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Weber and Kaufmann (1995) defined the similarity of wind fields using the wind vector itself. The similarity of wind vectors at the point of time A and B was defined by Eq. (1). Here, i means the node of wind vector (measurement or analysis grid point), and M refers to the total number of nodes.

In Eq. (1), the wind vector is normalized. In other words, the mean wind speed of a wind pattern is calculated according to Eq. (2), and the wind vector at each point of time is divided by the mean wind speed of the wind field by Eq. (3). Using normalization, the wind fields were classified according to the relative relationship of wind speed and wind direction in the wind field.

In this study, the similarity of wind vectors on two nodes A and B was expressed by the Euclidean distance defined by Eq. (4). Here, j indicates the time stamp of time-series wind vector and N denotes the total number of time stamps used to classify the wind regions.

To calculate the difference in wind vectors between two nodes, hourly wind speed and direction for 3 years were decomposed to the wind speed in the east-west direction (u) and wind speed in the north-south direction (v). The data were then normalized. First, the mean wind speed of each node for the past 3 years was obtained using Eq. (5).

The hourly wind speed components in the east-west direction and north-south direction were divided by the mean wind speed on each point (Eq. (6)).

The normalization reduces the distance between two nodes with different wind speeds but the same wind directions. Using the normalized wind vectors enables the wind regionalization using the wind direction and relative wind speed.

Cluster analysis is generalized into a hierarchical and a non-hierarchical method according to how the clusters are divided. The hierarchical method merges the groups that are close to one another until all of the groups are merged into one, whereas the non-hierarchical method divides the objects into K clusters by measure of similarity. Davis and Kalkstein (1990) and Davis and Walker (1992) developed a two-step cluster analysis method to classify weather patterns, and it was applied and reviewed in many studies (Jimenez et al., 2009; Jimenez et al., 2008; Kaufmann and Whiteman, 1999; Kaufmann and Weber, 1996). Based on their study results, the present study also applied the two-step cluster analysis method to surface wind regionalization.

The two-step cluster analysis is performed in two steps. The first step selects the appropriate number of clusters and initial clusters to be used in the next step through hierarchical cluster analysis. In hierarchical cluster analysis, there are several ways to merge the clusters according to how the distance between two clusters is defined. Davis and Walker (1992) applied the average linkage method. Weber and Kaufmann (1995) compared five linkage methods and confirmed that the complete linkage method was most appropriate. Jimenez et al. (2008) applied the complete linkage method for wind regionalization. The complete linkage method defines the distance between two clusters as the distance between two observations values with the longest distance in each cluster. This study defined the distance between two clusters using the complete linkage method and similarity defined at Eq. (4).

In this study, appropriate number of clusters was determined using the distance between two merged clusters, which was applied in the previous studies using the same similarity (Kaufmann and Whiteman, 1999; Kaufmann and Weber, 1996; Weber and Kaufmann, 1995). Generally, the appropriate number of clusters is determined using cluster validity indices in cluster analysis. There are several advanced indices that can be used, however, it is important to use appropriate indices depending on the kind of clustering method, similarity and data (Guerra et al., 2012). Meanwhile, a simple index is sufficient to evaluate our surface wind regionalization method. In hierarchical cluster analysis, the distance between two merged clusters keep increases as it merges two clusters with the minimum distance. If the distance greatly increases, it indicates that those two merged clusters are dissimilar; thus, merging must be terminated. The initial cluster is set to the clusters when the merging has to be terminated. The second step classifies the wind regions through non-hierarchical cluster analysis. It uses K-means cluster analysis, which is most widely used for non-hierarchical analysis. The distance between two sub-regions is defined with Eq. (4). The cluster count and initial clusters that must be set for K-means cluster analysis were set in the first step.

The first step sets the appropriate sub-region count through hierarchical cluster analysis. The distance between two clusters tends to be larger when two close clusters are grouped (Fig. 1). The points showing greater increase than the general trend are those with three clusters and eight clusters. Since the verification of the analysis result becomes much more complex when the wind regions are segmented into eight, this study derived three wind regions in Jeju Island, considering the intermediate wind system.

Fig. 1. 
Distance measure for each number of clusters in Jeju Island.

The second step derives the wind regions through K-means cluster analysis. On Jeju Island, three wind regions are classified around Mt. Halla at the center (Fig. 2). Different sub-regions appear in the south and north of Mt. Halla, with another sub-regions appearing from east to west. Such is consistent with the wind system on Jeju Island (Kim et al., 2008).

Fig. 2. 
Wind regions in Jeju island (cross symbols represent negative correlation of wind direction pattern over time with other nodes).

The three wind regions, numbered as [1], [2] and [3] in Fig. 2, contain 161, 310, and 101 nodes (grid cell size of 3 km²), respectively. Fig. 3 shows the wind roses at each sub-region. Although east-northeast is the prevailing wind direction of the sub-region [1] located north of Mt. Halla, various wind directions also appear. The eastern part of the sub-region [2], located east of Mt. Halla, has two prevailing wind directions, north and east, whereas the western part has one prevailing wind direction, i.e., north-northwest. Therefore, the prevailing wind direction of the sub-region [2] is north. The sub-region [3] located south of Mt. Halla has northeast as the prevailing wind direction.

Fig. 3. 
Wind roses of each wind region in Jeju Island.

To evaluate the meteorological validity of the regionalization method, similarity of wind characteristics under the same sub-region and difference of wind characteristics between different sub-regions were verified. Fig. 4, a box plot of mean wind speed by sub-regions, was presented to show the similarity and difference among sub-regions. The range of mean wind speed of each regions overlaps, but the sub-region [2] has a higher mean wind speed than the other sub-regions [1] and [3]. This can be interpreted that wind speeds of the sub-regions [1] and [3] are slowed down due to the existence of big topographic object, Mt. Halla, while wind speed spread of the sub-region [3] is wider than [2] due to a lee-side effect.

Fig. 4. 
Box plot of mean wind speed by wind regions.

Statistics of Weibull scale and shape factors representing wind speed distribution characteristics were compared by sub-regions to verify how these factors are uniform in a same sub-region. Thus, its mean and standard deviation were calculated by sub-regions (Table 1). The Weibull scale factor represents the mean value of wind speed distribution. Since the standard deviation of the Weibull scale factor was as fairly low as less than 1.0 m/s in all wind regions, the mean wind speed of the nodes in each sub-region can be judged to be similar. This is exactly corresponding to Fig. 4. The Weibull shape factor represents the dispersion of wind speed distribution from its mean value. A higher Weibull shape factor value means low dispersion of wind speed distribution from the Weibull scale factor which is the area mean of wind speed distribution. Because the ratio of standard deviation to its mean of the Weibull shape factor was less than 12% in all sub-regions, the wind speed variance of the nodes in each sub-region can be judged to be similar.

T-test was performed to check the statistical difference in mean values of Weibull scale factor in each sub-region at significance level of 5% (Table 2). The t-values between the sub-region [1] and [2] was - 13.1 and the sub-region [2] and [3] was +11.0, indicating clear difference in wind speed distributions. It should be noted that the sub-region [2] had much higher mean wind speed than those of the sub-regions [1] or [3] (see Table 1). However, the mean wind speed in the sub-region [1] and [3] was similar with t-value of - 0.5. In case of Weibull shape factor, the t-values between the sub-region [1] and [2] was - 12.4 and the sub-region [2] and [3] was 8.0, indicating clear difference in its distribution shape. Dispersion from the mean wind speed was lowest in the nodes of the sub-region [2] (see Table 1). Note, however, that the t-value between the sub-region [1] and [3] was 1.9, the lowest value, showing similar Weibull shape parameter. As such, the sub-region [1] and [3] has similar dispersion shape of wind speed distribution.

Table 3 summarizes the mean and standard deviations of wind direction in each sub-region. The calculation applied the method of obtaining mean value of wind direction and its standard deviation considering the circular (or cyclic) characteristics of angle (Berrens, 2009). According to the calculation, the prevailing wind directions of each sub-region [1], [2] and [3] were northeastern wind close to the east-northeastern wind, northern wind, and northeastern wind, respectively. Such is in accord with the wind rose of the center node in each sub-region as shown in Fig. 3. The standard deviations from the prevailing wind directions in the sub-regions [1], [2] and [3] were 31.3°, 37.6° and 18.9°, respectively, suggesting that the wind direction distribution in each sub-region was almost uniform. Table 4 shows the t-test results of wind direction distributions between sub-regions at significance level of 5% meaning that all three sub-regions showed different wind direction distribution statistically.

It is verified that the nodes regionalized in the same sub-region have similar wind characteristics but all sub-regions have different wind characteristics. No difference of mean wind speed between the sub-region [1] and [3] is attributed to the use of normalized wind vectors.

The correlation matrix of the time-series wind data in each sub-region is graphically depicted in Fig. 5. The columns and rows of unit correlation matrix become the nodes of each sub-region; different colors were painted according to the cross-correlation coefficient between the wind speed and direction of the nodes of the columns and rows. Red means strong positive correlation, whereas orange and yellow denote low correlation; blue means strong negative correlation. The circularity of the wind direction was reflected by applying the circular correlation coefficient formula suggested by Berrens (2009).

Fig. 5. 
Graphical depiction of correlation matrices by sub-regions.

Fig. 5 shows that most of the nodes exhibit strong positive correlation in the variation of time-series wind speed, but there are some areas of low or even negative correlation in the variation of time-series wind direction. Areas showing strong negative correlation with other regions are mostly located at the center part where the wind direction is diverging because of the conical shape of Mt. Halla and particularly such area in subregion [2] is located at the east side of Jeju Island, indicating difference between east side and west side of Jeju Island (Fig. 2). This meteorological characteristic is clearly shown as the wind vector field in Fig. 6 which are three-year averaged mean wind vectors. Due to the orographic characteristics of Jeju Island, mean wind vectors dispersing to four directions of north, east, west, and south centered on Mt. Halla peak. The central area of Mt. Halla can be considered a singular and unusual region.

Fig. 6. 
Mean wind vector field in Jeju island.

The correlation of the time-series wind speed was positively high, presenting that sub-regions are grouped with homogenous temporal wind speed variability. However, the correlation matrix of the time-series wind direction indicated that the temporal variability of wind direction is not homogeneous in the same sub-region. Although the wind regionalization method used in this study only reflected the variation of time-series wind speed but not that of time-series wind direction, the top of Mt. Halla could be separately regionalized from the correlation matrix of the wind direction. Also, east side and west side of Jeju Island could be separately regionalized. In this study, wind regionalization was done using 3 years of annual wind data, but by performing wind regionalization seasonally this division could be apparent.

At the first step, the value of distance measure at each number of clusters was examined to determine the appropriate number of wind regions (Fig. 7). A sudden and big increase appears when the number of clusters were five and seven. Fig. 8 shows the wind regions in Busan when it is divided into seven sub-regions. Considering the topographic complexity of Busan, seven wind regions would be appropriate rather than five.

Fig. 7. 
Distance measure for each number of clusters in Busan.

Fig. 8. 
Wind regions in Busan.

Fig. 9 shows the wind roses of each sub-region. The prevailing wind directions at the sub-region [1] and [7] are north-west, while it is north-northeast and southsouthwest at the sub-region [3]. The sub-region [2] in the high elevation has higher wind speed than subregions in the low elevation on inland such as [1]. Similar to the adjacent regions, the prevailing wind direction is northwest and north-northeast. The highest wind speed appears at the wind regions [5] and [6] located on the offshore. In the sub-region [4], including the Taebaek mountain ranges and the offshore, northerly wind is dominant. The rest of the sub-regions in the offshore have three prevailing wind directions, northeast, northwest and southwest.

Fig. 9. 
Wind roses of each wind region in Busan.

A comparison of the wind regionalization result and topographic features depicted as terrain elevation is suggested in Fig. 10. Along the regions with high elevation, homogeneous wind characteristics appear in the sub-region [2]. Valley and basin located between the high elevation regions have similar wind characteristics as the sub-region [3]. The sub-regions [1] and [7] presenting plains are divided by the mountains. The sub-region [4], including mountainous areas and offshore, show a local wind system, while the offshore area in the south of Busan is divided into two sub-regions [5] and [6]. Classifying wind regions based on hourly wind vectors improved on previous studies. The number of wind regions were smaller than previous study (Jung et al., 2007), i.e. 7 sub-regions instead of 9 subregions, but became more correlated to topographic features.

Fig. 10. 
Comparison of wind regions and terrain elevation (m) in Busan.

Fig. 11 displays mean wind vector field over Busan. Western wind is dominant in the sub-region [1]. The sub-region [7], Gimhae plane, adjacent to the subregion [1] but blocked by the mountains so that it shows different wind direction to the sub-region [1] (see also Fig. 9). South east wind is dominant in the sub-region [7]. The sub-region [6], including offshore area, is adjacent to the sub-region [7] but also partially blocked by the mountains. Different from the sub-region [7], prevailing wind direction is south. Another offshore region, the sub-region [5], has north-northwest as a prevailing wind direction and has higher wind speed than the sub-region [6]. This is because of the wind coming from the sub-region [2]. North-northwest wind is also dominant in the sub-region [2]. Wind direction in the sub-region [3] corresponds to the direction of valley and wind towards to the side that the elevation gets lower in the sub-region [4], i.e., southeast.

Fig. 11. 
Comparison of wind regions and mean wind vectors in Busan.