Computing Divergence from a Surface Network: Comparison of the ...

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Computing Divergence from a Surface Network: Comparison of the Triangle and Pentagon Methods JACQUELINE A. DUBOIS Research Experiences for Undergraduates Program, Oklahoma Weather Center, and University of Oklahoma, Norman, Oklahoma

PHILLIP L. SPENCER Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, and NOAA/National Severe Storms Laboratory, Norman, Oklahoma (Manuscript received 25 August 2004, in final form 18 December 2004) ABSTRACT Two methods for creating gridded fields of divergence from irregularly spaced wind observations are evaluated by sampling analytic fields of cyclones and anticyclones of varying wavelengths using a surface network. For the triangle method, which requires a triangular tessellation of the station network and assumes that the wind varies linearly within each triangle, divergence estimates are obtained directly from the wind observations and are assumed valid at triangle centroids. These irregularly spaced centroid divergence estimates then are analyzed to a grid using a Barnes analysis scheme. For the pentagon method, which requires a pentagonal tessellation of the station network and assumes that the wind varies quadratically within each pentagon, divergence estimates also are obtained directly from the wind observations and are valid at the station lying within the interior of each pentagon. These irregularly spaced divergence estimates then are analyzed to a grid using the same Barnes analysis scheme. It is found that for errorless observations, the triangle method provides better analyses than the pentagon method for all wavelengths considered, despite the more restrictive assumption by the triangle method regarding the wind field. For well-sampled wavelengths, however, the preanalyzed divergence estimates at the interior stations of pentagons are found to be superior to those at triangle centroids. When random, Gaussian errors are added to the observations, all advantages of the pentagon method over the triangle method are found to disappear.

1. Introduction Obtaining accurate estimates of the spatial derivatives of various quantities from observations is important in meteorology. For example, divergence, which is composed of a combination of spatial derivatives of the wind field, appears in numerous diagnostic equations such as those involving mass continuity, vorticity tendency, and moisture convergence. Therefore, the accuracy of the diagnoses of vertical motion, vorticity changes in mesoscale and synoptic weather systems, and areas imminently susceptible to severe convection, for example, depends on the accuracy of the divergence estimates.

Corresponding author address: Phillip Spencer, National Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069. E-mail: [email protected]

© 2005 American Meteorological Society

WAF867

Several techniques are available for creating gridded fields of divergence from wind observations. Traditionally, a finite-differencing scheme is applied to gridded fields of the horizontal wind components to yield gridded estimates of divergence. This traditional method remains popular today through ubiquitous software such as the General Meteorological Package (GEMPAK; Koch et al. 1983). Different variations of the triangle method (e.g., Bellamy 1949; Ceselski and Sapp 1975; Schaefer and Doswell 1979; Zamora et al. 1987; Doswell and Caracena 1988), all of which assume a linear variation of the wind across triangles created from the station network,1 estimate divergence directly from the wind observations. Any of a number of analysis schemes may be used to map these 1 Because of this common assumption, all of these variations of the triangle method are equivalent (Davies-Jones 1993).

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divergence estimates onto a grid. The pentagon method (Chien and Smith 1973), which assumes that the wind field varies quadratically across pentagons created from the station network, also estimates divergence directly from the wind observations. The pentagon method was developed as an attempt to alleviate the restrictive linearity assumption imposed by the triangle method. Although not overwhelmingly popular, triangle methods for calculating kinematic properties of the wind field have been used on occasion to study various weather phenomena. For example, Bosart and Sanders (1981) used a modification of Bellamy’s method to help diagnose the vertical motion field associated with the devastating Johnstown, Pennsylvania, flood of 1977; Zamora et al. (1987) applied the linear vector point function (LVPF) method to profiler observations to diagnose the upper-tropospheric divergence and ageostrophic winds associated with a well-developed trough and jet streak over Colorado; Stensrud and Maddox (1988) applied the line integral method to special soundings to diagnose opposing mesoscale circulations induced by two mesoscale convective systems; Menard and Fritsch (1989) and Brandes (1990) used the kinematic quantities calculated from triangles to document the structure and evolution of convectively generated mesovortices; Spencer et al. (1996) applied the LVPF method to wind profiler data to diagnose the structure of an amplifying and decaying baroclinic wave; and Roebber et al. (1998) used the triangle method to diagnose dynamical and moisture budget characteristics of a persistent low-overcast event. The pentagon method for creating gridded fields of kinematic quantities has received very little attention as a diagnostic tool. However, Ward and Smith (1976) and Chien and Smith (1977) used the pentagon method to compute divergence and other kinematic parameters to study the kinetic energy budgets associated with a period of repeated synoptic shortwave development and Hurricane Camille, respectively. Previous studies have shown that the triangle method for obtaining gridded divergence estimates generally is superior to the traditional method (e.g., Schaefer and Doswell 1979; Doswell and Caracena 1988; Spencer and Doswell 2001). Little is known, however, about the relative merits of the triangle and pentagon methods, although comparisons of analyses from the two methods to sensible weather (e.g., cloud cover and rainfall) by Chien and Smith (1973) and Portis and Lamb (1988) suggest that the pentagon method is superior. The purpose of this study is to compare further the triangle and pentagon methods for computing gridded

fields of divergence.2 In this study, however, a comparison between the divergence analyses and an analytically prescribed divergence field is used to quantify the effectiveness of each method. The traditional method is not considered here because its deficiencies relative to the triangle method are well established. In section 2, we describe how analytic observations are created. Section 3 provides a description of the triangle and pentagon methods. Section 4 describes how the irregularly distributed divergence estimates are analyzed to a grid. The results of the evaluation are presented in section 5. Finally, section 6 contains a summary and concluding remarks.

2. Creating analytic wind observations Wind observations are created by sampling analytic functions at the stations indicated in Fig. 1. The advantage of using analytic wind fields is that the true divergence values are known everywhere in the region, making it easy to compare the relative merits of the two methods. The stations in Fig. 1 represent a set of declustered surface observation locations whose average separation (⌬) is approximately 125 km. The horizontal components of the analytic wind field (ua, ␷a) are represented by the following equations: ua共x, y兲 ⫽ 10 cos

␷a共x, y兲 ⫽ 10 sin

冉

冉

冊 冉

2␲ 2␲ x ⫺ ␸x sin y ⫺ ␸y L L

冊 冉

冊

冊

2␲ 2␲ x ⫺ ␸x cos y ⫺ ␸y , L L

and 共1a兲 共1b兲

where L is the wavelength and ␸x, ␸y represent phase shifts. From (1), the analytic divergence (␦a) is easily derived as

␦a共x, y兲 ⫽ ⫺20

冉 冊 冉

冊 冉

冊

2␲ 2␲ 2␲ x ⫺ ␸x sin y ⫺ ␸y . sin L L L 共2兲

This represents a checkerboard pattern of cyclones and anticyclones.

3. Estimating divergence from irregularly distributed observations a. Triangle method The triangle method requires a triangular tessellation of the station network. To accomplish this, the De2 These methods may be used to estimate the other kinematic quantities, as well.

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FIG. 1. Declustered surface station network considered in the analysis.

launey triangulation scheme is used to create a set of nonoverlapping triangles (Ripley 1981). Four nearby stations compose a quadrilateral, of which competing triangular tessellations exist. The Delauney triangulation scheme selects the set of triangles with the largest minimum angle from the two competing tessellations. Also, the scheme requires that the nearest observation from any point within a given triangle be one of the vertices of that triangle. The triangular tessellation of the station network shown in Fig. 1 is presented in Fig. 2. This tessellation consists of 361 triangles whose average area is 6800 km2. The triangle method assumes a linear variation of the wind field within each triangle. Therefore, the horizontal wind components at the ith station composing a triangle (ui, vi) can be represented by a linear Taylor series:

冏

冏

冏

冏

ui共x, y兲 ⫽ uc ⫹

⭸u ⭸u ⌬xi ⫹ ⌬y ⭸x c ⭸y c i

␷i共x, y兲 ⫽ ␷c ⫹

⭸␷ ⭸␷ ⌬xi ⫹ ⌬y , ⭸x c ⭸y c i

and

共3a兲

共3b兲

where (uc, ␷c) represents the horizontal wind components at the triangle centroid located at (xc, yc), ⌬xi ⫽ (xi ⫺ xc), and ⌬yi ⫽ ( yi ⫺ yc). For each triangle, the system of equations is solved [using the method outlined by Davies-Jones (1993)] for the unknown gradients, from which divergence is computed.

b. Pentagon method The pentagon method requires a pentagonal tessellation of the station network, whereby each observing station (except those along the border of the domain) is enclosed within five nearby stations forming a pentagon. This tessellation was performed according to the algorithm described in the appendix.3 A few such pentagons are shown in Fig. 3. The complete tessellation consists of 165 pentagons whose average area is 31 000 km2. Following Chien and Smith (1973), the horizontal wind components at the ith station composing a pentagon (ui, ␷i) can be represented by a quadratic Taylor series

3

An alternative algorithm is found in Steinacker et al. (2000).

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FIG. 2. The Delauney triangulation of the observation network shown in Fig. 1. For clarity, some of the triangles along the outer edges are not drawn.

ui共x, y兲 ⫽ u0 ⫹ ⫹

冏 冏

冏

冏

⭸u ⭸u 1 ⭸2u ⌬xi ⫹ ⌬yi ⫹ ⌬x2 ⭸x 0 ⭸y 0 2!⭸x2 0 i

冏

⭸2u 1 ⭸2u ⌬xi⌬yi ⫹ ⌬y2 and ⭸x⭸y 0 2!⭸y2 0 i 共4a兲

␷i共x, y兲 ⫽ ␷0 ⫹ ⫹

冏 冏

冏

冏

⭸␷ ⭸␷ 1 ⭸2␷ ⌬xi ⫹ ⌬yi ⫹ ⌬x2 ⭸x 0 ⭸y 0 2! ⭸x2 0 i

冏

⭸2␷ 1 ⭸2␷ ⌬xi⌬yi ⫹ ⌬y2, ⭸x⭸y 0 2!⭸y2 0 i

共4b兲

where (u0, ␷0) are the horizontal wind components at the interior station located at (x0, y0), ⌬xi ⫽ (xi ⫺ x0), and ⌬yi ⫽ ( yi ⫺ y0). The system of 10 equations and 10 unknowns is solved to yield an estimate of the divergence at the interior station of each pentagon.

4. Analyzing divergence estimates to a grid The irregularly spaced divergence estimates at the triangle centroids and the interior of pentagons are analyzed separately to a 34 ⫻ 30 grid using a three-pass Barnes objective analysis scheme (hereafter BOA3; Barnes 1964, 1973; Achtemeier 1987, 1989). The BOA3 weighting function is given by 2

⫺Rk

wk ⫽ e共ci⌬兲2,

共5兲

599

FIG. 3. A sampling of the pentagonal tessellation of the observation network.

where wk is the weight of the kth observation, Rk is the distance between an observation and a grid point, ci is the smoothing parameter used during the ith analysis pass, and ⌬ is the average data separation. The nominal grid spacing is 40 km. Following the suggestion of Achtemeier (1989), we use a relatively large value of the smoothing parameter for the first pass (c1 ⫽ 1.75). The shape parameters for the two correction passes are chosen such that c2 ⫽ c3 ⬅ c2,3. To determine the value of c2,3 that provides the best analysis, as measured by the root-mean-square error (section 5a), c2,3 is varied from 0.5 to 2.5 in increments of 0.1. No “cutoff radius” is used in our analysis scheme.

5. Comparison of the triangle and pentagon methods The root-mean-square error (rmse) is used to quantify the results of the two methods. Each method is applied to four variations of the analytic wind field (L ⫽ 5⌬, L ⫽ 10⌬, L ⫽ 15⌬, and L ⫽ 20⌬) using various values of c2,3 as described in section 4. The rmse is calculated at the triangle centroids and interior stations of pentagons (i.e., before the BOA3 scheme is applied) as well as from the gridded analyses (i.e., after the BOA3 scheme is applied). We define the “best” method at each of the wavelengths as the method that produces the lowest rmse.

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FIG. 4. Contour plots of divergence from the (a) analytic function, (b) triangle method, and (c) pentagon method for L ⫽ 10⌬. The analysis parameters are c1 ⫽ 1.75 and c2,3 ⫽ 1.0. For these plots, ␸x ⫽ ␸y ⫽ 0. In (b) and (c), observation locations are indicated by an x. The rmse’s (s⫺1) for the analyses are shown in the upper-right corners of (b) and (c). The contour interval for all plots is the same.

a. Error estimation The rmse computed from an analysis is defined as follows:

rmse ⫽

冑

兺共␦

g

⫺ ␦a兲2

i,j

Ng

,

共6兲

where ␦g represents a divergence analysis, ␦a represents the analytic divergence at grid points, and Ng is the number of grid points in the verification domain. To avoid contamination of the rmse by boundary errors, the verification domain is limited to the innermost onehalf of the analysis domain. A similar equation is used to compute rmse’s at triangle centroids and at the interior station of pentagons.

b. Results using errorless observations Figures 4 and 5 illustrate the differences between the two methods for L ⫽ 10⌬ and L ⫽ 20⌬, waves

considered marginally well sampled and well sampled, respectively (Doswell and Caracena 1988). For both L ⫽ 10⌬ (Fig. 4) and L ⫽ 20⌬ (Fig. 5), the rmse values (shown in the upper-right corner of each analysis) indicate that the checkerboard field is represented best by the triangle method. The analyses from the pentagon method contain slight distortions that are not as readily apparent in the triangle analyses. Several conclusions may be drawn from Fig. 6, which compares the performances of the triangle and pentagon methods for various values of c2,3. First, as the wavelength increases, the rmse’s for both methods decrease. Errors generally decrease by more than an order of magnitude as the wavelength increases from L ⫽ 5⌬ to L ⫽ 20⌬. Clearly, for a given observing network, longer wavelengths are sampled better than shorter wavelengths, thus reducing the analysis error. We note also that the amplitude of the divergence field decreases as the wavelength increases (Fig. 7), which also

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FIG. 5. Same as in Fig. 4 except L ⫽ 20⌬, ␸x ⫽ ⫺␲/2, and ␸y ⫽ ␲/2.

partly explains the decrease in errors as the wavelength increases. Second, the rmse values have a U-shaped pattern, which is consistent with the findings of Barnes (1994a) and Smith et al. (1986). Generally, the lowest rmse values are associated with smoothing parameters in the range 0.8 ⱕ c2,3 ⱕ 1.0. However, at L ⫽ 5⌬, slightly lower values of c2,3 are required to minimize the rmse’s. When “excessively small” smoothing parameters are used (e.g., c2,3 ⬍ 0.8), an overfitting of the observations creates poor derivative estimates in data-void regions, yielding relatively high rmse values (e.g., Barnes 1994a; Spencer and Doswell 2001). On the other hand, with “excessively large” smoothing parameters (e.g., c2,3 ⬎1.4), the analysis is too smooth. In addition, increasingly large smoothing parameters cause boundary errors to creep toward the center of the domain, although we have attempted to reduce these errors by restricting our verification domain as mentioned in section 5a. Third, the average rmse’s of the preanalyzed diver-

gence estimates4 at triangle centroids and interior stations of the pentagons (labeled T and P, respectively, in Fig. 6) suggest that for undersampled and marginally well-sampled waves [L ⫽ 5⌬ and L ⫽ 10⌬, respectively; Doswell and Caracena (1988)], the divergence estimates from the triangle method are superior to those of the pentagon method, whereas for the well-sampled waves (L ⱖ 15⌬), the reverse is true. While all of the wavelengths considered are nonlinear, the L ⫽ 5⌬ and L ⫽ 10⌬ waves are the most nonlinear with respect to station separation (Fig. 8). For this reason, intuition suggests that at these wavelengths the pentagon method should be superior to the triangle method. Clearly this is not the case. Apparently, for L ⱕ 10⌬, the linearity assumption over the relatively small triangles (average area ⫽ 6800 km2) is superior to 4 These rmse’s refer to the average rmse values at the triangle centroids and interior stations of pentagons before the divergence estimates are analyzed to the grid via BOA3. Since errorless observations are used, these errors represent truncation errors.

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FIG. 6. Rmse’s as a function of c2,3 for the triangle (short-dashed curves) and pentagon (solid curves) methods for (a) L ⫽ 5⌬, (b) L ⫽ 10⌬, (c) L ⫽ 15⌬, and (d) L ⫽ 20⌬. All rmse’s have been multiplied by 106. The T in each of the plots indicates the average rmse of the preanalyzed divergence estimates at triangle centroids; the P represents the average rmse of the preanalyzed divergence estimates at the interior station of pentagons. For all curves, c1 ⫽ 1.75. Each of the curves represents averages of nine analyses whose phase shifts (␸x, ␸y) are (⫺␲/4, ␲/4), (0, ␲/4), (␲/4, ␲/4), (⫺␲/4, 0), (0, 0), (␲/4, 0), (⫺␲/4, ⫺␲/4), (0, ⫺␲/4), and (␲/4, ⫺␲/4).

the quadratic assumption over the relatively large pentagons (average area ⫽ 31 000 km2). For the larger waves (L ⱖ 15⌬), however, the reverse is true; namely, the linearity assumption over the relatively small triangles is inferior to the quadratic assumption over the relatively large pentagons. At these larger wavelengths, the nonlinearity with respect to station separation is much smaller (Fig. 8), but still large enough to allow superior analyses from the pentagon method. Figure 6 also indicates that after BOA3 is applied to

the irregularly spaced divergence estimates, the rmse minima from the triangle method are lower than the minima from the pentagon method for all wavelengths. Interestingly, for all wavelengths, the rmse’s of the pentagon analyses are greater than the rmse’s of the preanalyzed divergence estimates, whereas for wavelengths exceeding L ⫽ 5⌬, the minimum rmse’s of the triangle analyses are smaller than the rmse’s of the preanalyzed divergence estimates. We believe this to be a consequence of two factors. First, we find that not only does the triangle method involve more than

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better analysis from the triangle method.5 The nonuniformity ratio of the triangle centroid network is more than four times smaller than that of the pentagon network. Second, because the signs of the truncation errors of the divergence estimates at triangle centroids are somewhat randomly distributed in space (dashed lines in Fig. 9; Fig. 10), it appears that the application of BOA3 acts to reduce the effects of these errors. This is supported by the fact that nearly 70% of the triangle line segments in both Figs. 10a and 10b separate triangles whose truncation errors are of opposite sign. This cancellation of truncation errors is not as prevalent for the pentagon method, where the distribution of the signs of the errors is somewhat clustered and generally corresponds to the pattern of the divergence field (solid lines in Fig. 9; Fig. 11).

FIG. 7. Amplitude of the analytic divergence field as a function of wavelength (expressed as a multiple of the data spacing ⌬). The amplitudes have been multiplied by 105.

5 The nonuniformity ratio (Smith et al. 1986) is a measure of the spatial irregularity of an observing network. The nonuniformity ratio of a perfectly regular distribution of observations is zero.

twice the number of divergence estimates than does the pentagon method (361 versus 165), but the regularity of the spatial distribution of the divergence estimates, as measured by the nonuniformity ratio, favors a

FIG. 8. Relationship between the wavelength (expressed as a multiple of the data spacing ⌬) and the ratio of the magnitude of the sum of the nonlinear terms to the magnitude of the sum of the linear terms from the quadratic Taylor series representation of the wind field. As in Fig. 6, the curve represents averages of nine sets of analyses.

FIG. 9. Average distances from triangle centroids to the nearest centroid whose divergence error is of the same sign (top dashed curve) and of the opposite sign (bottom dashed curve) and average distances from interior stations of pentagons to the nearest interior station whose divergence error is of the same sign (bottom solid curve) and of the opposite sign (top solid curve). The distances are normalized by the respective average separation between divergence estimates (68 km for the triangle centroids and 125 km for the interior stations of pentagons). The wavelength is expressed as a multiple of the data spacing ⌬. These curves provide a measure of the degree of clustering of errors of like sign. The relatively large separation of the solid curves indicates that divergence errors of the same sign are much more clustered for the pentagon method than for the triangle method.

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FIG. 10. Signs of the differences between the analytic divergence at triangle centroids and the divergence computed from the triangle method for (a) L ⫽ 10⌬ and (b) L ⫽ 20⌬. Plus signs indicate that the computed divergence is greater than the analytic divergence. The analytic divergence used in (a) is the same as that presented in Fig. 4a and the analytic divergence in (b) is the same as that presented in Fig. 5a.

FIG. 11. Signs of the differences between the analytic divergence at the interior station of pentagons and the divergence computed from the pentagon method for (top) L ⫽ 10⌬ and (bottom) L ⫽ 20⌬. Plus signs indicate that the computed divergence is greater than the analytic divergence. The analytic divergence used in (top) is the same as that presented in Fig. 4a and the analytic divergence in (bottom) is the same as that presented in Fig. 5a.

Figure 12 indicates that for all wavelengths considered, larger triangle and pentagon areas are associated with larger truncation errors. Even a highly nonlinear field varies approximately linearly across small triangles so that the linearity assumption of the triangle method produces a reasonable estimate of the divergence. For larger triangles, the degree of nonlinearity of the flow within the triangle is larger, resulting in greater errors associated with the linearity assumption. A simi-

lar argument can be made for pentagon area versus error. For the smallest wave considered (L ⫽ 5⌬), even the smallest pentagons, with their less restrictive quadratic assumption, have truncation errors exceeding those of small triangles (Fig. 12a). Specifically, approximately 55 of the 361 triangles (⬃15%) have truncation errors less than those of the smallest pentagons. However, as the wavelength increases, an increasingly large number of

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FIG. 12. Rmse’s of the preanalyzed divergence estimates from the triangle (dashed curves) and pentagon (solid curves) methods as a function of the area of the triangle or pentagon for (a) L ⫽ 5⌬, (b) L ⫽ 10⌬, (c) L ⫽ 15⌬, and (d) L ⫽ 20⌬. The areas of the triangles are binned in 2000-km2 increments. Errors of all triangles associated with each bin are averaged to create the dashed curves. A similar procedure is used for the pentagons, except the bin increment is 8000 km2. As in Fig. 6, the curves represent averages of nine analyses.

pentagons have truncation errors less than those of the smallest triangles. For example, Fig. 12b indicates that for any pentagon whose area is less than about 18 000 km2, the quadratic assumption is better than the linearity assumption for all triangles; for L ⫽ 20⌬, the quadratic assumption is better for any pentagon whose area is less than about 38 000 km2 (Fig. 12d). This issue of the size of the polygon versus the truncation error is related to the “scale of the nonlinearity” discussion found in Doswell and Caracena (1988). Clearly there is a trade-off between the size of the polygon (i.e., which method is used) and the truncation error. The triangle method truncates more of the Taylor

series, yet the relative smallness of the triangles counterbalances—in a positive way—to some degree this more restrictive assumption. On the other hand, the pentagon method truncates less of the Taylor series, yet the relative largeness of the pentagons counterbalances—in a negative way—to some degree this less restrictive assumption.

c. The effect of random observation errors Up to this point, the observations have been assumed to be free of errors. This is a common assumption when analysis comparisons are made using analytic observations (e.g., Barnes 1994a,b; Smith and Leslie 1984;

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FIG. 13. Same as in Fig. 6, except that random, Gaussian observation errors have been included. See text for details.

Trapp and Doswell 2000). Although the use of errorfree observations provides a good platform for analysis comparisons, we believe that it is important to understand how observational errors affect the conclusions drawn from such comparisons. We now test the robustness of our results by introducing errors into the observations that represent those associated with imperfect instruments. Such errors usually are assumed to have a Gaussian distribution (Lorenc and Hammon 1988). Errors randomly drawn from a Gaussian distribution with a standard deviation (␴) of 0.25 m s⫺1 are added to each observation. The choice of ␴ ⫽ 0.25 m s⫺1 results in a median observation error of about 5%, a generous error for modern observing systems (e.g., Brock et al.

1995). Since the error distribution is Gaussian, about 68% of the observation errors have a magnitude less than 0.25 m s⫺1 and nearly 95% of the errors have a magnitude less than 0.5 m s⫺1. As expected, when random errors are added to the observations, rmse’s for both the triangle and pentagon methods generally increase (cf. Figs. 13 and 6). Increases in the rmse’s of the preanalyzed divergence estimates are substantially greater than the increases in the rmse’s of the divergence analyses, suggesting that the application of BOA3 to the irregularly distributed divergence estimates helps reduce the adverse impact of the random observation errors. In fact, when observation errors are included, the application of BOA3 to

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the irregularly distributed divergence estimates at the interior of pentagons reduces the rmse’s for all wavelengths, except for L ⫽ 5⌬. This was not the case when errorless observations were used (Fig. 6). As before, however, applying BOA3 to the irregularly distributed divergence estimates at triangle centroids reduces the rmse’s for most wavelengths. Earlier, we found that when errorless observations were used, the rmse’s of the preanalyzed divergence estimates at the interior of pentagons were lower than the rmse’s of the divergence estimates at triangle centroids for L ⫽ 15⌬ and for L ⫽ 20⌬ (see the relative positions of the Ts and Ps in Figs. 6c and 6d). However, when the observations include errors, the rmse’s of the preanalyzed divergence estimates at triangle centroids are lower than the rmse’s of the divergence estimates at the interior of pentagons for all wavelengths considered (Fig. 13). Finally, Fig. 13 suggests that larger smoothing parameters within the BOA3 scheme are required to minimize analysis errors when the observations contain errors. Clearly, less fit to the observations is desired with the introduction of observation errors.

6. Summary and conclusions This study has shown that for computing gridded fields of divergence from irregularly spaced, error-free wind observations, the triangle method outperforms the pentagon method for a variety of wavelengths, ranging from undersampled, to marginally well sampled, to well sampled. The preanalyzed divergence estimates from the pentagon method are superior to those from the triangle method for the well-sampled waves (L ⫽ 15⌬ and L ⫽ 20⌬), but are inferior for the undersampled and marginally well-sampled waves (L ⫽ 5⌬ and L ⫽ 10⌬, respectively). We find this to be an interesting—if not counterintuitive—result since the nonlinearity of the flow with respect to the station separation is greatest for the smaller waves and the stated purpose of the pentagon method is to capture a portion of that nonlinearity. Apparently, the linearity assumption over the relatively small triangles is superior to the quadratic assumption over the relatively large pentagons for the smaller waves; the opposite is true for the larger waves. We have found that the rmse’s of the gridded analyses from the pentagon method are higher than the rmse’s of the preanalyzed divergence estimates at the interior stations of pentagons. On the other hand, we found that the lowest rmse’s of the gridded analyses from the triangle method generally are lower than the rmse’s of the preanalyzed divergence estimates at the

centroids of triangles. We consider this behavior to be a consequence of both the number of irregularly spaced divergence estimates (361 for the triangle method and 165 for the pentagon method) and the smoothing effect of the BOA3 scheme on the somewhat randomly distributed truncation errors of the triangle centroid divergence estimates. These results indicate that when gridded analyses are not desired or not feasible, such as with a small network of observing stations, then the pentagon method for obtaining point values of divergence (or any of the other kinematic quantities) is a viable—if not preferable—option for large-scale phenomena. Otherwise, the triangle method appears to be the best option. When the observations contain random Gaussian errors, all advantages of the pentagon method over the triangle method that were found for errorless observations disappear. Namely, when the observations include errors, the divergence estimates at triangle centroids are superior to those at the interior of pentagons for all wavelengths considered. In addition, the gridded analyses obtained from the triangle method are superior to those obtained from the pentagon method, as well. Therefore, when real observations are used to compute kinematic quantities from a surface network, the triangle method appears to be the best option. Acknowledgments. A large portion of this work was completed while the first author participated in the Research Experiences for Undergraduates (REU) program at the National Severe Storms Laboratory (NSSL) during the summer of 2004 with the support of National Science Foundation Grant 0097651. This research also was supported under NOAA–OU Cooperative Agreement NA17RJ1227. We thank Daphne Zaras (NSSL) for coordinating the REU program and her assistant, Lance Maxwell. We wish to acknowledge Dr. Charles Doswell and Mr. Dave Watson for providing us the Delauney triangulation code and Dr. Kim Elmore for providing us matrix inversion software. Thanks to Dr. Harold Brooks for his suggested improvement to Fig. 9. Finally, we thank Dr. David Schultz, Dr. David Stensrud, and three anonymous reviewers for their helpful comments.

APPENDIX Pentagon Selection Algorithm The algorithm for defining pentagons around each observing station begins by sorting, for each station, the distances to each of the other stations. The 15 nearest neighbors to a given observing station are used to cre-

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ate all possible pentagons such that the given observing station is enclosed within each pentagon. A station lies within the interior of a pentagon if the sum of the angles generated by lines emanating from the interior station to each of the five stations composing the pentagon equals 360°. All pentagons meeting this criteria for a particular interior station are candidate pentagons. The best pentagon for each interior station is considered to be the candidate pentagon that minimizes 兺5i⫽1 di, where di is the distance from the interior station to the ith station composing the surrounding pentagon, subject to two constraints. The first constraint requires that the interior station be no closer than 0.25⌬ from any edge of the candidate pentagon, where ⌬ is the average data spacing. The purpose of this constraint is to keep the interior station close to the middle of the pentagon. If the interior station is closer than this distance to any edge, then the candidate pentagon is removed from further consideration. The second constraint requires that the angle nearest the interior station that is produced by connecting the interior station with two adjacent neighbors of a candidate pentagon be no less than 10°. The purpose of this constraint is to create pentagons whose stations are somewhat symmetrically placed about the interior station, a guideline used by Chien and Smith (1973). REFERENCES Achtemeier, G. L., 1987: On the concept of varying influence radii for successive corrections objective analysis. Mon. Wea. Rev., 115, 1760–1771. ——, 1989: Modification of a successive corrections objective analysis for improved derivative calculations. Mon. Wea. Rev., 117, 78–86. Barnes, S. L., 1964: A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3, 396–409. ——, 1973: Mesoscale objective analysis using weighted timeseries observations. NOAA Tech. Memo. ERL NSSL-62, National Severe Storms Laboratory, Norman, OK, 41 pp. ——, 1994a: Applications of the Barnes object analysis scheme. Part I: Effects of undersampling, wave position, and station randomness. J. Atmos. Oceanic Technol., 11, 1433–1448. ——, 1994b: Applications of the Barnes object analysis scheme. Part II: Improving derivative estimates. J. Atmos. Oceanic Technol., 11, 1449–1458. Bellamy, J. C., 1949: Objective calculations of divergence, vertical velocity and vorticity. Bull. Amer. Meteor. Soc., 30, 45–49. Bosart, L. F., and F. Sanders, 1981: The Johnstown flood of July 1977: A long-lived convective system. J. Atmos. Sci., 38, 1616–1642. Brandes, E. A., 1990: Evolution and structure of the 6–7 May 1985 mesoscale convective system and associated vortex. Mon. Wea. Rev., 118, 109–127. Brock, F. V., K. C. Crawford, R. L. Elliot, G. W. Cuperus, S. J. Stadler, H. L. Johnson, and M. D. Eilts, 1995: The Oklahoma

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Mesonet: A technical overview. J. Atmos. Oceanic Technol., 12, 5–19. Ceselski, B. F., and L. L. Sapp, 1975: Objective wind field analysis using line integrals. Mon. Wea. Rev., 103, 89–100. Chien, H., and P. J. Smith, 1973: On the estimation of kinematic parameters in the atmosphere from radiosonde wind data. Mon. Wea. Rev., 101, 252–261. ——, and ——, 1977: Synoptic and kinetic energy analyses of Hurricane Camille (1969) during transit across the southeastern United States. Mon. Wea. Rev., 105, 67–77. Davies-Jones, R., 1993: Useful formulas for computing divergence, vorticity, and their errors from three or more stations. Mon. Wea. Rev., 121, 713–725. Doswell, C. A., III, and F. Caracena, 1988: Derivative estimation from marginally sampled vector point functions. J. Atmos. Sci., 45, 242–253. Koch, S. E., M. desJardins, and P. J. Kocin, 1983: An interactive Barnes objective map analysis scheme for use with satellite and conventional data. J. Climate Appl. Meteor., 22, 1487– 1503. Lorenc, A. C., and O. Hammon, 1988: Objective quality control of observations using Bayesian methods. Theory, and a practical implementation. Quart. J. Roy. Meteor. Soc., 114, 515–543. Menard, R. D., and J. M. Fritsch, 1989: A mesoscale convective complex-generated inertially stable warm core vortex. Mon. Wea. Rev., 117, 1237–1261. Portis, D. H., and P. J. Lamb, 1988: Estimation of large-scale vertical motion over the central United States for summer. Mon. Wea. Rev., 116, 622–635. Ripley, B. D., 1981: Spatial Statistics. Wiley-Interscience, 252 pp. Roebber, P. J., J. M. Frederick, and T. P. DeFelice, 1998: Persistent low overcast events in the U.S. upper midwest: A climatological and case study analysis. Wea. Forecasting, 13, 640– 658. Schaefer, J. T., and C. A. Doswell III, 1979: On the interpolation of a vector field. Mon. Wea. Rev., 107, 458–476. Smith, D. R., and F. W. Leslie, 1984: Error determination of a successive correction type objective analysis scheme. J. Atmos. Oceanic Technol., 1, 120–130. ——, M. E. Pumphry, and J. T. Snow, 1986: A comparison of errors in objectively analyzed fields for uniform and nonuniform station distributions. J. Atmos. Oceanic Technol., 3, 84–97. Spencer, P. L., and C. A. Doswell III, 2001: A quantitative comparison between traditional and line integral methods of derivative estimation. Mon. Wea. Rev., 129, 2538–2554. ——, F. H. Carr, and C. A. Doswell III, 1996: Diagnosis of an amplifying and decaying baroclinic wave using wind profiler data. Mon. Wea. Rev., 124, 209–223. Steinacker, R., C. Häberli, and W. Pöttschacher, 2000: A transparent method for the analysis and quality evaluation of irregularly distributed and noisy observational data. Mon. Wea. Rev., 128, 2303–2316. Stensrud, D. J., and R. A. Maddox, 1988: Opposing mesoscale circulations: A case study. Wea. Forecasting, 3, 189–204. Trapp, R. J., and C. A. Doswell III, 2000: Radar data objective analysis. J. Atmos. Oceanic Technol., 17, 105–120. Ward, J. H., and P. J. Smith, 1976: A kinetic energy budget over North America during a period of short synoptic wave development. Mon. Wea. Rev., 104, 836–848. Zamora, R. J., M. A. Shapiro, and C. A. Doswell III, 1987: The diagnosis of upper tropospheric divergence and ageostrophic wind using profiler wind observations. Mon. Wea. Rev., 115, 871–884.