1. Introduction
Taiwan is an olive-shaped mesoscale island within the travel zone of tropical cyclones in the western Pacific. The north-northeast–south-southwest-oriented Central Mountain Range (CMR) on the island has an average elevation above 2 km (Fig. 1a). How does the CMR affect a tropical cyclone crossing the island? Many meteorologists have made their efforts to answer this question. Wang (1954, 1980), Brand and Blellock (1974), Chang et al. (1993), Shieh et al. (1996, 1998, 1999), and many others, carried out observational analyses in spite of data shortage. Chang (1982), Yeh and Elsberry (1993a,b), Lin et al. (2006), Jian and Wu (2008), and others did numerical model simulations. Wu and Kuo (1999) gave an excellent review of the earlier studies and made valuable suggestions on data and model improvements.
(a) Taiwan topography. (b) Some examples of discontinuous tracks (shown with 500-m contour) (from Wang 1980).
Citation: Monthly Weather Review 140, 1; 10.1175/MWR-D-10-05050.1
One of the observed typhoon phenomena caused by the CMR blocking is the discontinuity of a surface track, first discussed by Wang (1954), which may be described as follows. Before the surface center of a tropical cyclone crosses the CMR, a secondary cyclonic vortex or low pressure center may appear on the opposite side of the CMR. If the original surface center of the tropical cyclone vanishes during the crossing and the secondary center subsequently becomes the dominant surface center, the surface track of the tropical cyclone is discontinuous (see Fig. 1b for some examples). Otherwise, the surface track is continuous unless the whole tropical cyclone completely dissipates over the island. There were many discontinuous surface tracks, particularly among the westbound tropical cyclones, according to the research report by Shieh et al. (1998). The report gives a detailed collection of surface tracks of tropical cyclones in the Taiwan area from 1897 to 1996, based on hourly synoptic charts. Of 108 westbound tropical cyclone tracks crossing Taiwan during the 1897–1996 period, 43 were discontinuous.
Why did some of the cross-island cyclones have discontinuous surface tracks and the others have continuous surface tracks? One can say for sure that it is strong orographic blocking versus weak orographic blocking. This leads to the key question: what are the measurable factors or variables that control the degree of severity of orographic blocking and eventually the track continuity? There have been some answers to this question (Wang 1980; Yeh and Elsberry 1993b; Lin et al. 2005, 2006, and others). Wang (1980) extensively studied the terrain effects on the circulations and tracks of the typhoons in the Taiwan area based on the available data from 1949 to 1977 (including the tropical cyclones coming from the South China Sea). His answers to the key question may be summed up as follows: in the absence of other pronounced weather systems nearby, if a typhoon approaches the CMR from the east with an intercepting angle greater than 120° (i.e., coming from the south to southeast sector), its track will be discontinuous regardless of its intensity and dimension, unless the storm radius and center position of the typhoon are so coupled that the wind in the storm front approaches the northern tip of the mountain with an intercepting angle less than 20°, thus, making the formation of a leeside low impossible. The numerical model simulations of Yeh and Elsberry (1993b) show that vortices approaching the northern portions of Taiwan tend to track continuously around the northern end of Taiwan due to the deflection of the deep-layer mean flow and having no strong interaction with the barrier, and that discontinuous tracks predominate for vortices approaching the central and southern portions of Taiwan where severe distortions of the inner-core circulation occur. It is further demonstrated by their sensitivity studies that more intense and rapidly moving vortices are more likely to cross directly over the barrier and thus maintain a continuous tack. In short, the track continuity of a vortex depends on the location of landfall of the vortex and may also on the intensity and moving speed of the vortex. In Lin et al. (2005), six nondimensional parameters (Vmax/Nh, U/Nh, Vmax/fR, h/Lx, R/Ly, and U/fLx), composed of eight dimensional parameters (the maximum wind Vmax of the TC, the radius R of Vmax, the basic flow U, the Brunt–Väisälä frequency N, the Coriolis Parameter f, and the height h, width Lx, and length Ly of the mountain range), are named as prospective control parameters for the continuity (and deflection) of cyclone tracks across a mesoscale mountain range. In the dynamic framework used, six is the maximum number of independent parameters. These 6 parameters are used to review 16 tracks in previous studies (of which 14 are model simulations) and are further tested with an idealized no-latent-heating model by changing U, Vmax, R, and h between tests. In each test an idealized cyclone embedded in the basic flow U moves, from the initial location due east, across an idealized topography representative of the CMR. The final conclusion is that, based on idealized simulations of a westbound-moving cyclone, the cyclone track becomes a discontinuous (continuous) track and the cyclone experiences more (less) deflection with a combination of small (large) values of Vmax/Nh, U/Nh, R/Ly, U/fLx, and Vmax/fR, and large (small) value of h/Lx. With this conclusion, a conceptual model is proposed depicting three different responses of a westward-moving cyclone to weak, moderate, and strong blocking, where the degree of blocking is supposedly give by the combination of all the parameters mentioned in the conclusion, not clearly defined though. In other words, a weak blocking leads to a slightly northward deflection upstream and a continuous track; while a moderate (strong) blocking leads to a northward (southward) deflection upstream and a discontinuous track with a secondary vortex to the southwest (northwest) of the mountain range. Although all the parameters are considered relevant and no comparative importance of the parameters is mentioned in the conclusion, one parameter is clearly outstanding from the rest based on the reviewed cases and on the test simulations. Over the 16 reviewed cases, the parameter Vmax/Nh (called vortex Froude number, VFN) > 1.5 correctly separates all continuous tracks from all discontinuous tracks but one (see Lin et al.’s Fig. 3). The 38 idealized test simulations further confirm that cyclone track continuity (or discontinuity) is mainly controlled by the value of VFN, and is not sensitive to the other parameters (see Lin et al.’s Fig. 4). Quite clearly, the dominant control parameter of cyclone track continuity identified by these idealized model tests (i.e., the VFN) is clearly different from the above-mentioned ones identified by Wang’s observational studies (i.e., the typhoon’s approaching direction), and by Yeh and Elsberry’s numerical model simulations (i.e., the typhoon’s landfall location) mentioned above. Why? Further studies are therefore very much needed to put these diverse answers into the proper respective and thereby resolve their differences.
Typhoon track discontinuity is not just interesting academically, but also important in real life. By definition, a TC of discontinuous track does not maintain its original center structure while a TC of continuous track does. The percentage reduction in strength due to blocking is certainly more severe in a TC of discontinuous track than in a TC of continuous track. For a TC of discontinuous track, its structurally damaged and intensity-reduced flow needs time to be integrated with the new center. Before the integration is done the strength remains weak. A TC of continuous track, without serious structural damage, on the other hand, may still be strong after crossing the mountain and remains potentially destructive (Shieh et al. 1996). Moreover, great typhoon disasters in history were due to heavy rainfall on the west side of the CMR because the west side is always most populated. By a quick look at the rainfall records of the 628 typhoons affected Taiwan during the 1897–1997 period given by Shieh et al. (1999), the probability of heavy rainfall on the west side of the CMR is found significantly greater in a typhoon of continuous track than in a typhoon of discontinuous track. Let us take the westbound-crossing Taiwan typhoons between 1897 and 1997 (listed in Table 1) as a data sample. If we rank all the 628 historical typhoons by the rainfall amounts produced in the northern area in descending order, 12 typhoons in Table 1 will be ranked among the first 50, all 12 had continuous tracks. (The rest high-ranked typhoons either did not make landfall in Taiwan or did not move westward.) If the ranking is done by the rainfall amounts in the central and southern areas including the west slope of CMR, 26 typhoons in Table 1 will be ranked among the first 50, 22 except 4 had continuous tracks; the odds ratio is almost 4 to 1 favoring a typhoon of continuous track after adjusted by the size-ratio (65 continuous to 43 discontinuous).
Data of the tropical cyclones included in the study.
In this paper, by analyzing the historical track data of the westbound tropical cyclones crossing Taiwan, we identify the variables that are most influential to the track continuity of the cyclones and using them construct a model for diagnosing track continuity in advance. A description of the data is given in section 2. The data are analyzed in section 3. The primary controlling variables are identified and the results are compared with previous studies. The differences between the results of previous studies are explained. In section 4, the method of logistic regression (LR) is employed to quantify the dependence of track continuity on the variables and the results confirm the validity of the analyses of section 3. In section 5, an LR model for track continuity is built as a function of the variables and verified independently. Summary and further discussions are given in section 6.
2. Data selection
The surface tracks of all westbound-crossing Taiwan tropical cyclones, including tropical storms and tropical depressions, during the time period 1897–2009, except for the parabolic tracks leaving Taiwan on the east or northeast coast, are included in this study. From here on, the acronym “TC” for tropical cyclone will also be used for tropical storm as well as tropical depression for convenience. The track data during the period 1897–1996 are from Shieh et al. (1998), and the more recent track data from 1997 to 2009 are provided by the Typhoon Research Laboratory of the Central Weather Bureau of Taiwan. The track data before 1947 are mainly determined by synoptic analyses of surface observations and ship reports. For the ones after 1947, more data are used in the analyses, including upper-air observations and international typhoon track reports from the Japanese Meteorological Agency, Hong Kong Astronomy Observatory, and the Joint Typhoon Warning Center at Guam. For the most recent ones, satellite images and Doppler radar records are also included in the analyses. Along each TC track, the longitudinal and latitudinal position (x, y), the maximum surface wind, the radius of 15 m s−1 wind, and the sea level pressure at the center are estimated at 6-h intervals over the ocean and hourly over land. For each track the landfall location on the east coast, the moving velocity of the TC center and the time at which the radius of 15 m s−1 wind reaches the east shore can be deduced from the data. A TC track is called discontinuous if (i) a secondary center on the downstream side of the mountain coexists with the original center on the upstream side of the mountain and later replaces the original center to become the primary center of the TC, or (ii) the TC is destroyed completely by the mountain. A TC track is continuous if it is not discontinuous. In the following, we simply call the westbound cross-island TCs that have discontinuous surface tracks (i.e., D track) the D group, and those having continuous surface tracks (i.e., C track) the C group.
Generally speaking, with fixed topography, the continuity (or discontinuity) of a cross-island TC track is determined by the location of landfall, the steering flow, and the intensity, structure, and size of the TC. For each TC included in this study, these variables can be estimated from the track variables of the TC described above. However, since each track variable is a time series of 1 or 2 days in length, to pick multiple data points from the time series to represent the variable would overburden the study with excessive redundancy. We have to reduce the number drastically. The problem is that we do not know what data points in these time series are most relevant to the determination of track discontinuity. For practical purpose, suppose we uniformly sample only one (prelandfall) data point from each time series, what sampling time would be the best or most optimal for the diagnosing or prediction of track continuity? On the one hand, if we want to gain lead time in the future predictive application of the results of this study, it is better to sample the track variables at a time many hours before the track discontinuity became certain. On the other hand, for close relevancy to the process of orographic blocking, we do not want to sample the track variables when the TC is too far away from the CMR. In balance, we choose the sampling time for the track variables to be the time (t0) at which a TC is about to “affect” the east coast of Taiwan, that is, when the estimated radius of 15 m s−1 wind was about to reach the east coast in less than 1 h. For every typhoon track included in this study, the time t0 (date and hour) and the values of the track variables at time t0 are listed in Table 1, where PMN, VMX, R15, and SPD are the sea level center-pressure (in hPa), the maximum wind (m s−1), the radius (km) of 15 m s−1 wind, and the moving speed (km h−1) of the TC at time t0, respectively. The location of landfall is represented by YLF (the distance from the southern tip per the island’s length), ANG is the direction (in degrees from the north) from the landfall location to the TC center at time t0, and GRP is the type of the track (0 for C track and 1 for D track; see Fig. 2 for a schematic illustration of the variables).
Schematic illustration of the track variables. The t0 is the time at which the radius of the 15 m s−1 wind of a TC reaches the east coast of Taiwan. At t0, the radius of the 15 m s−1 wind, the maximum wind, the sea level pressure at the center, and the moving speed of the center of the TC are to be called R15, VMX, PMN, and SPD, respectively. The location of landfall is denoted by YLF, which is the distance from the southern tip of the Island to the landfall location divided by the length of the Island. ANG is the direction from the landfall location to the TC center at t0 measured by degrees from the north. Center locations before and at t0 are denoted by solid circles, while those after t0 are denoted by open circles.
Citation: Monthly Weather Review 140, 1; 10.1175/MWR-D-10-05050.1
3. Explanatory variables
The track variables selected in section 2 are expected to be important to diagnosing track continuity. In this section we first look into the relationship between track continuity and each of these variables one by one statistically and then present them together in various combinations. Comparisons with previous studies are made.
a. The location of landfall–proxy for a finite barrier
The effects of a mountain range on a given TC obviously depend on the height and shape of the mountain range. For a not-very-long mountain range, the length is also a very important factor. Giving the actual dimensions and profile of the CMR and the closeness of the CMR to the east coast, the location of landfall of a given TC determines what part of the CMR the flow has to climb over, how steep the slopes are, how rugged the ridgeline that the climbing flow has to overcome, and how much of the low-level flow can go around the mountain’s southern or northern tip. Therefore, the YLF of a TC is a single nondimensional proxy for the three-dimensional parameters, height, width, and length combined, and is expected to be very important to the track continuity of the TC. To see the importance, we first divide the YLF data into C and D groups and use boxplot (or the box and whisker diagram) to display the overall data structure of each group (Fig. 3a). In a boxplot, the central box spans the first quartile to the third quartile [i.e., the interquartile range (IQR)]. The bold bar in the box denotes the median. The top and bottom thin bars (whiskers) denote the maximum and minimum values, respectively, if no outliers exist. Outliers are either 3 × IQR or more above the third quartile or 3 × IQR or more below the first quartile. If either type of outlier is present the whisker on the appropriate side is taken to 1.5 × IQR from the quartile rather than the maximum or minimum, and individual outlying data points are displayed as circles. Clearly displayed in Fig. 3a, the locations of the two medians are well separated by one-third of the island’s length. A three-quarter majority of D group is in the south and one-half of C group is in the far north. Only a single D-group outlier (B45) has YLF greater than 0.8. The rough north south separation between the C-track and D-track vortices described in Yeh and Elsberry (1993b) generally fits the picture.
(a) The boxplot of YLF. The central box spans the first quartile to the third quartile. The bold bar in the box denotes the median. The top and bottom thin bars (whiskers) denote the maximum and minimum values, respectively, if no outliers exist. Outliers are displayed as circles. (b) The estimated probability of track discontinuity as function of YLF and its quadratic approximation (see section 4b).
Citation: Monthly Weather Review 140, 1; 10.1175/MWR-D-10-05050.1
To estimate the probability of C track (or D track) as a function of YLF, we divide the east coast into 10 segments of equal distance. For each segment we count the TC tracks in Table 1 whose YLF values fall into the segment, then, to estimate the probability of a C track we divide the number of C tracks in the segment by the total number of C tracks plus D tracks in the segment. The results are shown in the histogram in Fig. 3b, where the probability of track discontinuity p is plotted. The probability of track continuity is simply 1 − p. The histogram displays two very important features. First, the probability of track continuity depends greatly on the value of YLF. Starting from 1 at the northern tip the probability reduces southward only mildly along the first one-fourth of the coast, then drops sharply to a value only slightly above 0.5 and reduces gradually again until reaches the lowest value of about 0.2, then increases sharply passing 0.5, and remains at about 0.7 along the rest of the coast (the southernmost one-fifth). Second, with respect to the midpoint of YLF, the probability shows a remarkable north–south asymmetry. The point of maximum blocking shifts markedly southward. The blocking is nearly zero for a TC making landfall near the northern end of the CMR, but clearly not so for a TC making landfall in the southernmost segment. The estimated probability distribution divides the east coast into four sections: the northernmost section where all tracks are continuous, the inner-northern section where the majority of tracks are continuous, the central section where the majority of tracks are discontinuous, and the southern section where the majority of tracks are continuous again, with the dividing points at about YLF ~0.9, 0.62, and 0.17, respectively.
The north–south asymmetry can be explained by the north–south asymmetry of the cyclonic circulation, because when a TC is moving westward against a mountain, the winds in the northern part blow against the mountain but the winds in the southern part blow away from it. Thus, a TC centered about a half R15 length south of the highest mountain peak likely feels the greatest blocking and a TC centered at the southern tip of the mountain, having the winds in its northern part blow upslope, still feels strong blocking unless mitigated by other factors. But a TC centered at the northern tip of the mountain, with its northern part flow completely unblocked, is always allowed to pass continuously. It is noted that the asymmetry depicted in Fig. 3b is related to the asymmetries displayed in the conceptual model of Lin et al. (2005). See a further discussion in section 6.
b. The importance of cyclone’s intensity and size
Three quantities are conventionally used to characterize a TC: its maximum wind (VMX), pressure at the center (PMN), and storm radius (~R15). The three quantities well characterize the intensity and physical dimensions of a symmetrical vortex. In a TC–CMR encounter, these parameters are important for different physical reasons. The size parameter is important as discussed at the end of section 2a and further discussed later in section 6. For a given barrier, the intensity parameter generally measures the ability of maintaining the internal structure and the resiliency to recover from low-level destruction. Since the three quantities are coherent in a TC and all TCs are generally similar in structure, data of the three variables for any given sample are well linearly correlated and similar in structure as can be seen in the boxplots in Fig. 4 and as shown in later discussion in this subsection. The influence of this similarity will be demonstrated and discussed later in section 5.
As in Fig. 3a, but for (a) PMN, (b) VMX, and (c) R15.
Citation: Monthly Weather Review 140, 1; 10.1175/MWR-D-10-05050.1
The importance of the variable VMX in diagnosing track continuity is suggested by the sensitivity studies of Yeh and Elsberry (1993b) and confirmed by the idealized simulations of Lin et al. (2005) via VFN. For the constant Brunt–Väisälä frequency adopted by Lin et al. 0.01 s−1, and the average height of the CMR, the VFN of Lin et al. is Vmax multiplied by a constant, and Vmax directly translates to VMX. The VMX data in Table 1 plotted separately for the C and D tracks in Fig. 4b show an approximate one-quarter shift between the two IQRs, indicating that the track variable VMX is important to the separation between the two track groups. To see the dependence of track continuity upon VMX more clearly, we adopt the same method used in section 3a. We divide the VMX values into five ordered (right closed) bins with dividing points at 21, 32, 50, and 61 m s−1, corresponding to the categories of very weak, weak, moderate, strong, and superstrong typhoons, respectively. The counts in the five categories are 11, 22, 71, 15, and 12, respectively. Before we present the results, we note that the count in the middle bin is much more than the counts of the other categories since the TCs crossing Taiwan are mostly moderate. Intuitively, we expect the superstrong typhoons to cross the CMR continuously without much damage to their structures, the very weak ones to be fatally blocked, and the moderate ones to pass either continuously or discontinuously most likely depending on other factors. A big count in the middle bin helps to smooth out the effects of other factors and thereby let the effect of VMX stand out clearer, as long as there are enough counts in the other bins. Somewhat as expected, the estimated probabilities of the C track are 0.18, 0.55, 0.58, 0.80, and 0.91 in the very weak, weak, moderate, strong, and superstrong category, respectively. If we combine the two strong categories as well as the two weak categories to make only three bins, the estimated probabilities of the C track will be 0.42, 0.58, and 0.85, respectively. Both sets of statistics show that the probability of the C track generally increases with increasing vortex intensity while the probability of the D track (1 − the probability of the C track) decreases with increasing vortex intensity. This is in a general agreement with the suggested intensity separation between the C and D tracks in the sensitivity studies of Yeh and Elsberry (1993b). It is also in agreement with the idealized simulations of Lin et al. (2005). Based on Fig. 4b and the distribution of the above-estimated probability, the most optimal point for separating the C from the D tracks (the point of 0.5 probability for track continuity) is close to VMX = 32 m s−1. We note that, according to Fig. 4b, the optimal separation point would be about VMX = 40 m s−1 if the data samples for the C and D tracks are equal in size, instead of 78 to 53 favoring the C track. For the Brunt–Väisälä frequency N = 0.01 s−1, as in Lin et al. (2005), and the average height of the CMR h = 2.5 km, VMX = 32 m s−1 corresponds to VFN = 1.28. This VFN value is pretty close to the value obtained by Lin et al. (2005). However, wide differences exist when we make further comparison. Corresponding to the transition zone defined by the (1.2, 1.6) VFN interval, the (30, 40) interval in VMX contains about one-third IQR of the C tracks and one-half IQR of the D tracks (see Fig. 4b). Outside the transition zone, about one-quarter of the C tracks and nearly one-and-a-half of the D tracks would be misclassified. The number of misclassified cases may decrease if the Brunt–Väisälä frequency is estimated case by case. Nevertheless we cannot say that the data in Table 1 support the conjecture that the TC track continuity is mainly controlled by the value of the VFN. Furthermore, from the distribution of our estimated probabilities presented above, 18% of the tracks in the weakest category are continuous and 9% of the tracks in the superstrong category are discontinuous, no clear transition zone can be defined for the data. Why? The reasons are discussed later in section 3d.
As is commonly known, the variables PMN (Fig. 4a) and VMX (Fig. 4b) are very much negatively correlated with a linear correlation coefficient of −0.93. If the PMN values are divided accordingly in five intensity categories, the probability estimates (not shown) do not show significant changes from those mentioned above. The close linear correlation between the two variables makes the combined diagnosing capability of the two approximately equal to the capability of either one alone. These statistics based on 131 real TCs offer the strongest practical evidence for the idea that both the maximum wind and the pressure drop at the center stands equally well for the intensity of a TC.
As revealed in Fig. 4, the linear correlation between VMX and R15 is not as good as that between PMN and VMX; it is 0.75. To assess the importance of R15 in diagnosing track continuity, we adopt the same method used in the above. We divide the R15 values in Table 1 into five ordered (right closed) intervals with dividing points at 100, 190, 300, and 360 km, corresponding to the categories of very small, small, moderate, large, and superlarge typhoons, respectively. The dividing points are chosen to make the intervals somewhat equivalent to the five VMX categories and to have enough tracks in every interval. The counts in the five intervals are 13, 28, 48, 31, and 15, respectively. The estimated probabilities of the C track are 0.37, 0.57, 0.57, 0.68, and 0.80 in the five intervals, respectively. We find from these numbers that the dependence of track continuity on R15 is largely overlapping with that of VMX and seemingly somewhat less impressive (to be quantitatively assessed later in section 5).
c. Approaching direction and speed
We now examine the two track variables ANG and SPD. According to Wang (1980), the angle between the long axis of the CMR and the approaching direction of a TC is very important to the track continuity of the TC. Wang’s conclusion is based on all the TC data between 1949 and 1977. His data is only partially overlapped with ours. Let us verify his conclusion with the westward-moving TC tracks between 1897 and 2009 listed in Table 1. To do that, we adopt the same method used in the above. We divide the ANG values in Table 1 into six (left closed) ordered sectors with evenly separated divisions at 90°, 105°, 120°, 135°, and 150°. The corresponding counts in the order are 8, 10, 33, 33, 37, and 10. The estimated probabilities of C-track are 0.38, 0.90, 0.73, 0.67, 0.46, and 0.30 for the six ANG sectors, respectively. These numbers roughly show that the probability of track continuity generally increase with increasing ANG for ANG < 90, peaks at about ANG = 105, then reduces and becomes 0.5 after ANG =135 and reaches the lowest value for ANG > 150. The variation is clearly nonlinear and largely quadratic. Since the direction of the long axis of the CMR is about 20°, this means that a right-angle encounter of the barrier and the vortex mostly produces a C track while a sharp angle encounter most likely leads to a D track. This is in agreement with the analysis of Wang (1980). In fact, ANG is more influential on track continuity in the period between 1949 and 1977 that Wang studied than in the whole period. The angular interval of ANG that favors the D track is also in general agreement with the special area identified by Chang et al. (1993) for the center of TC that induces leeside secondary low. Studying orographic effects on typhoons over Taiwan, Chang et al. (1993) concludes that the leeside secondary low develops only when a typhoon center is located in a special area, which covers southeastern Taiwan and an ocean area to the east-southeast. On the basis of the above simple statistical analysis, it is advantageous to replace ANG by a modified variable DIR defined by the absolute value of (ANG − 110), or |ANG − 110|, where 110 approximates the direction normal to the long axis of the CMR. The variable DIR approximately takes care of the quadratic dependency of track continuity on ANG (to be confirmed by quantitative assessments in section 4a). The boxplots in Figs. 5a,b show that the modified variable DIR is better than the variable ANG in linearly diagnosing track continuity. The DIR data in the period between 1949 and 1977 that is a part of the data used in Wang (1980) are also plotted in Fig. 5c for a comparison with Fig. 5b. Indeed, the variable DIR is an excellent discriminator of track continuity in that subperiod, much better than the DIR over the whole period. Why is that? A statistical explanation is given in section 3d.
As in Fig. 3a, but for (a) ANG, (b) DIR, (c) DIR in the period of 1949–77 only, and (d) the component of SPD perpendicular to the major axis of the CMR.
Citation: Monthly Weather Review 140, 1; 10.1175/MWR-D-10-05050.1
It is plausible that rapidly moving vortices suffer less terrain blocking. According to Lin et al. (2005), the smaller the basic-flow Froude number U/Nh, the stronger the orographic blocking. The numerical sensitivity studies of Yeh and Elsberry (1993b) demonstrate that more intense and rapidly moving vortices are more likely to cross directly over the barrier. All of these seem to suggest that the track continuity of a vortex depends on the moving speed of the vortex. To check for such dependence in the data collected in Table 1, the SPD data in the C and D groups are plotted separately (not shown) and compared. Furthermore, the SPD data in the C and D groups are projected onto a line perpendicular to the long axis of the CMR (to be called XSPD) and then similarly plotted. In the boxplots of SPD, the C and D plots have almost identical medians and extremely similar structures except that the data of the the C group spread wider. The same is true in XSPD plots (Fig. 5d). The kind of structural shift between the boxplots of the C and D groups clearly seen in the boxplots of YLF, DIR, and VMX, as evidence of dependence, does not exist in the boxplots of SPD or XSPD. The wider spreading of the C plot at the upper end indicates that there are more C tracks than D tracks in the speed range, so it may be said that very rapidly moving TCs are likely to have a C track, in agreement with Yeh and Elsberry (1993b). By the same token, the wider spreading of the C plot at the low end indicates that slow-moving TCs are also likely to have a C track, which is puzzling and appears at odds with all that was just said above. This puzzle may related to the fact that the data sizes (see Table 1) are 78 to 53 (almost 3 to 2) favoring the C track, because more data would spread wider. To know more about this, we look into the XSPD distribution of both groups. As before, we divide the range of XSPD into seven intervals of equal width with break points at 5, 10, 15, 20, 25 and 30 km h−1. The mean and medium XSPD of both C and D groups are all close to the low end of the midinterval. The counts in the individual speed intervals, from low speed to high speed, are 2, 15, 19, 17, 13, 8, and 4 for the C group and 1, 8, 14, 21, 7, 1, and 1 for the D group, respectively. There are more C tracks than D tracks in each of the intervals except the midinterval. If we wanted to, we could shift the upper break point of the middle interval a little bit from 20 km h−1 to make the C tracks a majority even in the midinterval; that is to say, the only majority status of the D track in the midinterval is not certain. The calculated C track probabilities in the intervals are 67%, 65%, 58%, 45%, 65%, 89%, and 80%, respectively. Why is the C track favored almost without exception in all speed intervals? It is because there are much more C tracks than D tracks in the whole data (almost 3:2 as pointed out above). The 3:2 size ratio favoring the C track is the realization, over the 1897–2009 time period, of the overall control by the local physical–climatological environment. This C track advantage is not clearly seen when we examine a variable that has stronger control over track continuity, because the control of the variable prevails. It becomes clear only when there is no other significant control present. Then the question arises: if XSPD is really not in control how do we explain that the probability of the C track in the two intervals that XSPD > 25 km h−1 are evidently higher than that in the other intervals? To look for answers, we need to examine all other characteristics associated with those TCs in the two intervals. Of the 12 C-group members in the two high-probability intervals, 7 approach from east with ANG between 110° and 120°, 6 are with YLF > 0.675, and 6 are moderate to superstrong typhoons with VMX > 37.5 m s−1. Later in section 5, a parametric statistic model that excludes XSPD as a predictor will be able to classify the 12 TCs with only one error, as compared with two errors if XSPD > 25 km h−1 is used to do the classification alone. In short, we may conclude by saying that the track continuity of a TC is insensitive to XSPD and the high C-track probability seen in the top two XSPD intervals can be attributed to other variables coinciding with high XSPD. However, this in no way denies the important influence of moving speed (or the basic flow) on track deflection before landfall that affects YLF as well as ANG (or DIR); thus, it does not really contradict the results of Yeh and Elsberry (1993a,b). The only difference is that our focus is different here (see the second paragraph in section 6). More related discussions are given in the following tree-based analysis and the quantitative analysis in section 4.
d. A tree-based analysis
By looking into the relationship between the track continuity and the individual track variables, we have found approximately how the probability of track discontinuity depends on the variables individually. However, track continuity is determined by the combined effect of these factors. To see how all the six explanatory variables, not just YLF, DIR and VMX, work together, the data in Table 1 is fitted by a classification tree using the statistical software package Recursive Partitioning and Regression Trees (RPART; Therneau and Atkinson 1997) that closely follows Breiman et al. (1984). The package consists of a set of routines that recursively split the data (the root node) into a tree of many nodes (partitions), prune off the unreliable nodes, and select the best tree for displaying the classification rules. Each splitting is made by finding the explanatory variable and its split point that most reduces the measure of node impurity under the Gini rule. During the growing process, if a node is pure (i.e., all members are of the same kind) or the number of its members is smaller than a chosen minimum (MINSPLIT), it becomes a leaf-node (requiring no further splitting) and is classified according to the majority of its members. The splitting process stops when there are no more nodes to split. Setting MINSPLIT = 2, the 131 TC tracks in Table 1 can be perfectly fitted by a tree with 27 binary splits, which has 27 + 1 leaf nodes (not shown). The tree-growing algorithm makes the best split at the root node where there are the largest number of cases and, hence, a lot of information. Each subsequent split has fewer and less representative cases to work with. Toward the end, as the nodes getting smaller and smaller in size, the real signal left at a node may become drowned by the ever-existing random noises and the split made may then become meaningless and not applicable to a new sample and the tree becomes overgrown. Thus, a large tree needs pruning to ensure that the tree is small enough to avoid fitting random variation and large enough to avoid systematic biases. The well-established methodology is the cost-complexity pruning. According to Breiman et al. (1984), the set of rooted subtrees of a large tree that minimize the cost-complexity measure (a function of fitting error and size of the subtree) is itself nested. This can be utilized to find the optimal trees by a sequence of snip operations on the large tree, producing a sequence of trees from the size of the large tree down to the smallest tree with just the root node. For our full-grown tree of 28 leaf nodes, there are 7 nested trees in the set [see the table in Fig. 6 for their number of splits and the corresponding complexity parameters (CP)]. To select the best tree from the set, we choose to have a sevenfold cross validation (CV) done, along with the tree-growing process, to measure the errors of individual rooted subtrees. It involves randomly dividing the original data into seven subsets of (nearly) equal size, setting one portion aside as a test set, constructing a tree for the remaining six portions and evaluating the tree using the test portion. This is repeated for all portions and an estimate of the error is evaluated. Adding up the error across the seven portions gives the CV error rate. The calculated CV error rate (XERROR) and its standard deviation (XSTD) of each nested tree are given in the CP table in Fig. 6, together with the tree number, associated CP, number of splits (NSPLIT), and the fitting error relative to that of the root node (REL ERROR).
(top left) The CP table. Listed for the seven nested rooted subtrees of the full-grown tree are the CP, the number of splits (nsplit), the relative fitting error (rel error), the relative CV error (xerror), and the standard deviation (xstd) obtained by a sevenfold CV. (bottom left) Tree 3: the tree recommended by the 1SE selection rule and yielding the minimum CV error. (right) Tree 4: the tree also yielding the minimum CV error. The selected tree is tree 4 without the two XSPD splits enclosed by the dashed lines. The label just above the root (top) node or an intermediate node tells the chosen variable and the splitting point that applies to the left child node in the binary split. The left (right) number written just below an end node is the number of continuous (discontinuous) tracks in the node.
Citation: Monthly Weather Review 140, 1; 10.1175/MWR-D-10-05050.1
It is clear from the CP table (Fig. 6) that the XERROR decreases from tree 1 to tree 3 and then increases monotonically after tree 4. Tree 3 and tree 4, both yielding the minimum CV error, are shown in Fig. 6. Tree 4 has 7 splits and 14 misfits (with a relative fitting error 14/53 ~ 0.26). There are 20 more splits in tree 7 (the full-grown tree) than in tree 4. All of them are unreliable judging by the monotonic increasing of the CV error. In fact, the 20 splits altogether only reduce 14 misfits, meaningless even judged by common sense. Thus, all the reliable splits are already included in tree 4. Tree 3 is a subtree of tree 4, with the 4 splits at YLF = 0.17, VMX = 44.5, XSPD = 22.9, and 11.5 snipped off. According to the one standard error (1SE) rule, the best tree is the smallest tree within 1SE to the tree yielding the minimum CV error. The 1SE choice is evidently tree 3 based on the CP table. On snipping off the two splits at XSPD = 22.9 and 11.5 (thereby getting rid of the apparent puzzle that high XSPD > 23 km h−1 and low XSPD < 11 km h−1 both favors the C track), the choice is in agreement with the result of section 3c and later sections 4 and 5. On snipping off the other two splits at YLF = 0.17 and VMX = 44.5 m s−1, the 1SE choice is rather excessive. The split at YLF = 0.17 roughly differentiates the effect of the south tip from that of the inner southern section. The split at VMX = 44.5 m s−1 (coupled with the split at VMX = 23.75) defines the upper (and lower) end of a VMX transition zone as can be clearly seen in Fig. 4b. Both may be retained in the selected tree, at least for purpose of easy physical interpretation. The exact locations of splitting may be different in samples. The retaining of the VMX split at 44.5 m s−1 has no net effect on fitting the current sample.
To interpret the selected tree (tree 4 without the two XSPD splits enclosed by the dashed lines), it is important to remember from the tree-growing methodology that the importance of a splitter is ranked by the order of its selection, because at each splitting the most decisive one is always chosen. From the selected tree we see the following: 1) the landfall location (YLF), the approaching direction (ANG), and the intensity of a TC (represented by VMX) are ranked 1, 2, and 3 in determining the track continuity of the TC, respectively. This agrees with the quantitative assessment in section 4a. 2) For TCs with YLF on the northern coast (YLF ≥ 0.675), nearly 90% have a C track, factors ANG and VMR are generally inconsequential there. This well agrees with Yeh and Elsberry (1993b). 3) For TCs approaching from the south-southeast sector and with YLF < 0.675, 80% have a D track regardless of their intensity, agreeing with Wang (1980). In terms of dynamics, a TC center can hardly survive in a slant encounter with the high mountain where the center structure can be very much distorted and where the pressure center can be separated from the circulation center (Yeh and Elsberry 1993b). 4) The intensity factor represented by VMX plays a dominant role only when a TC approaching the CMR from the east (ANG < 132.5 or DIR < 22.5) and with YLF on the central or southern part of the coast (YLY < 0.675). This explains why VFN appears so outstanding as a controlling factor for the idealized simulations of Lin et al. (2005), but much less effective for all the data in Table 1 (as pointed out in section 3b), because all the TCs in the idealized simulations come from due east, while the real TCs in Table 1 were from all directions between northeast and south-southeast. 5) In the central or southern coastal section (YLF < 0.675) where the core of cyclonic flow is blocked by the high ridge and side escaping is nearly impossible, the center of a very weak TC (with VMX < 24 m s−1) will inevitably be destroyed. For stronger TCs, 79% will have a C track even with YLF in the central or southern section. 6) It is interesting to note that the left branch from the ANG split at 132.5 is clearly dominated by the node where the small-angle encounters (ANG < 132.5) couple with strong intensity. If more TCs of strong intensity are involved in small-angle encounter in a sample, the control of track continuity by ANG (or DIR) estimated from the sample will be seen enhanced. In the 1949–77 subperiod, nearly 69% of all the cases on the left branch of the ANG split are coupled with VMS > 45.5 m s−1, while in the whole period only 40% of all the cases on the left branch of the ANG split are coupled with VMS > 45.5 m s−1. This explains why DIR is seen more impressive in controlling track continuity in the subperiod 1949–77 (Fig. 5c) than in the whole period (Fig. 5b).
To close the section, a note is in order. Either a numerical modeling or an observational study just presents the picture it sees. A model simulation with simplifications and/or limited testing range may narrow down its variability range and thereby may misrepresent the whole picture. For example, the dynamic framework of assuming a one-dimensional basic flow U and starting the TC in due east in Lin et al. (2005), may have limited the variability of their idealized simulations (see the related discussion in section 3b and point 4 in the above paragraph). Similarly, an observational analysis with insufficient data or a statistical study of limited sample, including this one, is likely to present a biased picture as demonstrated in Fig. 5d and explained by point 6 in the above paragraph. From a broad view, the seemingly conflicting results existed in the previous studies mentioned in the introduction are explainable and can be put into proper respective, as demonstrated by the above tree-based analysis.
4. Quantitative assessment using LR
Unlike ordinary least squares (OLS) regression, LR does not require normally distributed variables nor assume homoscedasticity. For a given set of data, the model parameters, c and (bi, i = 1, I), are determined by maximum likelihood estimation (MLE; an iteration procedure), such that by maximizing log likelihood (LL), the picked values of the model parameters make the dataset “most likely” (e.g., see Hosmer and Lemeshow 2000). The −2 LL (twice log-likelihood), or the deviance, reflects the significance of the unexplained variance in the dependent variable and has approximately a chi-square distribution, and hence can be used for assessing the significance of LR.
The most frequently used test statistic in LR is the likelihood ratio test for comparing the fit of two nested models, which is simply the deviance difference between the two models, a measure of the improvement of fit by adding the predictor(s). The likelihood ratio test may be applied to assess the overall model fit, in which case the deviance difference is between the null model (the model with the constant only, or zero predictor) and the overall model, or to test the significance of an individual predictor, in which case the deviance difference is between the models with and without the predictor (Fox 1997; Johnston and DiNardo 1997).
a. Comparative importance of individual variables
The dependency of track continuity (DTC) on a track variable may be quantitatively estimated by using the variable in a LR model fitting the GRP data in Table 1. The fraction of deviance explained by the variable, or the difference between the null deviance and the residual deviance of the model per the null deviance, gives a quantitative measure of the DTC on the variable as the model sees it. The calculated results are listed in Table 2. In agreement with section 3, YLF, DIR (or ANG), and VMX (and PMN) are the leading factors in controlling the track continuity of a TC crossing Taiwan, while the role of SPD (or XSPD) is negligible.
Deviances explained by single-variable models.
b. Nonlinear dependency
The analysis in section 3a has shown that the DTC on YLF, as depicted by the histogram in Fig. 3b, is highly nonlinear. To model the nonlinear dependence on YLF (by a parabola with a southward-shifted vertex), a second-order term and a linear term are both needed. Using both terms in a LR model, the reduction in the residual deviance (the explained deviance) is more than doubled and the fraction of deviance explained (FDE) jumps from 0.0909 to 0.2093 (see Table 2 and the fitted curve in Fig. 4b). Can we further improve the fitting by adding more high-order terms to the single-variable models listed in Table 2? We find that adding higher terms makes no significant deviance reduction at the 10% level. In general, interaction between predictors may also contribute to the reduction of deviances significantly, but in the present case, no interactive term between the six variables contributes significantly at the 10% level.
c. Duplicated or overlapped dependency
It has been shown in section 3b that PMN and R15 are correlated to VMX with linear correlation coefficients of −0.93 and 0.75, respectively. Clearly, we should not include both PMN and VMX in the set of control variables. To test whether R15 should be included in the set, we use the dataset to build a LR model with linear predictors VMX and R15. The model residual deviance is 164.75, which is 93.2% of the null deviance of 176.80. The deviance explained by the predictors is 6.8%. These numbers are practically the same as the model of a single predictor VMX (see Table 2). The addition of R15 makes no significant contribution. If we further add PMN, it will also make no significant contribution. This test and the high linear correlations mean that the variations in VMX, R15, and PMN from one TC to another in the sample are statistically similar, as somewhat revealed by the boxplots in Fig. 4, even though they may in fact significantly differ or produce outliers in some individual cases. This would rather be expected from the dynamic similarity of TCs.
The quantitative assessments in this section may be summed up as follows: 1) the quantitative analyses confirm the results presented in section 3; 2) a quadratic term is required to approximate the nonlinear dependency of track continuity on YLF; 3) the YLF variable explains much more deviance of the probability of track continuity than any of the other included variables (the variables DIR, VMX, and PMN are nearly equal in importance); and 4) either VMX or PMN is enough to represent all of VMX, PMN, and R15.
5. An LR model
In this section, a model is built to further demonstrate the overall validity of the above analyses and to examine the collective capability of the three variables in diagnosing the track continuity. The track data in Table 1 are divided into two parts: The 64 tracks between 1944 and 1996 are used as the training sample for model building, and the rest of the 67 tracks either before 1944 or after 1996 are used to verify the model.
a. Parsimonious model(s)
The usual way to assess the model’s success is to count the numbers of correct and incorrect classifications based on the default cut probability of 0.5. Of all the 64 track classifications made by the model, 57 tracks are correctly classified, giving an 89% overall accuracy. Separately, 79% of the 24 D tracks and 95% of the 40 C tracks are correctly classified. Looking at it another way, of the 21 tracks that are classified to be D track, 19 are correct, the hit rate is 90.5%; of the 43 tracks that are classified to be C track, 38 are correct, the hit rate is 88.4%. The scores are pretty high except in classifying D tracks.
To assess the success of the model more closely, we next examine the actual modeled probabilities of individual cases. In Fig. 7, the middle solid vertical line separates the group of observed D tracks from the group of observed C tracks for viewing convenience, the modeled probabilities of the individual cases of the training sample are plotted according to case number from cases 26 to 65 (limited by two vertical dashed lines) and from cases 98 to 121 (limited by the other two vertical dashed lines). The upper (lower) horizontal dashed lines are the default cut probability 0.5 plus (minus) the RMS prediction error (0.1275) estimated by the leave-one-out CV method (Davison and Hinkley, 1997). The rest of the figure is explained later in the verification section. For cases 26–65 and 98–110, most calculated probabilities are seen considerably away from the default cut probability (shown by the horizontal solid line). Ten cases have probabilities between the two horizontal dashed lines on both sides of the default cut probability (or within the RMS error range); eight cases are still correctly classified (three C tracks and two D tracks almost misclassified), and two (one C track and one D track) are misclassified. Five cases (one C track and four D tracks) are in serious errors (outside the RMS error range), including one D-track outlier (Mary of 1965) very close to the line of zero probability. To sum up, 49 of the 64 cases (77%) are out of the RMS error range and correctly classified.
Fitted or predicted discontinuity probability of individual tracks. The vertical solid line separates the C tracks (0) from the D tracks (1) and the dashed vertical lines separate the cases used for model building (the middle part in each group) from the cases used for model verification. The horizontal solid line denotes the cut probability 0.5. The two horizontal dotted lines give the estimated RMS error range.
Citation: Monthly Weather Review 140, 1; 10.1175/MWR-D-10-05050.1
The confident (outside the RMS error range) success rate (77%) is not remarkable. Can the model be further improved by adding more predictors? Based on the results shown in section 4, the answer is: no predictors in Table 2 or their interactive terns can be added to reduce the residual deviance significantly at the 10% level. This is the parsimonious model we can get unless new control variables not in Table 1 are found.
b. Model verification
Now we apply the model to the 67 tracks separated for model verification. Based on the default cut probability 0.5, 82% of all the 67 tracks are correctly predicted. Separately, 79% of the D tracks and 84% of the C tracks are correctly predicted. Of the 29 tracks that are predicted to be D track, 79% are correct. Of the 38 tracks that are predicted to be C track, 84% are correct. To compare with the fitting in the above subsection, the success rate in predicting the D-track group is about 10% points lower, so is the hit rate of the tracks predicted to be the D track, but both are still at good levels.
The predicted probabilities of the individual cases are plotted in Fig. 7 according to case number from cases 1 to 25, from cases 66 to 97, and from cases 122 to 131. There are nine cases between the two RMS error bars with three wrong predictions. Nine cases are seriously wrong, of which two C tracks (B21 and Morakot of 2003) and three D tracks (B36, B45 and Talim of 2005) may be considered outliers. To summarize, 49 of the 67 cases (73%) are out of the RMS error range and correctly predicted. The overall score of the predictions is only slightly less than that of the fitted cases. The appropriateness of the model and moreover the combined power of the three control factors are confirmed.
6. Discussion and summary
Generally speaking, the variables or parameters determining the track continuity of a vortex crossing a barrier varies with the ratio of the scale (radius) of the vortex to the length of the barrier, R/L. 1) For large R/L, the case is trivial and there will be no track discontinuity. 2) For R/L much smaller than 1 and the vortex not near either end of the barrier (or an infinite barrier), two situations exist. (i) If the basic (steering) flow is horizontally uniform and nearly perpendicular to the barrier, the track continuity will be controlled by the VFN (as defined in Lin et al. 2005). This is evident from the idealized simulations of Lin et al. (2005) in spite of using a mesoscale mountain there. If a very long mountain were used, the dominance of VFN would only be enhanced. (ii) If the basic flow is not nearly perpendicular to the barrier, then the approaching angle of the vortex comes in as another factor in determining the track continuity. The right main branch of tree 3 in Fig. 6 is a fair approximation. 3) For R comparable to L, such as a TC crossing the CMR, then the north–south asymmetry comes in as a result of the finiteness of the barrier and the situation becomes much more complicated. If a comprehensive numerical simulation study is carried out, in addition to the potential control parameters expressed in combinations of the dimensions of the mountain and the intensity (and size) of the TC, such as the parameters defined in Lin et al. (2005), the list of test parameters should also include a position parameter representing the starting location of the testing TC (owing to the nature of an initial-value problem) and the basic flow should not be fixed in one direction. Based on the discussion in section 3d, we expect that the simulation results will depend on the starting latitudinal location of the TC and the direction of the basic flow. In other words, it is highly likely that the six-dimensional parameter space defined by Lin et al. (2005) is only a proper substructure of the whole solution structure of the problem.
For our statistical study, to efficiently find out a shortlist of the most important factors controlling the track continuity of a TC crossing the CMR, we realize that the TC’s landfall is pivotal. Without a landfall, the problem of track continuity of a TC would not even come up. Thus we may cut into the heart of the problem by asking two separate questions: “what parameters control the TC’s track continuity after landfall?” and “what parameters control the trajectory of a TC before landfall?” The first question focuses on the blocking and damage of the TC center by the mountain after landfall while the second question focuses on the diversion of the TC’s route forced by the mountain before landfall. The second question is partially answered in Lin et al. (2005) (see the introduction) where the idealized simulations are also about the upstream deflection of a TC. Since the initial location of a testing TC is not treated as a testing parameter and all testing TCs in the idealized simulations are from due east steered by a basic flow U, the results describe only a part of the picture. In real situations listed in Table 1, the TC in question may come from all directions between the northeast and the south-southwest. Further studies are needed. We leave the second question to a later study.
Focusing our current study on the first question, our results have been presented in the above sections. To sum up, we have found, based on the track data listed in Table 1, that the track continuity of a westbound cross-Taiwan TC depends mostly upon the landfall location (YLF), the approaching direction (DIR), and the intensity (VMX) of the TC. The dependence on YLF is nonlinear and remarkably asymmetric with respect to the midpoint of the east coast and may be well approximated by a quadratic function of YLF (sections 3a, 4b, and Table 2). The dependence of track continuity on TC intensity and size may be represented by a linear function of VMX (sections 3b, 4c, and Table 2), and the dependence of track continuity on the approaching angle may be approximated by a linear function of DIR (section 3c and Table 2). A nonparametric tree-based analysis shows the prevailing ways that the three controlling factors work together and helps to explain and resolve the differences among the existing observational analysis and numerical simulations (section 3d and Fig. 6). The findings are further confirmed by a LR model of track continuity, built with the three variables, that gives a predictive success rate over 80% (section 5). In short, this paper serves the purpose of furnishing a statistical picture of the track continuity phenomenon of TCs crossing Taiwan, complementary to previous numerical simulation and observational studies.
It is obvious but we still like to emphasize that the root of the nonlinear dependence of track continuity on YLF is the fact that the horizontal dimension of TCs (R) and the length of the mountain range (L) are comparable. Obviously, if L approaches to infinite (and if the shape of the mountain is regular), as in case 2 above, or, if L is considerably less than R, as in case 1 above, the track continuity will not be dependent on the landfall location at all. Precisely because of the finite scale of the mountain, the orographic blocking is minimal at both ends of the mountain and has a maximum somewhere in between. The discussion given in section 3a only points out the most basic physics but goes no further. It is the coupling of the mesoscale of the mountain range and the north–south asymmetry of cyclonic circulation that leads to the dynamic processes such as the various interactions between the TC’s outer flow and the mountain depicted in the conceptual model of Lin et al. (2005) and the distortion of the TC’s inner structure, the generation of ventilation flow, and the tracking around the northern tip described in Yeh and Elsberry (1993a,b).
In section 5 we have seen that out of 131 tracks there are few outliers markedly contradictory to the ruling of the model. The presence of outliers is clear evidence that the influences not represented by the selected three factors may once in a while become critical in determining the track continuity of a TC. That is rather expected because only the statistically significant factors are included in the model.
Acknowledgments
Comments by three anonymous reviewers helped improve the quality of the manuscript. The supply of track data between 1997 and 2009 by the Typhoon Research Lab of the Central Weather Bureau of Taiwan is greatly appreciated. The statistical computations are done using the free statistical software R (version 2.6.1) released by the R Foundation for Statistical Computing, Vienna, Austria. This research is supported by the Central Weather Bureau of Taiwan.
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