STOCHASTIC SIMULATION OF TROPICAL CYCLONE TRACKS IN THE NORTHWEST PACIFIC REGION WITH CLASSIFICATION MODEL

With the development of tropical cyclone risk assessment in large regions^{[1, 2]}, many full track simulation methods have been proposed to synthesize several full tropical cyclone tracks over the whole area. Vickery et al. used the negative binomial distribution to fit the annual occurrence number. The autoregression model was used to simulate tropical cyclone tracks, and model coefficients were determined with the least square method. The intensity model was used as the lysis model of tropical cyclone. This model can satisfactorily simulate tropical cyclone tracks in the Atlantic region.

James and Mason ^[5] also used the same autoregression model in the Coral Sea of Australia. Li and Hong ^[6] recently simplified this simulation method and applied it in the Northwest Pacific region ^[7]. The historical tropical cyclone tracks were separated into easterly and westerly headed parts to consider the movement characteristics of tropical cyclone tracks. Powell et al. ^[8] and Emanuel et al. ^{[9, 10]} used Markov process to simulate tropical cyclone tracks. Rumpf et al. ^{[11, 12]} divided tropical cyclone tracks into different disjoint classes on the basis of the spatial characteristics of tropical cyclone tracks, and each class was fitted by using probability density function. Therefore, tropical cyclone tracks can be simulated with different probability density functions. Yang et al. ^[13] used a probabilistic model of stochastic track simulation to simulate the tropical cyclone tracks. Hall et al. ^[14] developed an empirical track model that utilizes the statistical characteristics of historical tropical cyclone tracks in the Atlantic region. Subsequently, Yanekura et al. applied this model to the Northwest Pacific region and considered the effects of El Niño and La Niña phenomena. The statistical dynamic model was also developed and used to simulate tropical cyclone tracks (Emanuel et al. ^[9]; Chen and Duan ^[18]). In summary, tropical cyclone tracks in the Atlantic region can be satisfactorily simulated by most of the above-mentioned tropical cyclone track models. Given the complexity of spatial distribution of tropical cyclone tracks in the Pacific region, the same conclusion cannot be obtained in this region with the above-mentioned tropical cyclone track models.

To solve this problem, a classification model, which integrates the methods used by Rumpf et al. ^[11] and Hall et al. ^[14], is developed to simulate tropical cyclone tracks in the Northwest Pacific region. Section 2 discusses the history of the tropical cyclone tracks in the Northwest Pacific region. Sections 3 and 4 introduce the classification model and use them to simulate the tropical cyclone tracks in the Northwest Pacific region. Section 5 compares and analyzes the simulation results.

The historical tropical cyclone data from 1949 to 2016 obtained from China Meteorological Administration are used in this study ^[19]. The dataset consists of information on each tropical cyclone track, including date, time, longitude, latitude, intensity category, central pressure, and two-minute maximum wind speed. To eliminate the influence of tropical depression, we discard tropical cyclones with an intensity less than tropical depression. In addition, each tropical cyclone data only retains non-zero two-minute maximum wind speed records. Finally, 1, 854 tropical cyclones are used.

The tropical cyclone tracks in the Northwest Pacific region have strong complexity in spatial distributions. To improve simulation quality, tropical cyclone tracks are classified into five classes on the basis of the movement characteristics and steering positions in conjunction with the researchers'personal experience, as shown in Fig. 1. The five classes include the following: 1) Class 1 refers to the tropical cyclones whose tracks head west in an almost straight line; 2) Class 2 includes the tropical cyclones whose tracks head northwest in an almost straight line; 3) Class 3 stands for the turned tropical cyclones whose turning point locates in latitudesgreater than 30° N; 4) Class 4 contains the turned tropical cyclones whose turning point locates in latitudes between 20° N and 30° N; 5) Class 5 contains the turned tropical cyclones whose turning point locates in latitudes less than 20° N. With the five different tropical cyclone classes categorized, all subsequent steps for tropical cyclone track simulation are separately performed for each class. Specifically, the proportion P_i (i = 1, …, 5) of each historical tropical cyclone class and the corresponding frequency interval $\left[ {\sum\limits_{i = 0}^{i - 1} {{p_i}} \sum\limits_{i = 0}^i {{p_i}} } \right](i = 1, \ldots , 5) $ are first calculated, in which p_o is specified as zero. For each tropical cyclone track simulation, a random number r (0 < r < 1), obeying uniform distribution, is generated with Monte Carlo simulation method. The corresponding tropical cyclone class is obtained on the basis of the interval in which it falls.

Figure 1.  Tropical cyclone classifications.

In genesis model, the kernel probability density function (KPDF) is used to simulate the annual occurrence number and genesis positions of tropical cyclones. The basic KPDF f(x) is shown as follows:

where x is the estimated vector, x_i is the ith sample vector, n is the sample size, h is the bandwidth, K(·) is the kernel density function, and the Guassian kernel function is used in this study.

The basic KPDF presented in Eq. (1) is used to simulate the annual occurrence number. Given that h is a key to accurate estimation of the KPDF, the one-dimensional biased cross-validation method proposed by Scott et al. ^[20], as shown in Eq. (2), is used to calculate the optimal h of the annual occurrence number.

where Δ_ij = (x_i-x_j)/h. x_i and x_j are the ith and jth typhoon annual sample numbers, respectively.

A three-dimensional KPDF is used to estimate the space-time probability density of tropical cyclone genesis positions. The space-time of tropical cyclone genesis positions is composed of the following three dimensions: latitude, longitude, and time. The three-dimensional KPDF and its biased cross-validation method (Sain et al.^[21]) used in this study are given as follows:

where h₁, h₂, and h₃ are the optimal bandwidths of the latitude, longitude, and time, respectively. x_lat, x_lon, andx_time are the simulated genesis positions. x_{lat, i}, x_{lon, i}, and x_{time, i} are the ith historical genesis positions. S_lat, S_lon and S_time are the mean square error of latitude, longitude, and time, respectively. Δ_ijk = (x_ik-x_jk)/h_k (k = 1, 2, 3). x_ik and x_ik are the ith and jth normalized variables in the kth dimension, respectively.

To consider the periodicity in the time dimension of tropical cyclone genesis positions with a period of 0-366 days^[18], we expand the historical tropical cyclone genesis positions thrice in the time dimension to consider the influence of last year and next year genesis positions on the probability densities of tropical cyclone genesis positions this year. In addition, given that the three dimensions of tropical cyclone genesis positions have different quantities and units, the historical tropical cyclone genesis positions should be normalized to avoid one or more dimensions dominating the others in the probability densities of tropical cyclone genesis positions.

By setting the longitude direction as x-axis and the latitude direction as y-axis, the traveling model based on the means and mean square errors of tropical cyclone tracks in latitude and longitude directions proposed by Hall and Jewson^[14] is used and is given as follows:

where x and y are the simulated 6 h displacement in longitude and latitude directions, respectively. $ {\bar x}$ and ${\bar y} $ are the mean 6 h historical displacements in longitude and latitude directions, respectively. S_{x_rms} and S_{y_rms} are the mean square errors in longitude and latitude directions, respectively. x_i and y_i are the historical 6 h displacements in longitude and latitude directions, respectively. ζ_x and ζ_y are the normalized coefficients in longitude and latitude directions, respectively. ${\tilde x} $ and ${\tilde y} $ are the normalized simulated displacement in longitude and latitude directions, respectively. f_x and f_y are the correlation coefficients in longitude and latitude directions, respectively. s_x and s_y are deviation factors, and s² = 1 - f². e_x and e_y are free errors. σ_x and σ_y are weighted variances within the corresponding scale range of the current point in each direction. L is the length scale. N is the sample size. Subscript n represents the specific nth step. d_i is the great circle distance.

The lysis of tropical cyclone is linked to its intensity, but the intensity model has not yet been developed. Thus, the termination probability model

where p(r) is the termination probability at current position r. Θ_i is the termination factor, when position i is the final point of a tropical cyclone; Θ_i = 1, otherwise Θ_i = 0.

After p(r) is calculated, a random number between 0 and 1 is immediately generated. If p(r) is greater than the random number, then this point becomes the final point of tropical cyclone track. To control the length of tropical cyclone track, simulation step is limited within 100.

The simulation of tropical cyclone tracks can be summarized as follows: 1) Generate the start point of tropical cyclone track on the basis of the genesis model. This point is the first current point of the simulation track; 2) Determine the tropical cyclone class of the simulation track; 3) Obtain the second point of tropical cyclone track with distances x and y that typhoon moves in latitude and longitude directions in the first step with the equations $ x=\bar{x}+\varepsilon_{x} S_{x_{-r m s}}$ and $y = \bar y + {\varepsilon _y}{S_{y\_rms}} $, respectively, where $ {\bar x}$ and ${\bar y} $ are the mean 6 h historical displacements in longitude and latitude directions of the first point, respectively, e_x and e_y are the free errors, and S_{x_rms} and S_{y_rms} are the mean square errors; 4) Obtain the subsequent points of tropical cyclone track with the traveling model; 5) For each point of the simulation track, the lysis model is used to determine whether the typhoon is terminated. If terminated, then the simulation track is completed. Otherwise, simulation step 4) is continued until the typhoon is terminated, and a completed simulation track can be obtained; 6) Repeat the above steps until enough completed tropical cyclone tracks are simulated.

To obtain the annual occurrence number of tropical cyclone with Eq.(1), the optimal h of annual occurrence number is first obtained with the China Meteorological Administration's 68-year best-track dataset by minimizing the one-dimensional Gaussian kernel cross validation of Eq.(7), as illustrated in Fig. 2(a). The BCV has a minimum value when h is equal to 3, and the probability density function of annual occurrence number can be obtained with this minimum value. Then, annual occurrence numbers of 68 years of tropical cyclones can be generated, as displayed in Fig. 2(b), and repeated 30 times. The average value of 30-fold-simulated annual occurrence number is 1, 892 and approximately 2.07% difference compared with the historical data shown in Fig. 2(c). Thus, this probability density function can be used to generate the annual occurrence number of tropical cyclones.

Figure 2.  Annual occurrence number of tropical cyclones.

The latitude, longitude, and time of tropical cyclone genesis positions are determined with the three-dimensional kernel density function shown in Eq. (8). The optimal h for the three-dimensional normalized samples obtained from Eq. (9) are 0.25, 0.15, and 0.05, respectively. Then, many tropical cyclone genesis positions can be obtained with the fitted three-dimensional kernel density function. A total of 1, 854 typhoon genesis positions are simulated and compared with the historical data, as shown in Fig. 3. Comparisons of the simulated and historical typhoon genesis positions in three-dimensional directions are illustrated in Fig. 4. Only the simulated genesis positions from 30° to 40° in latitude, 120° to 140° W in longitude, and 0 to 50 days in time are greater than 10% compared with the corresponding historical statistics. This result is due to the few historical tropical cyclones that happened in these regions, and the randomness of genesis model may generate a relatively high error. In general, the simulated genesis positions with the present genesis model are similar to those in the historical data and can be used in the simulation.

Figure 3.  Comparison of historical and simulated typhoon genesis positions.

Figure 4.  Comparisons between simulated and historical genesis positions in three dimensions, and the simulation results are the average results of 30 times simulations.

The lysis model is used to determine whether the simulation track disappears or not. Fig. 5 shows the comparison of the historical and simulated end points of tropical cyclone tracks from 1949 to 2016. Then, the end points of the 30 sets of tropical cyclone simulation tracks are performed, and the correlations of end points between the historical and each set of simulated tropical cyclone tracks are analyzed. The averaged correlation value is 0.9879. Therefore, this lysis model can be used in end point simulation.

Figure 5.  End points comparison of tropical cyclone tracks.

In traveling model, the mean and mean square error of 6 h historical displacements in longitude and latitude directions are determined on the basis of the historical tropical cyclone tracks cantered on the current point within L. The optimal L can be determined using the Jackknife method. A minimum simulation error at approximately 250 km is used as optimal L. The mean 6 h displacements in latitude and longitude directions are computed as illustrated in Fig. 6(a) and 6(b), respectively. Most tropical cyclones move from low latitude to high latitude, and only few tropical cyclones move toward the low latitude. In low latitude regions, most tropical cyclones move from the southeast to the northwest. In high latitude regions, most tropical cyclones move from the southwest to the northeast.

Figure 6.  Mean and mean square error spatial distribution of 6h displacements.

Most tropical cyclones steer in a movement direction between 15° N and 30° N. The mean square errors of 6 h displacements in latitude and longitude directions are computed, as displayed in Fig. 6(c) and 6(d), showing the complex characteristics of tropical cyclone tracks in many regions in the latitude and longitude directions. The mean square errors in peripheral areas are small due to the few tropical cyclone tracks appearing in these regions. In most regions, the high latitude region has a large mean square error. Fig. 7 shows the 6 h displacement distributions at a low latitude location (15° N, 130° E) and a medium latitude location (30° N, 140° E). The overall movement direction trend of tropical cyclone is toward northwest in low latitude location and northeast in middle and high locations. The probability distributions at both locations conform to the Gaussian distribution. The same conclusion can also be obtained in the latitude direction. This result demonstrates the reasonable use of Eqs. (10) and (11) as the traveling model.

Figure 7.  Displacement distributions at low and middle latitude locations.

A total of 1, 854 tropical cyclone tracks are simulated using the historical genesis positions from 1949-2016 in the Northwest Pacific region as the simulation start point. The historical and simulated tropical cyclone tracks are shown in Fig. 8. The movement trends and characteristics of historical tropical cyclone tracks can be well captured by the simulated tropical cyclone tracks.

Figure 8.  Comparisons of historical and simulated tropical cyclone tracks.

To quantitatively evaluate the classification model performance, the spatial density of each set of tropical cyclone tracks in the Northwest Pacific region (i. e., number of 6 h tropical cyclone positions per area) is computed. The average spatial density of the 30-set ensemble of tropical cyclone tracks is computed and compared with the historical results, as illustrated in Fig. 9. Fig. 9(a) and 9(b) shows that the basic spatial distribution features of the historical tropical cyclone tracks can be well captured by the simulated tropical cyclone tracks. Although additional historical tropical cyclone tracks can be found in the tropical cyclone concentration regions, only few differences can be found in most regions. Fig. 9(c) shows the mean square error of the spatial density of tropical cyclone tracks. The value is greater than 10 in tropical cyclone concentration regions and less than 5 in most other regions. Fig. 9(d) presents the z-score value, that is, the historical distribution minus the average simulated distribution divided by the historical distribution. The z-score values in most regions fall into the range from - 0.3 to 0.3, which indicates that the simulation results are satisfactory. By contrast, the z-score values in few regions are high by more than one because these regions have few historical or simulated tropical cyclones.

Figure 9.  Spatial density distribution of 6 h tropical cyclone tracks.

The evaluation of latitude and longitude crossings is performed through the count comparisons of tropical cyclones crossing different latitude and longitude lines. In this study, comparisons of latitude and longitude crossings are performed in four different scenarios. Fig. 10(a) and 10(b) shows the counts of tropical cyclones crossing the 10°N: 10°: 50°N latitude lines. The number of crossings increases by 100 each time along the direction of increasing vertical coordinate to avoid overlapping of the drawing curves. This treatment is also used in other cases. Fig. 10(c) and 10(d) illustrates the counts of tropical cyclones crossing the 100° N: 10° : 180° E longitude lines. In most regions, the simulation spread is encompassed by the historical crossings. This result is consistent with the underestimation in the spatial density distribution of tropical cyclones in tropical cyclone concentration regions.

Figure 10.  Latitude and longitude crossings (solid line: historical; dash line: simulated).

The landfall count of tropical cyclone for the concerned region is important for tropical cyclone risk assessment. A coarse model of China's coastline is constructed using 16-line segments, as shown in Fig. 11 (a), and the coordinates are presented in Table 1. Fig. 11 (b) illustrates the historical and simulated landfall counts along China's coastline. The trend of simulated landfall count along the coastline is close to the historical landfall count, but relatively large differences appear in segments 3, 10, and 16. Fig. 11(c) shows the z-score value of landfall counts along the coastline. Three peak values appear in segments 10, 12, and 14 because the landfall counts in these segments are small, resulting in a relatively large deviation.

Figure 11.  Comparison of the landfall counts.

A classification model used for the stochastic simulation of tropical cyclone tracks in the Northwest Pacific region is proposed and applied to the simulation of tropical cyclone tracks from 1949 to 2016 in this region. The following conclusions are obtained through analysis:

(1) Based on the movement characteristics and steering positions of tropical cyclones, our research classifies the tropical cyclone tracks in the Northwest Pacific region into five categories.

(2) The consistency between the simulated and historical genesis and lysis positions indicates the reliability of the genesis and lysis models.

(3) The tropical cyclone tracks have been simulated with the traveling model by using L, mean, and mean square error of 6 h displacements.

(4) By using different quantitate analysis methods, comparisons of the spatial density distribution, latitude and longitude crossings, and landfall count between simulated and historical tropical cyclone tracks have been performed.

(5) Although the simulated tropical cyclone tracks have a relatively large difference compared with the historical data in certain regions, the tropical cyclone tracks can be satisfactorily simulated in most regions. This finding demonstrates the reliability of the classification model used in the stochastic simulation of tropical cyclone tracks.