Improving Accuracy of Mapping Cyclone Numbers and Frequencies

748 MONTHLY WEATHER REVIEW VOLUME 130

NOTES AND CORRESPONDENCE

Improving the Accuracy of Mapping Cyclone Numbers and Frequencies

OLGA ZOLINA AND SERGEY K. GULEV P. P. Shirshov Institute of Oceanology, Moscow, Russia

20 October 2000 and 1 August 2001

ABSTRACT The uncertainties associated with the mapping of cyclone numbers and frequencies are analyzed using the 42-yr winter climatology of cyclone tracks derived from 6-hourly NCEP±NCAR reanalysis. Tracking is performed using an automated procedure, based on computer animation of the sea level pressure ®elds. Uncertainties in the mapping result from an incomplete catchment of cyclones by the grid cells: the coarse temporal resolution of data causes fast-moving storms to skip grid boxes. This introduces error into estimates of cyclone frequency, with cyclone counts systematically underestimated. To minimize these biases, it is possible to simulate higher temporal resolution of the storm tracks by linear interpolation applied to the original tracks. This simple procedure reduces bias in estimates of both cyclone frequencies and numbers and enables quantitative estimation of errors. Errors in cyclone counts are estimated over the Northern Hemisphere for different time resolutions and different grids. Standard errors in cyclone frequencies increase from 5%±15% to as much as 50% as the original temporal resolution of the storm tracks decreases from 6 to 24 h. Mapping the results on circular grids reduces these errors. Appropriate grid sizes for different temporal resolutions of the storm tracks are recommended to minimize the uncertainties in storm occurrence mapping.

1. Introduction found that normalization by latitude results in biases in cyclone occurrence and recommended the mapping of Cyclone activity is usually characterized by cyclone raw frequencies or use of equal-area grids (e.g., Bal- frequency maps derived from sea level pressure (SLP) lenzwieg 1959). However, Hayden (1981a,b) only con- data using automated storm tracking algorithms (e.g., sidered the latitudes from 25Њ to 45ЊN, for which the Alpert et al. 1990; Le Treut and Kalnay 1990; Murray area change of a latitude±longitude box is less than 25%. and Simmonds 1991; Koenig et al. 1993; Hodges 1994; Sinclair 1994, 1997; Serreze 1995; Serreze et al. 1997; In the high latitudes, mapping becomes very sensitive Blender et al. 1997; Sinclair and Watterson 1999). to the cell size. In Arctic cyclone climatologies (Serreze Blender and Schubert (2000) recently studied the impact and Barry 1988; Serreze et al. 1993, 1997; Serreze of different space±time resolutions on the results of cy- 1995), it is a common practice to use an equal-area grid clone tracking and found that a coarser temporal reso- based on the Lambert polar stereographic projection, lution may result in 10%±50% biases in cyclone counts when the grid cells are referenced to the Cartesian co- (when the resolution decreases from 2-hourly to 24- ordinate system (Thorndike and Colony 1980). How- hourly). Decreasing spatial resolution from T106 to T42 ever, in the midlatitudes and subtropics the actual con- introduces approximately 30% bias in the number of ®guration of cells of such grids becomes very different cyclones. However, accuracy of storm tracking is not from that for the high latitudes and may not necessarily the only source of uncertainty in the analysis of cyclone provide an effective catchment of cyclones. This is the activity. The procedures involved with mapping cyclone second mapping problem, ®rst quoted by Taylor (1986). occurrences can also introduce error. To account for the varying density of the cyclone tracks For latitude±longitude grids, there has been much de- of different directions, Taylor (1986) introduced the so- bate on the choice between the so-called equal-area grids called effective cross sections, which require the use of and scaling the results with latitude. Hayden (1981b) gerrymander-type grids, which are impractical. An al- ternative approach is to use circular cells, ®rst recommended by Kelsey (1925) and used for mapping cyclone Corresponding author address: Sergey Gulev, P. P. Shirshov In- frequencies in the Southern Hemisphere by Sinclair stitute of Oceanology, RAS, 36 Nakhimovsky Ave., 117851 Moscow, Russia. (1994). Circular networks are very effective for the con- E-mail: [email protected] sideration of regional climatologies. However, on a

᭧ 2002 American Meteorological Society MARCH 2002 NOTES AND CORRESPONDENCE 749

FIG. 1. An idealized example of estimating storm frequencies (n and ni) and numbers (nc and nci) for the 10 idealized boxes from the storm tracks with original temporal resolution (n, nc) and simulated 1/6 temporal resolution (ni, nci). global scale they may result in variable spatial smooth- observer within the box during the chosen time interval ing at different latitudes. Some authors convert the raw (e.g., month, season), the cyclone frequency n. The frequencies into percent frequencies (Keegan 1958; Ser- number of cyclones that pass the box during the same reze et al. 1993). In this case, a given cell is associated time (multiple entries have to be ignored) will be de®ned with a percentage of the total number of lows over the as the number of cyclones nc. These measures corre- hemisphere. However, the use of the percent frequencies spond to the cyclone density and track density, respec- creates dif®culties in studying interannual variability in tively, used by Sinclair (1994). The relationship between the cyclone activity. these two is not trivial. Cyclone frequency is in¯uenced The third problem of cyclone frequency mapping is by both number of transients and their velocities, where- the dependence of the catchment of cyclones on the as number of cyclones is less affected by the propagation temporal resolution of storm tracks. No grid (be it rect- velocity. If we consider a single cyclone tracking, the angular, circular, or other) can account fully for fast- derived cyclone frequency (the so-called partial fre- moving cyclones that pass one or more grid cells during quency) is equal to the time of the cyclone's passing one time step. These biases tend to underestimate storm through the box. It has an accuracy of Ϯ2t. counts. Murray and Simmonds (1991), Sinclair (1994), If the typical cyclone migration during one time step and Chandler and Jonas (1999) interpolated 12-hourly is comparable to the box size, both cyclone frequency track points to 6-hourly spacing to account for this prob- and the number of cyclones will be biased. Biases will lem. Nevertheless, a general approach to minimizing increase for fast cyclones (boxes 4 and 9 in Fig. 1) and and understanding this bias has yet to be well developed. for the cyclones ``cutting'' the box corners (box 7). Our study will quantify these biases for different map- Biases can be minimized if we simulate a higher tem- ping geometries and will suggest a simple procedure for poral resolution by linear interpolation between the their minimization. In section 2 we formulate the prob- storm-track points. This procedure has to be applied to lem using an idealized example. Section 3 describes the the results of the tracking and not to the initial SLP data and the method used to perform storm tracking. ®elds; that is, it does not imply any assumptions about Results of the quantitative estimation of the mapping the development of the SLP ®eld during a t-hourly pe- uncertainties associated with the resolution of storm riod and does not change the storm tracks. Linear in- tracks for different grids are presented in section 4. Sec- terpolation results in the same tracks but with t/6 tem- tion 5 summarizes the results and discusses their pos- poral resolution (Fig. 1). The estimates of cyclone fre- sible applications. quencies and numbers from the interpolated data (ni and nci, respectively) are less biased (their accuracy is Ϯt/3). To compare the frequencies derived from the original 2. Formulation of the problem and interpolated data, partial frequencies have to be An idealized example, summarizing the problem, is scaled with the cyclone lifetime. Table 1 shows the fre- shown in Fig. 1. Two cyclones (I and II) are tracked in quencies derived from the original and interpolated data. the 10 neighboring boxes for the original temporal res- The coarse temporal resolution results in biases of both olution t, which varies in practice from 6 to 24 h for signs, which vary in magnitude from 50% to 100% of different analyses and models. We will term the number the mean values. If we enlarge the box size (holding of the pressure minimum events, counted by an Eulerian the time step constant), the biases reasonably become 750 MONTHLY WEATHER REVIEW VOLUME 130

TABLE 1. Partial and total cyclone frequencies for idealized boxes in Fig. 1. Original temporal Simulated 1/6 Box No. resolution temporal resolution Difference 1 1/6 ϭ 0.16 3/31 ϭ 0.09 0.07 2 1/6 ϩ 1/5 ϭ 0.37 6/31 ϩ 1/25 ϭ 0.23 0.14 3 2/6 ϭ 0.33 10/31 ϭ 0.32 0.01 4 0 4/31 ϩ 1/25 ϭ 0.18 Ϫ0.18 5 2/6 ϩ 1/5 ϭ 0.54 8/31 ϩ 2/25 ϭ 0.34 0.2 6 1/5 ϭ 0.20 3/25 ϭ 0.12 0.08 7 0 6/25 ϭ 0.24 Ϫ0.24 8 1/5 ϭ 0.20 5/25 ϭ 0.20 0 9 0 2/25 ϭ 0.08 Ϫ0.08 10 1/5 ϭ 0.20 5/25 ϭ 0.20 0 1 ϩ 2 ϩ 6 ϩ 7 2/6 ϩ 2/5 ϭ 0.73 9/31 ϩ 10/25 ϭ 0.69 0.04 4 ϩ 5 ϩ 9 ϩ 10 2/6 ϩ 2/5 ϭ 0.73 12/31 ϩ 10/25 ϭ 0.78 Ϫ0.05 Total 2.00 2.00 0 smaller. In the case of a very large box (say, the Atlantic hourly (0000 and 1200 UTC) and 24-hourly (0000 UTC) Ocean, or the Northern Hemisphere), the biases are resolution. This procedure has been applied to the storm equal to zero. However, such large cells do not allow tracks derived from the original resolution and not to detailed mapping of regional cyclone numbers and fre- the SLP data; that is, it did not affect the accuracy of quencies and provide only large-scale estimates, which the tracking itself. are, however, more accurate than the box averages. To get the reference estimates of cyclone frequencies Thus, mapping cyclone frequencies and numbers re- and numbers, the original 6-hourly and simulated 12- quires either analysis of the relationship between the and 24-hourly storm tracks were linearly interpolated grid size and temporal resolution or interpolation be- on time steps of 10 min (36 points per one 6-hourly tween time steps for a given grid geometry. step). This guarantees nearly negligible biases. A typical cyclone migration during 10 min varies from 3 to 15 3. Data and preprocessing km, which is much smaller than the dimensions of the grid used to map the results. We examined three quasi- We used the results of storm tracking performed (Gu- rectangular grids with the basic dimensions south of lev et al. 2001) for the 42 winter seasons (January± 70ЊNof5Њϫ5Њ,5Њϫ10Њ, and 10Њϫ10Њ (hereinafter March) from 1958 to 1999 from the 6-hourly National called grids 1, 2, and 3; see Fig. 2). North of 70ЊNwe Centers for Environmental Prediction±National Center used specially designed grids with a larger box size. For for Atmospheric Research (NCEP±NCAR) reanalysis example, the Arctic counterpart of the 5Њϫ5Њ basic (Kalnay et al. 1996) SLP ®elds using the software of grid starts with the circle rounded at 88ЊN and then uses Grigoriev et al. (2000). This software is based on a the boxes with longitudinal dimensions of 15Њ (88Њ± computer analysis of the SLP ®elds and simulates the 80ЊN), 12Њ (80Њ±75ЊN), and 10Њ (75Њ±70ЊN). All cyclone manual procedure but makes it faster and less dependent frequencies and numbers for every grid were then nor- on the subjective view and mistakes of a particular op- malized by latitude. To estimate the uncertainties for a erator. The cyclone centers are de®ned automatically as circular grid, we performed the averaging for circular a minimum pressure with respect to the eight neigh- cells of radius of 555 km, as recommended by Sinclair boring points, and then the tracking is carried out by (1994). an interactive procedure. The output of the tracking is the coordinates, time, and corresponding SLP values. For use with the Grigoriev et al. (2000) software, we 4. Results interpolated 2.5Њ SLP ®elds onto a 181 ϫ 181 polar To assess the problem, we ®rst examined character- stereographic grid, using a modi®ed version of a pro- istics of cyclone propagation that are important for the cedure of Akima (1970). Tracking errors, estimated estimation of mapping biases. Figures 3a and 3b show from the outputs of different operators for the selected climatological winter maps of the mean and maximum years, were less than 5% (Gulev et al. 2001). Although cyclone propagation velocities, computed from the 6- the NCEP±NCAR reanalysis provides 6-hourly SLP hourly tracks. Mean cyclone propagation velocity (Fig. data, a coarser time resolution (12 and 24 h) is com- 3a) varies from a minimum of 25 to a maximum of 75 monly used for cyclone analysis. Thus, the resolution km hϪ1. The largest mean velocities are observed in the of the NCAR SLP archive (Trenberth and Paolino 1980) central Paci®c and over the eastern American coast. The is 12±24 h. Stein and Hense (1994) computed their regions of the slowest mean cyclone velocities are as- storm counts using daily data. To estimate the uncer- sociated with the subpolar Paci®c and Arctic basins. The tainties in mapping storm tracks of coarser temporal highest maximum velocities are ranged from 100 to 120 resolution, we derived datasets that correspond to 12- km hϪ1 and are associated with the subtropical Paci®c, MARCH 2002 NOTES AND CORRESPONDENCE 751

FIG. 2. The arrangement of the grid cells for grids 1 (thin lines) and 2 (bold lines) used in this study.

American continent, and the North Atlantic midlatitudes by the same cell (rt). Figure 3e shows the spatial dis- (Fig. 3b). Figures 3c and 3d demonstrate the mean cy- tribution of the climatological ratio, clone propagation vectors and the characteristics of their directional steadiness, which was estimated in percent R ϭ͗rm6 /rt͘ϳUT/L, (1) for 45Њ directional classes. Average cyclone propagation for grid 1. If the ratio R is less than 1, the biases in directions are similar to climatological geostrophic cyclone frequencies and numbers are negligible. For the winds. They are primarily eastward over the Gulf areas where R is greater than 1, the biases associated Stream and Kuroshio Current and turn to the northeast with the poor catchment of transients by the grid cells over the open midlatitudinal and subpolar Atlantic and will increase. Figure 3e shows that grid 1 is not effec- Paci®c. In the Arctic the preferential propagation of tive, even for the 6-hourly storm tracks in most of the cyclones is to the northeast. Figure 3d shows the dom- locations. The highest expected errors are observed in inating eastward and northeastward cyclone propagation midlatitudes and in the Arctic basin. North of 30ЊN, R over the North Atlantic and North Paci®c. However, varies from 1 to 4, indicating that the size of the grid some cyclones propagate in other directions. Over other cells in high latitudes was more or less properly selected. regions, the scatter in the cyclone propagation directions For instance, the use of the 5Њ grid north of 70ЊN results is higher. Thus, it is dif®cult to select a single direction, already in R values of about 20±30. determining the so-called effective cross section (Taylor As a basis for further analysis we show ®rst the winter 1986), that guarantees the effective catchment of lows. cyclone frequencies and numbers (Figs. 4a,b) computed It is important to introduce the relationship between for grid 1 using results of cyclone tracking interpolated a length scale L, a timescale T, and a speed U. From onto 10-min steps. They exhibit the number of pressure the cyclone propagation velocity we derived for every minima and the number of cyclones per winter per grid cell an estimate of the length of cyclone migration 218 000 km2 (square of the 5Њϫ5Њ box at 45ЊN). Gen- during 6-hourly time step (rm6). It can be compared eral features of these charts were discussed in Gulev et with the actual length of the cyclone trajectory captured al. (2001). The highest frequencies are associated with 752 MONTHLY WEATHER REVIEW VOLUME 130

FIG. 3. Winter climatological (a) mean and (b) maximum cyclone propagation velocities (km h Ϫ1), (c) climatological cyclone propagation vectors (km h Ϫ1), (d) directional steadiness of the cyclone propagation for the selected regions (%), and (e) ratio R for grid 1. Contour intervals are (a) 10 km hϪ1, (b) 20 km hϪ1, and (c) 1. MARCH 2002 NOTES AND CORRESPONDENCE 753

FIG. 4. Winter climatological cyclone frequency (events per winter), referenced to (a) 6-hourly time steps and (b) the number of cyclones (cyclones per winter), (c) cyclone frequency normalized with the cyclone lifetime, and (d) the spatial distribution of the absolute errors in the cyclone frequencies for 6-hourly resolution and grid 1. the Aleutian low and Icelandic low. The maxima of the Figure 4d shows the absolute difference between the number of cyclones are located over the Gulf Stream normalized cyclone frequencies, derived from the in- and Kuroshio Current, where they are higher than in the terpolated and the original 6-hourly storm tracks for grid Aleutian and Icelandic low regions. Cyclone frequencies 1. The spatial distribution of the differences shows that in Fig. 4a are referenced to 6-hourly time steps. To coarse temporal resolution results in both positive and compare cyclone frequencies derived from the data with negative biases in the storm frequencies, which lie with- different temporal resolutions, we show in Fig. 4c the in 15% of the mean normalized cyclone frequency (Fig. normalized (with respect to the cyclone lifetime) winter 4c). They do not exhibit any regular pattern. If the points cyclone frequency, which will be used as a reference of a coarse-resolution track are not captured by a grid for future comparisons. cell, the frequency is underestimated. However, when 754 MONTHLY WEATHER REVIEW VOLUME 130

FIG. 5. (a), (c) Relative error in the normalized cyclone frequencies and (b), (d) absolute error in the number of cyclones (cyclones per winter) for the (a), (b) 6-hourly and (c), (d) 12-hourly temporal resolution for grid 1. Contour intervals are (a) 5, (b) 1 cyclone per winter, (c) 10, and (d) 2 cyclones per winter. the coarse-resolution track points are marked in the grid the estimates of the number of cyclones derived from cell, the actual frequency, normalized with the cyclone 10-min tracks with the raw cyclone numbers for 6-hour- lifetime, may be higher than for that derived from the ly and 12-hourly tracks. Corresponding estimates for high-resolution tracks. This is easily visible from an 12- and 24-hourly temporal resolution but for grid 2, idealized example in Fig. 1. Thus, the uncertainty in are shown in Figs. 6a±d. Cyclone frequencies demon- cyclone frequencies appears random. strate the highest standard errors over continental North Figure 5 shows the standard errors in the normalized America (in the Great Lakes and Rocky Mountains re- cyclone frequencies for 6-hourly (panel a) and 12-hourly gions), the Mediterranean, and in the Arctic basin, where (panel b) storm tracks for grid 1, estimated in percent they are larger than 15% for 6-hourly temporal reso- of the frequencies derived from the interpolated (on 10- lution and grid 1. Over the other regions, standard errors min resolution) storm tracks. Figures 5c and 5d compare vary from 3% to 10%. For 12-hourly resolution and grid MARCH 2002 NOTES AND CORRESPONDENCE 755

FIG. 6. Same as Fig. 5 but for the (a), (b) 12-hourly and (c), (d) 24-hourly temporal resolution for grid 2.

1, standard error increases up to 25% over continental mation regions can be underestimated by 10%±20%. North America and the storm formation regions and up Relative errors grow considerably over the central At- to 15% in the areas of Aleutian and Icelandic lows. For lantic and central Paci®c as well as in the Icelandic and 12-hourly temporal resolution and grid 2 (Fig. 6a), the Aleutian lows, reaching 50%. The use of 12-hourly res- uncertainties grow up to 20%±30% over the main North olution for grid 1 (Fig. 5d) results in a 20%±40% un- Atlantic and North Paci®c storm tracks and in the Arctic derestimation of the number of cyclones in storm for- basin. Relative standard errors in the cyclone frequen- mation regions and in 30%±80% biases in open-ocean cies derived from 24-hourly data (Fig. 6c) may be higher regions and in the Arctic. Errors for grid 2 and a 12- than 50% for the midlatitudinal areas and in the Arctic. hourly resolution (Fig. 6b) are slightly higher than those If we consider the number of cyclones, coarse-reso- obtained for grid 1 and a 6-hourly resolution. The use lution tracking results in systematic underestimation of of a 24-hourly resolution in combination with grid 2 the cyclone counts. For a 6-hourly resolution and grid (Fig. 6d) results in biases that are 5%±15% higher than 1 (Fig. 5b), the number of cyclones in the storm for- for the 12-hourly tracking in grid 1. 756 MONTHLY WEATHER REVIEW VOLUME 130

where ␦(nc) is the absolute error in the number of cyclones. A typical number of cyclones in the main storm track areas is from 5 to 18 per winter per box for grid 1. Thus, R greater than 2.5 results in higher than 20% underestimation of cyclone counts. For coarser resolutions, the exponential dependence stands but the biases in the number of cyclones become 60% higher for 12- hourly resolution and nearly 100% higher for daily resolution. Use of grid 2 decreases the error in the number of cyclones by 1.4±2.6 times but again holds the exponential dependence valid. It is interesting to estimate whether the use of circular networks (Sinclair 1994; Sinclair and Watterson 1999) can decrease the biases in mapping cyclone numbers. We derived the estimates of the number of cyclones for a circular network with a radius of 555 km (5Њ at the equator). This grid was used by Sinclair (1994) for a cyclone climatology in the Southern Hemisphere. In terms of the catchment area, such a grid corresponds to approximately 10Њϫ10Њ boxes in low latitudes. As for the latitude±longitude grids, for circular cells we esti- FIG. 7. Scatterplot between the ratio R and absolute error in the mated the number of cyclones from 6-hourly tracks and number of cyclones for 6-hourly temporal resolution and grid 1. from the tracks interpolated onto a 10-min resolution. Spatial patterns of cyclone counts for this grid become much smoother. Multiple counts of cyclones in the over- Ratio R determined by (1) shows how effective the lapping neighboring circular cells have the effect of a chosen relationship between the grid size and temporal smoothing operator. Figure 8a shows the absolute errors resolution of tracks is. Figure 7 demonstrates the scatter in the number of cyclones derived for a given circular between this ratio and the error in the number of cyclones grid from 6-hourly storm tracks. Underestimation of the for 6-hourly resolution and grid 1. The underestimation cyclone numbers is within 5%±7% of mean values for of the number of cyclones grows exponentially with R, most regions. For the 12-hourly resolution (Fig. 8b), and the relationship can be approximated as biases increase up to 10%±15% of the mean values. The ␦(nc) ϭ 0.049 exp(1.238R), (2) largest biases are observed in the midlatitudes. These

FIG. 8. Absolute error in the number of cyclones (cyclones per winter) for the (a) 6-hourly and (b) 12-hourly temporal resolution for a circular grid with radius of 555 km. MARCH 2002 NOTES AND CORRESPONDENCE 757

TABLE 2. Relative standard errors (%) in cyclone frequencies (n) and relative absolute errors in cyclone counts (nc) for different temporal resolutions of storm tracks and different networks.

Time Grid 1 Grid 2 Grid 3 resolution (h) n nc n nc n nc 6 7.4 19.5 5.6 10.9 4.2 5.9 12 10.1 32.8 9.1 19.6 7.8 11.8 24 14.0 52.2 11.2 23.9 9.9 13.5 estimates can be compared very roughly with our results estimation of the number of cyclones derived from the for grid 3 (10Њϫ10Њ boxes). Biases estimated for 6- 6-hourly data is about 20%, 11% and 6% for grids 1, hourly tracks and grid 3 are 1.5±3 times higher than 2, and 3, respectively. For a coarser temporal resolution, those for a circular network. For the 12-hourly reso- this uncertainty grows and reaches a level of 50% for lution, this ratio is from 1 to 2. Thus, a circular averaging the daily data. The use of a circular averaging geometry geometry provides in general more effective cyclone results in the mean underestimation of the number of catchment than do the rectangular boxes. However, the cyclones of 3.9% for 6-hourly tracks and 9.8% for 12- results derived for this geometry are in¯uenced by in- hourly tracks. These are 1.2±1.5 times smaller than for adequate spatial smoothing, especially in high latitudes. grid 3, which roughly can be taken as a reference for comparison with averaging within the radius of 555 km. 5. Summary and discussion Different authors use different latitude±longitude grids for mapping the results of cyclone tracking. Taylor Uncertainties inherent in cyclone frequencies and (1986) reviewed the gridbox sizes used in different stud- numbers for different grid cells have been analyzed us- ies. The typical grid size used is approximately 2Њϫ5Њ ing storm tracks of different temporal resolution. The (e.g., Hayden and Smith 1982), although some clima- relationship between the temporal resolution of storm tologies are derived for 2Њϫ2Њ (e.g., Zishka and Smith tracks and the grid size determines the level of the ran- 1980; Trigo et al. 1999) and even 1Њϫ1Њ (e.g., Colucci dom errors in cyclone frequencies and a systematic un- 1976). Approximately one-half of the climatologies derestimation of the number of cyclones. To minimize were performed for the 5Њϫ5Њ grid (e.g., Dickson and these biases, we simulated a higher temporal resolution Namias 1976; Whittaker and Horn 1981). Taking into of the storm tracks by a temporal interpolation of storm account that most results were derived from 12-hourly tracks, which gives the least biased estimates of both or coarser resolution, we can conclude that the char- cyclone frequencies and numbers. For the rectangular acteristics of cyclone activity reported by many authors grid with the basic grid size of 5Њ and a 6-hourly tem- were in¯uenced by the uncertainties in cyclone fre- poral resolution, random errors in cyclone frequencies quencies and numbers. Sickmoeller et al. (2000) re- may be higher than 15% and an underestimation of the number of cyclones can range from 10% to 20%. The cently used very large boxes (of about 5Њ latitude) for uncertainties grow considerably for coarser temporal the 6-hourly data. Chandler and Jonas (1999) used 8Њ resolutions. The use of a circular averaging geometry ϫ 4Њ grid over the Northern Hemisphere and partly decreases the uncertainties but does not allow us to accounted for the cyclones that pass the boxes faster avoid them fully. than during 12 h. However, even their results at high If the tracking is performed on the basis of 6- to 24- latitudes are in¯uenced by the uncertainties. Thus, if the hourly data, one has to be careful with the selection of original resolution of tracking is 6 or 12 h and no post- the optimal grid for mapping cyclone frequencies and processing is applied, one should use at least a 5Њϫ numbers. Table 2 summarizes the results for the three 10Њ grid to minimize the uncertainties to the level of most frequently used temporal resolutions and the three 10%. The use of ®ner than 5Њϫ5Њ grids (say, 2.5Њϫ rectangular grids on a hemispheric scale. Already for a 2.5Њ) may result in very high uncertainties in both cy- 6-hourly resolution the uncertainties are not negligible. clone frequencies and numbers. The use of circular av- The average random error in the cyclone frequencies is eraging geometry is an advantage, because it gives 1.5- 7.4% for grid 1. For the grids with larger boxes (grids times smaller uncertainties with respect to the rectan- 2 and 3) the uncertainties decrease but still remain at gular networks. However, even for these networks, bi- the 5% level. Temporal resolution of 12 h implies for ases are not negligible and an interpolation of the storm grid 1 a relative standard error of 10%. It decreases by tracks is required, especially for the averaging radius 10%±30% for the grids with larger boxes. The use of smaller than 555 km. Moreover, multiple cyclone counts daily data results in uncertainties of 14% for grid 1 and may result in the spatial smoothing of patterns, espe- about 10% for grid 3. Variations between different ba- cially in high latitudes. sins are not large, although there is a general tendency Interpolation, applied to the storm tracks, minimizes for decreasing uncertainties in the Arctic. The under- the level of uncertainties inherent in the gridded cyclone 758 MONTHLY WEATHER REVIEW VOLUME 130

FIG. 9. Effective time step (h) for the interpolation of storm tracks, which provides estimates of cyclone numbers with accuracy higher than 5% for grid 1. Contour interval is 1 h. numbers and frequencies. It makes the mapping pro- for the most effective catchment of cyclones. For limited cedure practically independent of the original temporal areas (e.g., the North American east coast), a circular resolution of the data used. It does not mean, of course, averaging geometry can be used. The procedures jus- that this procedure can improve the accuracy results of ti®ed in our study will improve the accuracy of mapping the original analysis. Clearly, numerical storm tracking of cyclone numbers and frequencies for such a grid, and algorithms demonstrate higher skill for high-resolution will help to quantify uncertainty. Our study may also data. The procedure recommended here also helps to have implications for analysis of climate variability of account partly for the problem of the gridcell orientation cyclone characteristics. The cyclone life cycle may have (Taylor 1986). Simulation of very high temporal reso- pronounced interannual-to decadal-scale variability, as- lution of the storm tracks provides an effective catch- sociated with the changes in the intensity of synoptic ment of cyclones even for cells that are not properly processes on different scales (Gulev et al. 2000). Gulev oriented with respect to the preferential directions of the et al. (2001) found that during the period from 1958 to cyclone propagation. Note that circular geometry also 1999 there was a general tendency of increasing deep- avoids this problem. Figure 9 shows the effective time ening rate, decreasing cyclone lifetime, and increasing step that can be used for interpolation to provide esti- number of relatively slow cyclones over the Northern mates of cyclone numbers with an accuracy of about Hemisphere. Changes in the cyclone velocities may pro- 5% for grid 1, which guarantees that 95% of all cyclones duce time-dependent biases, if the time resolution used passing the boxes will be marked at least once within is inappropriate with respect to the grid size chosen for the box. This map has been derived from the actual the averaging of results. estimates of rm and rt for 42 winters from 1958 to 1999. For most regions, a 2±3-hourly resolution is appropriate. Acknowledgments. We thank Glenn White of NCEP The main Atlantic storm-track area requires a somewhat (Camp Springs) and Rolland Sweitzer of CDC (Boulder) ®ner resolution in comparison with the Paci®c. who provided the NCEP±NCAR reanalysis data. Dis- The development of this study can go in several di- cussions with Thomas Jung and Eberhard Ruprecht of rections. It is important to determine the optimal grids IFM (Kiel, Germany), Peter Koltermann of BSH (Ham- MARCH 2002 NOTES AND CORRESPONDENCE 759 burg, Germany) and Mark Serreze of University of Col- application to meteorological data. Mon. Wea. Rev., 122, 2573± orado (Boulder) were very important. We greatly ap- 2586. Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Re- preciate the criticism of two anonymous reviewers. analysis Project. Bull. Amer. Meteor. Soc., 77, 437±471. Their suggestions helped to improve the manuscript. Keegan, T. J., 1958: Arctic synoptic activity in winter. J. Meteor., This work was supported by the EU INTAS Foundation, 15, 513±521. Brussels (Grant 98-2089), the Volkswagen Stiftung Kelsey, K., 1925: A new method of charting storm frequency. Mon. Wea. Rev., 53, 251±252. (Hannover, Germany) under the North Atlantic Oscil- Koenig, W. R., R. 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