicate that, although there are magnitude issues associated with both methods, the line-integration approach provides a better match to the pattern of vorticity (i.e., large calculated gradients are located where the analytic field has large gradients). Additionally, the study found that the benefits of using a line-integration approach over a Barnes interpolation increased with increasing wavelengths, with higher wavelengths being better sampled.

Bourassa and Ford (2010) implemented Green’s theorem using polygonal regions on Quick Scatterometer (QuikSCAT) wind data to calculate vorticity. In their study, regular polygons, having a diameter of a given number of points, are generated on a gridlike set of winds (gridlike in that there are two distinct dimensions: across swath and along swath) and missing or error-prone data are accounted for by changing the shape of the polygon. Using this approach, they demonstrated that regions defined by larger polygons of higher order can significantly reduce the uncertainty in vorticity calculations, although the reduction levels off for polygon diameters beyond four grid spaces. At minimum, this approach requires some prior knowledge of the distribution of observations and it is preferable that the observations occur on some variety of grid allowing for easy polygon generation. Their study identifies three sources of error: random error (i.e., random observational noise), truncation error (due to the assumption that the wind varies linearly between observation points), and representation error (caused by the validation and calibration of QuikSCAT’s instruments with relatively low-resolution buoy data). Of these, the truncation error provides the largest source of error in the vorticity calculations. Although lack of prior knowledge of the distribution prevents the use of this approach with observations that are not taken in a predictable pattern (e.g., dropsondes), their conclusion that increasing the order and size of the bounding polygon decreases the uncertainty of the calculation remains valid and is exploited here.

The present study describes a line-integration method using polygonal regions that can be applied in situations where the data are not distributed according to a predictable pattern. Section 2 briefly explains the application of Green’s theorem for calculating spatial derivatives and provides details on the construction of the polygonal regions. An analysis of the accuracy of the method is given in section 3. A summary and some concluding remarks are given in section 4.

## 2. Methodology

### a. Calculation of spatial derivatives

*C*and having area

*A*,where

*Q*and

*P*are arbitrary variables and Δ

*x*and Δ

*y*are the zonal and meridional components of the edge lengths, respectively. By assigning the proper variables and signs to

*Q*and

*P*in (2), any spatial derivative can be calculated. Table 1 provides the appropriate values and resulting equations for vorticity, divergence, and deformation.

Green’s theorem approximations of spatial derivative equations for vorticity, divergence, and deformation, where *u* and *υ* are the zonal and meridional components of the wind, respectively.

### b. Polygonal region construction

The polygonal regions used in the method described here are an extension of the triangular regions used by previous methods and are constructed using a triangle tessellation. More specifically, a Delaunay triangle tessellation is used due to a number of beneficial properties (Lee and Schachter 1980). Of greatest convenience to scientific study is the property that the Delaunay triangulation on any given set of points is unique, allowing for easy reproduction of results. Additionally, this type of triangulation maximizes the minimum angle of each triangle, meaning that the greatest number of triangles will be regular, or approach being regular. The benefit of having regular, or near-regular, triangles is twofold: it not only encourages edges to be of similar length (and thus have similar weighting in a line integration), but it also encourages shorter edges,^{1} which reduces the two-dimensional scale on which the calculation will apply. It is important to note that while maximizing the minimum angle encourages shorter edges overall, a Delaunay triangulation does not guarantee a minimization of triangle edge length (Lloyd 1977).

As mentioned previously, the polygon construction process begins with a Delaunay triangle tessellation, an example of which is given in Fig. 1a. For each observation, the set of all other observations that share an edge, referred to as an observation’s neighborhood, is located (Fig. 1b). The neighborhood polygon for an observation is defined by its neighborhood vertices, as depicted in Fig. 1c. The mean wind is then calculated for each edge (Fig. 1d) and decomposed (Fig. 1e) into its zonal and meridional components. Using the equations provided in Table 1 and repeating the process for each observation, a field of values can be calculated. An example of the calculated vorticity field is provided in Fig. 1f. It is worth noting that, in instances where the observation lies on the outer edge of the tessellation (i.e., on the convex hull), the observation is used alongside its neighborhood vertices to reduce the frequency of triangular neighborhood polygons.

For comparison, Fig. 2a depicts the vorticity field calculated from the same data as the example shown in Fig. 1, but using triangular regions. Note the sharp gradients and extreme values that appear in Fig. 2a but are absent in the polygon-based calculations (Fig. 1f). Figure 2b displays the same vorticity field but assigns the calculated value to the corresponding triangle region and reveals a patchwork pattern of alternating positive and negative vorticity extremes. Careful examination also reveals that the region of strong negative vorticity at the west end of the domain is associated with a triangle that has a very small area.

## 3. Evaluation of accuracy

*u*,

*υ*, and

*L*have the same definitions as in (1). Figure 3 is an example of the wind fields used and the corresponding analytic solutions for the spatial derivatives are provided in Fig. 4. As an in-depth comparison between using a Cartesian Barnes interpolation approach and a triangle-based line-integral approach is provided by Spencer and Doswell (2001), this study limits the comparison to using polygonal and triangular regions in a line-integration approach.

Since the method proposed here was developed in order to analyze dropsonde wind observations (e.g., Helms and Hart 2012), the error analysis provided here distributes the artificial observations along a number of flight paths in addition to a random distribution of observations. In cases based on flight paths, the observations are positioned by first evenly distributing them along the flight path and then applying a random two-dimensional perturbation. In the case of the random distribution, the observations are positioned on an evenly spaced grid and then a random perturbation is added to each, as per Spencer and Doswell (2001). Example observation distributions are shown in Fig. 5. For each flight pattern, the path, from which the evenly distributed observation locations are randomly perturbed, is plotted with a thick line. Also included in Fig. 5 are the observation locations (asterisks at the triangle vertices) as well as the corresponding Delaunay triangle tessellation.

### a. Random observation distribution

The simplest of the observation distributions used to test the proposed method is one in which the observation locations have been randomly perturbed from a grid. This distribution is identical in construction to that used by Spencer and Doswell (2001), excepting that significantly fewer observations are used in order to remain consistent with a typical research flight. An example of such a distribution is depicted in Fig. 5a.

The vorticity fields that result from performing the line-integral method calculations on triangular and polygonal regions with a random observation distribution are depicted in Figs. 6a and 6b, respectively. Note the sharp gradients and extreme values present on the edges of the triangle-based analysis are not present in the polygon-based analysis. These values are caused by calculations on long, narrow triangles often found near the edge of the observation domain (see Fig. 5), which have relatively small areas and give overwhelming weight to the most distant observation. Although most studies that rely on triangular regions remove these long, narrow triangles, the polygonal region approach used here retains the data that would otherwise be thrown out. While the polygonal regions reduce the errors caused by these triangles, through reducing the influence of individual observations, caution should be exercised when sharp gradients are present. Also noteworthy is the difference in magnitude between the analytic vorticity (Fig. 4a) and the calculated vorticities (Figs. 6a,b). The primary drawback of using Green’s theorem for calculating spatial derivatives is that the extrema are muted, an effect that is more pronounced when polygonal regions are used.

The root-mean-square error (RMSE) is calculated over the entire domain (Fig. 7a), as well as over an inner domain (Fig. 7b) that excludes the outer 250 km of the 1000 km × 1000 km domain. As noted above, the long, narrow triangles are primarily found along the edge of the observation domain (which extends to approximately 100 km from the outermost edge of the full domain), and by excluding the outer 250-km edge from the RMSE, the effects of these triangles are minimized in the inner-domain RMSE calculations. Although the zonal and meridional wavelengths are defined here to be equal, it is expected that features lacking this symmetry would result in a degree of accuracy that lies somewhere between the accuracies resulting from symmetric features of each of the two component wavelengths. These plots were generated using 100 sets of 30 observation positions, and calculations were performed for 100 wavelengths evenly spaced between 10 and 1000 km. Informal testing suggests that these results should be applicable to datasets with as few as 10 observations. Note the reduction in the full-domain RMSE when using polygonal regions exceeds an order of magnitude for features with a wavelength greater than 150 km. As previously noted, this vast improvement is due to an increased robustness to long, narrow triangles. As is expected, the inner-domain RMSE indicates only a small difference in accuracy, relative to the full domain. Since the polygons are allowed to overlap and each observation point has a corresponding polygon, the number of polygons will always be close to the number of triangles for the same set of observations. As a result, the density of the calculated values, each taken to be located at the corresponding region centroid, will be similar for both polygon- and triangle-based calculations. Besides issues related to the long, narrow triangles, the primary difference between fields calculated with triangles and those calculated with polygons is seen in the size of the smallest resolvable features. This difference is manifest in an abrupt loss of accuracy seen at extremely short wavelengths occurring at slightly higher wavelengths when using polygons than when using triangles. It should be noted that, although this difference is observed in the RMSE plots, which are composites of 100 sets of observations, the difference is unlikely to be of practical importance in an individual case due to its dependence on the exact distribution of the observations. While it appears that the vorticity consistently experiences the greatest improvement, this is likely a result of the calculated vorticity, divergence, and deformation fields having different average magnitudes. Normalizing the RMSE by the mean absolute value of the corresponding variable at each wavelength (Fig. 8) results in an almost complete removal of the spread in RMSE values among each of the two methods analyzed. For the purposes of this study though, the unnormalized RMSE is sufficient to demonstrate the benefits of using the polygonal regions over triangular regions and will be used throughout the rest of this paper.

### b. Distributions based on flight paths

Three patterns of flight paths are used to gauge the accuracy of the method described here. The first, a lawnmower flight path (e.g., Fig. 5b), consists of a series of parallel flight legs that are usually oriented in the north–south direction. The lawnmower flight path is primarily used in poorly organized systems that may have multiple important features and tend to cover a relatively large area. The second pattern, a square-spiral flight path (e.g., Fig. 5c), involves the aircraft flying a series of concentric squares that are typically centered on a feature of interest. Square-spiral flight paths are well suited to weakly organized systems where a single core feature is more easily identifiable (e.g., weak tropical cyclones). In highly organized, approximately axymmetrical systems (e.g., mature tropical cyclones), flights usually occur in either a butterfly pattern (consisting of three, equally spaced, long flight legs stretching from one side of the system to the other and intersecting at the center of circulation; see Fig. 5d) or a figure-four pattern (which is similar to the butterfly pattern except for having only two long flight legs). As both the butterfly and figure-four flight paths give similar error analyses and generate similar observational distributions, only the analysis of the butterfly pattern is included here.

Figures 9 and 10 depict the RMSE errors that occur with a lawnmower flight path and a square-spiral flight path, respectively. The similarity of the errors associated with these distributions is likely a result of the similar distribution of observations each flight path generates. As evidenced by Figs. 5b and 5c, the triangle tessellations generated with each of these patterns are composed of triangles of a consistent shape. In contrast, the random distribution (Fig. 5a) results in triangles with shapes ranging from the problematic, long, narrow triangles to approximately equilateral triangles. In both patterns, the full-domain, triangle-based RMSE is one to one and a half-order of magnitude larger than the polygon-based RMSE for wavelengths beyond 150 km. At shorter wavelengths, the features are no longer resolved by the larger polygonal regions and, as such, the difference in accuracy is diminished. For the inner-domain RMSE, the difference in the error between the triangular and polygonal regions is much smaller and, as a result, is wavelength dependent. This indicates that the polygon-based approach provides a large gain in accuracy near the edge of the observation domain with very little loss of accuracy in the center of the domain.

RMSE plots using a butterfly pattern flight path (Figs. 11a,b) show noticeable differences from those using lawnmower (Fig. 9) and square-spiral (Fig. 10) pattern flight paths. The RMSE values display a much greater level of noise for this pattern than the others. This noise suggests that the calculations are much more sensitive to the relative distribution of the observation points and of the extreme values of the actual wind field. Figures 11c and 11d show the RMSE plots depicted in Figs. 11a and 11b with a five-point smoothing applied. The smoothed RMSE plots reveal that the accuracy improvements, seen with the lawnmower and square-spiral flight paths, also occur to a much lesser extent with the butterfly flight path. The periodicity seen in these plots, for the butterfly pattern in particular, is a result of an interaction between the feature wavelength and the shape of the flight path. At all wavelengths, an extreme value of vorticity and divergence is always located at the center of the domain. This is accomplished by shifting the nondivergent and irrotational wind fields prior to making the calculations. As the wavelength changes, the extrema that are not located at the domain center will at times be located in regions of relatively high-density observations. When this occurs, the accuracy improves due to better representation of the extreme values and the gradients that occur between them. As the extrema move into regions of relatively low-density observations, the extrema become muted and the gradients between them become weaker. Since the derivative quantities rely on an accurate representation of these gradients, the accuracy of the calculations suffers due to gradient underestimation. Despite any apparent relationship between accuracy, wavelengths, and the flight path patterns, the reader is cautioned against using the RMSE figures presented here as a reference for determining minimum resolvable wavelengths as, in reality, flight paths can and will vary widely from the idealized paths used here.

## 4. Summary

A new line-integral method is proposed for calculating spatial derivatives from nonuniform observations. This method uses polygonal regions instead of the triangular regions used by previous studies. The polygon approach results in a decrease in RMSE in comparison to a triangle approach for a variety of observation distributions and feature wavelengths. In some cases, this decrease in error reaches an order of magnitude. The polygonal regions are constructed using a Delaunay triangle tessellation, which results in a unique tessellation for a given set of points and has the benefit of being easily automated. This allows the method described here to be used to generate a large number of easily replicated analyses. By using polygonal regions, the method is more robust to the long, narrow triangles that result in large errors in triangle-based calculations. While these error-prone triangles are usually removed from triangle-based analyses, the use of polygonal regions allows these data to be retained. Furthermore, when these long, narrow triangles are not present, there is very little difference between polygon- and triangle-based methods in terms of RMSE. Finally, it is worth noting that this method can be used to calculate any spatial derivative and is not limited to those involving wind measurements.

A number of opportunities exist for improving upon the present method that may warrant further research into their impacts on the accuracy of the method. The present method does not make any special considerations for the overlapping areas of polygons and the calculated values are typically displayed as point values located at the geometric centroid of their respective polygons. In actuality, the calculated values are representative of the entire area of the polygon. Since the polygonal regions are, in most cases, required to overlap by construction, this appears to present a contradiction in which a single point can have several calculated values assigned to it. As the multiple calculated values at each point represent spatial averages of vorticity over different-sized areas, no actual contradiction exists. While there exist a number of possible ways to reconcile the presence of multiple values at a single point (e.g., deconvolution techniques and inverse-area-weighted averaging), their examination is outside the scope of this study. Regardless, the overlapping quality of the polygons implies that mass is not conserved, posing an issue for budget calculations that use fields derived with this method. An additional area of possible improvement is found in that the present method does not impose any requirements on the polygons included in the calculations; the addition of a minimum polygon order could be imposed to further improve accuracy at the expense of increasing the scale of the minimum resolvable features.

The authors thank Mark Bourassa and Vasubandhu Misra of The Florida State University for their feedback on the thesis on which this work was based. Additionally, the authors thank two anonymous reviewers for their helpful comments on this manuscript. Data support for the PREDICT field experiment is provided by NCAR/EOL under sponsorship of the National Science Foundation. This work was funded through NASA Genesis and Rapid Intensification Processes (GRIP) Grant NNX09AC43G.

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^{1}

By maximizing the minimum angle, the total of the other angles must see a corresponding decrease. This results in the angles and edge lengths becoming closer in size to one another. There are two possible methods by which this can occur: the short edges can grow or the longer edges can shrink. Increasing the length of the short edges requires constructing a larger triangle that will have data points located within its bounds. As a Delaunay triangulation is constructed such that no point may lie within a circle that circumscribes a triangle in the tessellation, the first method is not a viable solution. Hence, the longer edges must shrink, resulting in smaller triangle sizes in general (but not necessarily for each individual triangle).