1. Introduction

As populations in coastal areas continue to rise and real estate and infrastructure become more concentrated there, the risk of damage to human life, real estate, and infrastructure due to strong winds, storm surges, and high waves brought by tropical cyclones (TC) is expected to increase in the future. Bengtsson (2001) suggested that society’s capacity for disaster prevention against tropical cyclones could now be weaker than before because massive hurricanes did not occur between the 1960s and 1990s, yet the population in the United States increased steadily during that time. His suggestion was based on the work of Goldenberg et al. (2001), who focused on the long-term trend in the occurrence frequency of massive hurricanes during the period from 1950s to the early 2000s. It was not until August 2005 that a massive TC developed, with Hurricane Katrina developing up to a category 5 storm on the Saffir–Simpson hurricane scale. Katrina brought severe damage to the southeastern part of the United States. In addition, Hurricane Sandy inflicted massive damage along the east coast of the United States in October 2012.

Our knowledge on TC systems in terms of cyclogenesis factors, developmental processes, and the prediction of their movement is still limited, despite the risk of damage being high. The intensity, tracks, and occurrence frequency of TCs are necessary for estimating the risk of damage on a local and national scale. One major problem is the small amount of observational data on TC systems, both in terms of number of events and details regarding their internal mechanism. According to Henderson-Sellers et al. (1998), although the most reliable observation method is based on the use of reconnaissance airplanes, this method is expensive even for isolated events, and therefore global observation is impossible. On the other hand, global observation with satellites is a powerful approach and has the potential to become the dominant observation method, although using airplanes and in situ observations are still necessary for correcting observational data. However, global satellite-based observation of TCs began in 1966, and therefore even the information on the frequency of TCs is unreliable without considering the difference between data collected before and after the introduction of satellites (Knutson et al. 2010).

Other factors contributing to the limited knowledge of TC systems include the limitations to predicting the development process of TCs from the moment of their genesis on the basis of numerical simulations. The cyclogenesis mechanism is associated with large variations in both spatial and temporal scales because it depends on local physical and thermodynamic conditions, such as local atmospheric shear flow and distribution of water vapor content under the global Earth system. Therefore, global simulations with high resolution (10–50 km) are necessary for proper TC analysis for risk assessment of TCs on a global scale. For example, based on simulations with a resolution of 180 and 60 km, Bengtsson et al. (2007) suggested that a resolution of at least 60 km is necessary for simulation of TCs. Furthermore, comparing the results for global calculations with resolutions of 20, 60, 120, and 180 km, Murakami and Sugi (2010) also concluded that TC simulations require a resolution of at least 60 km, and that annual variations in TC frequency as well as climate changes show a correlation with the resolution of simulation. However, the detection of long-term trends is not realistic at present, even though we can conduct simulations at ultrahigh resolution with grid cell size of 10–20 km (Oouchi et al. 2006). This is because the global simulation of TCs for hundreds of years in the future at such high resolution would require enormous computational resources. Also, although general circulation models (GCMs) are being actively developed by different organizations, each model has a specific set of biases for TC reproducibility in the present climate. For example, Yokoi et al. (2009) compared the results obtained with eight GCMs for the North Pacific Ocean and suggested that the cyclogenesis characteristics of each GCM are different for the mean annual frequency of TC genesis, and that they depend on the reproducibility of the monsoon trough.

Other factors include the lack of quantitative knowledge with regard to the influence of global climate change caused by natural factors or anthropogenic environmental modification caused by greenhouse gases. For example, Goldenberg et al. (2001) showed evidence of long-term fluctuations in intense hurricane frequency in the North Atlantic Ocean and the Caribbean Sea. We must face the influence of the long-term trend of TC activity on disaster risk assessment even though the amount of TC data is small. In recent years, many researchers have analyzed the influence of global climate change by using GCMs results considering several greenhouse gas emission scenarios. Details of such analyses are presented in Henderson-Sellers et al. (1998), Emanuel et al. (2008), and Knutson et al. (2010). According to recently acquired knowledge, the number of tropical cyclones is expected to decrease whereas the occurrence frequency of intense TCs is expected to increase on a global scale. Furthermore, Murakami and Wang (2010) showed that the locations of cyclogenesis areas are likely to move in the future. We must reduce the uncertainty of future projections and distinguish them from natural variability to quantitatively assess the impact of anthropogenic factors.

Actually, storm surge damage is very sensitive to not only TC intensity but also TC track. Therefore, a shortage of observation data of TCs and the diversity of TC tracks are severe problems. One approach for increasing the number of events (TCs) for a particular region is to use of a stochastic downscaling method that can generate numerous artificial TC datasets that cannot be obtained with a physical model. This stochastic method, which is based on Monte Carlo simulations, is referred to as the stochastic tropical cyclone model (STCM). In the STCM, the sequential development of TCs is calculated statistically from given statistical parameters of TC data.

The assessment of extreme events is often based on extreme value analysis in various engineering fields (e.g., coastal engineering, river engineering, and wind engineering). The extreme values of TC intensity and storm surge around coastal areas has been estimated with extreme value analysis using historical observational data. However, estimating the return values for 50–100 yr in the future from short historical records entails considerable uncertainty because the number of TCs arriving at a particular region is small, and the damage depends on the TC track. Impact assessment analysis with the STCM can provide important information for the prediction of extreme values with consideration of climate change.

Several STCMs have been proposed thus far. One widely adopted method is known as a multiple linear regression (MLR) model, in which TC parameters such as minimum sea level pressure (SLP), translation speed, and their rates of change are expressed as sums of polynomial equations and random terms. The coefficients of the polynomial equations are determined by autoregression analysis of given data. The following equation is a general expression of an example of an MLR model:

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Here, ϕ is an objective TC parameter, ψ and θ are explanatory variables, the A_i terms are coefficients of the MLR model, and ε is a random residual component. The subscript t denotes a time step. For example, Vickery et al. (2000), James and Mason (2005), Hall and Jewson (2007), and Graf et al. (2009) suggested similar MLR models, although their particular choices of model expression, explanatory variables, and MLR model order were different. An MLR model assumes that the local TC parameters can be mainly determined from some correlation between the parameters (in the form of a polynomial expression), with an additional random factor following a Gaussian or some other type of distribution.

The characteristics of MLR models are divided into a deterministic part and a random residual component part. In MLR models, statistical variation in TC data is small if the deterministic part is dominant. Conversely, if the residual term is dominant, the result of the MLR model is similar to a random walk simulation. Furthermore, the seasonal variability of TCs and extreme TC events cannot be accounted for by MLR models if all TC data are used for calibration of the model coefficients. The reason for this is that the random component has been approximated by a simple PDF such as Gaussian distribution in most MLR models. To resolve the limitations of MLR models, classification of TC data was attempted in many studies. For example, Zhang and Nishijima (2012) recently presented a method for selection of proper variables for MLR models and a method for classification of TC data for the northwestern Pacific (NWP). However, these classifications were performed in an arbitrary manner, suggesting that the development of a generalized MLR model for global simulation is difficult.

An alternative to MLR models is the translation model approach. Emanuel et al. (2006) developed a method that can sequentially determine the TC parameters (translation speed and direction) by using probability density functions (PDFs) of their rates of change estimated from given data. The model they proposed regards PDFs as functions of the parameter values at previous steps, with the exception of minimum SLP, which is calculated deterministically from averaged synthetic environmental data. In addition, in the same paper, they proposed another model that can be used for calculating TC movement from averaged synthetic environmental data. Furthermore, Rumpf et al. (2007) developed a similar method that can estimate sequential changes in TC parameters (translation speed, direction, and maximum wind speed) by using PDFs of their rates of change estimated from given data. The model Rumpf et al. proposed assumes only the PDF of the maximum wind speed as a function of the value at the previous time step. An STCM using TC development as a function of the value at the previous time step based on temporal correlation is known as a Markov chain model (Emanuel et al. 2006). In this paper, we denote this type of model as a temporal correlation PDF model for convenience. The following equation is a general expression of the temporal correlation PDF model:

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Here, x is the location of the target point. In the temporal correlation PDF model, the range of TC statistical variation depends on the shape of the estimated PDF. The temporal correlation PDF model is more flexible than an MLR model, in which a large portion of rates of change are calculated deterministically. However, the accuracy of the model depends on the given approximated PDFs, and therefore the particular method of approximation of the appropriate PDFs from observational data, which tend to be insufficient, plays an important role. Emanuel et al. (2006) presented a method for expanding the sampling scale in both spatial and temporal directions to minimize the impact of data shortage. Unfortunately, this procedure is not well documented. For example, neither Emanuel et al. (2006) nor Rumpf et al. (2007) describe in detail how they estimated the relevant PDFs. In addition, Emanuel et al. (2006) showed that weak hurricanes are overestimated and intense hurricanes are underestimated in their model. Therefore, the temporal correlation PDF model must be improved to obtain an accurate quantitative prediction of TC parameters.

In previous studies, an STCM was developed for specific ocean basins, and as yet there are no global STCMs. One major reason for this may be the difficulty of proper universal classification of TC data for MLR models. However, global simulation of TC using STCM-implemented annual global TC variation is necessary for the assessment of climate change. In addition, the unified evaluation of reproducibility of TC parameter by using same model in different basins is important for the future development of an STCM. By the way, Emanuel et al. (2006) showed that the simulation of a single TC life cycle requires 15 s because their models require environmental conditions to be calculated to estimate the minimum SLP. The computational cost of their STCM is lower than that of GCMs. But it is still nonnegligible in the case of a long-term global simulation. Furthermore, previous temporal correlation models are not suitable for applying to areas where the amount of TC data is small. From the standpoint of generalization and improvement of STCMs, standard verification is also required for each individual basin.

In this study, we propose and verify a global STCM (GSTCM) with temporal correlation PDFs that can be applied to areas with scarce TC data. This paper is organized as follows: First, in section 2, we present an overview of GSTCMs and the characteristics of TC data. Next, section 3 contains details of the temporal correlations between TC parameters and methods for estimating the PDF of each TC parameter. In particular, we discuss the estimation of PDFs for calculating rates of change of TC parameters. Section 4 provides verification of numerical results as compared with observational data and discusses further improvements and applications. Section 5 presents a discussion of the results and our conclusions.

2. Overview of GSTCMs

The entire process of development and decay of TCs (from the cyclogenesis phase to the cyclolysis phase) is recreated in a GSTCM. The International Best Track Archive for Climate Stewardship (IBTrACS) v02r01 as provided by the National Oceanic and Atmospheric Administration was used as a source of data for calibration of the statistical model coefficients. IBTrACS v02r01 is a global observational dataset covering the period from the 1800s to 2008 [the latest version of IBTrACS (v03r04) was released recently (Knapp et al. 2010)]. This archive contains ensemble mean data from observations performed by different institutions using various methods, where observational data from different sources are unified. Figure 1 shows the annual variation in the number of cyclogenesis events at each basin and the global number of cyclogenesis events as calculated from IBTrACS data. Each line denotes the moving average value for 3 yr. The tendency of the number of events to increase each year is due to not only the natural variation but also the development of observational system, as mentioned in the preceding section. As can be seen from this figure, global TC observation started in the 1950s. Furthermore, according to Knutson et al. (2010), global satellite-based TC observation started in 1966, therefore, the quality of TC data is not homogeneous in the 1950s–60s. With this in mind, although TC data collected since the 1950s were used for model calibration in this study, the PDFs of cyclogenesis points were determined from data collected after 1966 because homogeneity in global was improved by satellite observation. Note that continuous effort has been made for improve global homogeneity of TC data since 1966. Figure 1 shows that the number of cyclogenesis events is highest in the NWP, followed by the southern Indian Ocean (SIO), the northeastern Pacific (NEP), and the North Atlantic (NAT).

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Annual variation in the number of cyclogenesis events in each basin—NAT, South Atlantic (SAT), NWP, NEP, SP (South Pacific), NIO, and SIO—and the global number of cyclogenesis event from 1950 to 2008.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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It is notable that the annual number of cyclogenesis events at each basin fluctuates wildly and the maximum of their standard deviation is about 10. Moreover, long-term variations in the annual number of cyclogenesis events become obvious from the calculated moving averages, although verification is difficult because of the shortage of observational data. Goldenberg et al. (2001) pointed out that such variations arise under the influence of global-scale natural climate oscillations, such as El Niño–Southern Oscillation (ENSO) and the North Atlantic Oscillation. In strict terms, we must incorporate the effects of these multidecadal variations into the GSTCM. However, the correlation of these long-term variations cannot be fully evaluated because the observation period is too short. On the other hand, Fig. 1 shows the annual variation in the number of global cyclogenesis events, where the global number of cyclogenesis events around 1970 was clearly higher compared to the present. The variation of global cyclogenesis events is smoother and more stable than that of each basin. With these in mind, in this study the number of global cyclogenesis events is determined by considering its variance statistically, without directly giving the variation in the number of cyclogenesis events. Subsequently, the annual number of cyclogenesis events is distributed across all basins while maintaining a constant average ratio of the number of cyclogenesis events at each basin.

Figure 2 shows an overview of the GSTCM. IBTrACS includes the longitude and latitude of the TC center, the minimum SLP, and the maximum wind speed acquired at intervals of 6 h. In this study, we modeled three TC parameters, namely translation direction and speed and minimum SLP, by means of stochastic modeling. The translation direction and speed are calculated from the longitude and latitude of the TC center for each instant. In this GSTCM, the track of the TC center and the TC parameters related to it are statistically simulated. However, maximum wind speed data were not used, because the modeling wind speed required additional statistical model as a function of TC radius. Therefore we only modeled minimum SLP as representative parameter of TC intensity. On the basis of the IBTrACS reliability index, we excluded some portions of the minimum SLP data from the model because of low reliability, and the missing data were generated by linear interpolation.

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Overview of the GSTCM.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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The GSTCM can be divided into three major parts: cyclogenesis part, development part, and cyclolysis part. In our model, all given model parameters were calculated in advance using a 3° grid and the actual Monte Carlo simulation was conducted with a 1° grid where interpolated statistical values were used.

The first stage of the GSTCM is the cyclogenesis part. The annual global number of cyclogenesis events, the locations of cyclogenesis areas, and the initial TC parameter values are determined statistically in this order. The PDF of the annual global number of cyclogenesis events can be approximated by a lognormal distribution with a mean of 4.725 and a standard deviation of 0.134 (Fig. 3). The Pearson chi-square test applied with a significance level of 1% did not reveal any significant difference between the approximated PDF and the PDF of observational data. Every run for determining the global number of cyclogenesis events is independent, and data for consecutive years have no physical meaning, although the effects of annual variation are reflected in the PDF of the annual global number of cyclogenesis events as collective data. The locations of cyclogenesis areas are determined statistically based on global cyclogenesis frequency distribution estimated using the Monte Carlo method with a 1° grid. As a result, the component ratio of the sum of PDFs of the cyclogenesis events at each basin corresponded to the mean ratio of the number of cyclogenesis events at each basin (Fig. 1). In fact, the TC activity changes depending on the season (e.g., Menkes et al. 2012). Although it is possible to incorporate seasonal effects, they are in a trade-off relation with the reliability of the annual mean. Therefore, we did not split the number of cyclogenesis events by month. The initial TC parameter values are determined statistically based on the PDF of each parameter as estimated from observational data. In this case, we assume that the translation direction follows a normal distribution, whereas the translation speed and the minimum SLP follow a lognormal distribution. Figure 4 shows the results of a lognormal approximation of the translation speed and the difference in the minimum SLP from the peripheral pressure at certain regions. The crosses denote observational data, showing a close agreement with the approximated distribution, which is denoted with a broken line. However, the minimum SLP shows a discrete distribution because many mean values of observational data varied by an order of 1 hPa.

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Exceedance probability of the global number of cyclogenesis events with lognormal distributions (times signs: data; dashed line: lognormal distribution).

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Exceedance probability of initial TC parameters (times signs: data; dashed line: lognormal distribution).

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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The subsequent development process in the GSTCM provides the rates of change of TC parameters as functions of their values at the previous time step. For example, the temporal correlation between minimum SLP and its rate of change is shown in Fig. 5. This figure shows sample data points obtained with a 3° grid centered at 33°N, 138°E. The plots and contour lines denote observational data and the approximations of their joint PDFs, respectively. There is a local temporal correlation between the minimum SLP and its rate of change. If the minimum SLP at the previous step is low (which indicates a strong TC), the rate of change is likely to be high (the strong TC is likely to weaken). This is reasonable because TCs generally tend to weaken in midlatitude regions. Some fluctuation can be seen in the plausible rate of change of the minimum SLP at previous steps. The joint PDFs, which are denoted by contour lines, are approximated considering this fluctuation. At the development stage, realizations of the rate of change are generated by Monte Carlo simulation. These realizations are simulated following conditional PDF given the parameter value at the previous step, where the conditional PDF is calculated from the joint PDF by

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There are several candidates for an approximation method for joint PDFs, but relevant details are presented in the following section. The TC parameter values at the next time step are determined by multiplying the rate of change with the time step Δt. A Δt = 3 h is taken as the basic time step; however, if the TC moves to a neighboring grid cell within one time step, Δt is reduced, and the parameters are recalculated automatically.

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An example of temporal correlation between minimum SLP and its change rate, and their joint PDF: the black solid lines represent the ridge lines of the joint PDF. Red circles represent the observational data.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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The final part of the GSTCM is the evaluation of cyclolysis conditions. However, the transition from TCs to extratropical systems was not considered in this research, because the condition of transition phase are difficult to determine by using only TC track data such as minimum SLP correctly. For example, Hart (2003) showed that three parameters (the lower-tropospheric thermal asymmetry, the lower-tropospheric thermal wind, and the upper-tropospheric thermal wind) are useful for the criteria of the cyclone phase. However, it is difficult to extract these parameters from reanalysis data of tropical cyclones over the world at this moment. In this stochastic model, the distinction of tropical cyclone and extratropical cyclone was not included. As a result, we have simply used the term cyclolysis for convenience in this paper. The model determines whether a TC disappears every time the TC moves to a neighboring grid cell. The criteria for a cyclolysis event are as follows:

1. TC moves into an area where no TCs have arrived in the past.

2. The minimum SLP of the TC reaches 1015 hPa.

3. The conditions for a cyclolysis event are determined statistically based on a cyclolysis probability R_lysis estimated from observational data.

Actually, local environment SLPs are different from 1015 hPa. But the maximum value of SLP in the final stage of the TC was 1015 hPa in IBTrACS data, therefore this value was used as a maximum threshold. In case that one of these three conditions is satisfied, the corresponding TC is considered to have disappeared, and the calculation is terminated. However, if a single judgment was performed for criterion 2, the ratio of weak TCs in the simulation result was larger than actual. With this in mind, even if criterion 2 was satisfied, recalculations from the initial conditions were conducted a maximum of 5 times. As a result, most TCs disappeared by criterion 3. The probability of a cyclolysis event R_lysis is determined as the ratio of the number of cyclolysis events N_lysis to the total number of TCs N_arrival arriving at each area. However, in the actual simulation, R_lysis is determined empirically using the following equations considering the insufficient number of observations:

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Although in this model the entire TC development process is determined statistically, some TCs move toward the equator and decay because most TCs occur at low latitudes and initial movement is unstable. Therefore, if TCs move across the equator, recalculation is conducted from the initial conditions as an additional criterion. Moreover, translation directions fitting inside μ ± 2σ (here, μ and σ are the mean and the standard deviation for the present grid) are allowed, and outliers are recalculated up to 10 times at each time step. If more recalculations are conducted, a tentative PDF is prepared from the mean and the standard deviation of the present grid, and the next translation direction is determined by using a Monte Carlo simulation. In reality, such extraordinary processes occur in less than 0.1% of the cases and can be ignored. If the TC track describes a closed curve, recalculation is conducted. The number of observed TCs describing a closed track is not zero. However, such TCs are rather rare, and therefore they are omitted intentionally from this model.

3. PDF model using temporal correlation with TC parameters

As mentioned in the previous section, in a GSTCM the PDFs of the rates of change of TC parameters are determined statistically as functions of the parameter values at the previous time step by using joint PDFs. Therefore, it is important to examine how the joint PDF of each parameter can be approximated accurately to improve the reproducibility of the model. The easiest method of approximation is based on a histogram of the rates of change corresponding to TC parameter values at the previous time step. This method is not adequate in cases of insufficient TC data because limited data would be separated again at each area. However, the method considering the continuity of data by using principal component analysis (PCA) could estimate joint PDF from all TC data that passed through a particular area. Accordingly, in this study we propose a method for smooth approximation of joint PDFs by using PCA of local TC data.

First, appropriate virtual variables are created in PCA to obtain the temporal correlation between a TC parameter ϕ and its rate of change . In this study, not only the first principal component α, but also the second principal component β is calculated as part of a simplified algorithm considering the change in correlation variance. The values of α and β are obtained from the following equations:

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Here, and are the mean values of the TC parameter and its rate of change, and and are eigenvectors calculated from the variance–covariance matrix of ϕ and , respectively. Assuming a normal distribution of the principal components α and β, the joint PDF is approximated as a superposition of two normal distributions around two principal axes. Therefore, the joint PDF is determined as a function of six explanatory variables. Here, and are the mean values of α and β, and σ_α and σ_β are their standard deviations, respectively.

As an example of the joint PDF for the minimum SLP, P is presented using contour lines in Fig. 5. First, we compared results using normal and logarithmic values of TC parameter ϕ. As a result, it was shown that use of lognormal values is better for estimation of joint PDF of minimum SLP and translation speed. Therefore, logarithmic values are used for statistical modeling of them. The ridge lines of this joint PDF correspond to the principal component axes. The parameterized joint PDF can be considered to represent an appropriate distribution verified by substantial observational data. However, if we consider the distribution in greater detail, the distribution of observational data in P–ΔP space can be divided into two parts, where ΔP is the rate of change of the minimum SLP. One part is distributed with high intensity near the point (P, ΔP) = (1000, 0) and the other part shows a wide distribution near the point (P, ΔP) = (960, 0.5) in Fig. 5. In that case, if a unimodal distribution is used for the joint PDF, the approximation is not as close to the actual data. Considering a bimodal distribution, we conduct cluster analysis of TC data to obtain two clusters before approximation of the joint PDF with PCA. A joint PDF is prepared from each cluster, after which the bimodal joint PDF is synthesized by assigning a weight R_c to each cluster, where

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Here, N_c is a the number of the respective cluster data samples, and N_all is the number of all data samples. However, a unimodal distribution is assumed when the number of TC data samples is less than 30. In this model, a bimodal joint PDF was applied only when the result of cluster analysis showed that corresponding data could be divided into two groups. If the data for the corresponding area could not be separated after the cluster analysis, a unimodal joint PDF was applied. Therefore, the bimodal joint PDF model also contains a unimodal joint PDF. Furthermore, if the number of TC data samples is less than 5, an approximation of the joint PDF is not calculated, and the result for the nearest point is used. We apply the K-mean method to the cluster analysis algorithm with the criterion of Euclidean distance. The initial center of each cluster is selected 50 times at random in a single clustering procedure. Finally, the result for which the sum of Euclidean distances is minimal is adopted as the cluster candidate.

Figures 6 and 7 show a comparison of the unimodal and bimodal joint PDFs of the minimum SLP at latitudes from 6° to 45°N and longitudes from 135° to 138°E. These regions are used for verification in NWP, where TCs are generated frequently. We estimate the minimum SLP at the entry and exit points at each grid cell by using linear interpolation. The entry value is the minimum SLP in the previous time step, and the difference between entrance and exit values is defined as the rate of change. As can be seen from this figure, the bimodal joint PDF shows one cluster peak near 1000 hPa at all latitudes, and another peak appears at a lower minimum SLP. Here, we refer to these two clusters as first and second for convenience. The peak joint PDF value of the first cluster is larger than that of the second cluster because the number of observations belonging to the first cluster is greater at low latitudes near the equator. This trend becomes weaker at higher latitudes. The two cluster peaks are almost equal around 25°N (Fig. 7, example 4). After that, the peak value of the first cluster is larger than that of the second cluster at higher latitudes. The second peak of the cluster is around 970 hPa at low latitudes, but its peak moves in the direction of lower minimum SLP at higher latitudes. Furthermore, the variance of the minimum SLP of second cluster increases as the latitude increases. The second cluster peak shifts in the direction of higher minimum SLP if the latitude is higher than 20°N. On the other hand, the second cluster peak of the minimum SLP rate of change is around −0.5 hPa h⁻¹ at low latitudes (a negative rate of change means that the TC becomes stronger). The peak of the PDF moves to a positive direction in the rate of change when the north latitude is above 15°N. For example, the second cluster peak is around 1.5 hPa h⁻¹ at 38°N. However, the peak of the minimum SLP rate of change in the first cluster does not change, although the variance of the first cluster does. This sequence of joint PDFs changes (cluster ratio R_c as well as joint PDF peak position and variance) correspond to the development and decay process of TCs. Based on the above discussion, the meridional development of TCs can be summarized as follows. First, the ratio of undeveloped TCs is high at low latitudes, and TCs slowly increase in strength at higher latitudes. Developing TCs become strong quickly as their minimum SLP is lower. The ratio of developing TCs increases at higher latitudes and the variance of the minimum SLP of TCs also increases. However, after TCs reach about 25°N, strong TCs weaken rapidly. The variance of the minimum SLP of TCs becomes smaller, and TCs start to decay.

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Unimodal joint PDF of minimum SLP and its change rate at different latitude ranges. Red circles represent observational data.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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As in Fig. 6, but for a bimodal joint PDF.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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In the unimodal model, the peak position and the variance of the joint PDF depend on the latitude. Also, there is a transition between the development and decay phases of TCs around 25°N. However, the capability of the unimodal joint PDF to account for observational data is inferior to that of the bimodal joint PDF. Particularly, in a GSTCM with the unimodal joint PDF model, weaker TCs decay at low latitudes. This is a severe problem in the simulation of the early development process of TCs. This problem is evident from the fact that the ridge line of the unimodal joint PDF over 1000 hPa is an upward-sloping curve at low latitudes (Fig. 6, examples 1 and 2). In contrast, the ridge line of the bimodal joint PDF over 1000 hPa is almost flat, and the mean rate of change is below 0 hPa h⁻¹. Therefore, the bimodal model tends to intensify weaker TCs at low latitudes. In summary, the bimodal model is qualitatively superior to the unimodal model.

In addition, a similar comparison is conducted for translation direction and speed, respectively. Figures 8 and 9 show the joint PDF of the translation direction θ (°) and its rate of change Δθ (° h⁻¹) at latitudes from 6° to 45°N and longitudes from 135° to 138°E. It should be noted that the values of the translation direction wrap around at 360°, and therefore it is difficult to perform PCA directly from observational data for θ_o for angles are distributed around 0° and 360°. With this in mind, we calculate normalized data by using the following equation at each region:

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Here, represents the mean value at each region and θ represents normalized data. Therefore, the mean of θ is 180° in PCA. PCA and subsequent calculations are conducted with this θ. Therefore, the observational data are distributed around θ = 180 in Figs. 8 and 9. In strict terms, this normalization operation cannot produce accurate PCA results, but it is acceptable as an approximate solution because the variance of the translation direction of TCs is within a certain range in each region. Figures 8 and 9 show that this normalization operation gives an appropriate approximation of the joint PDF results for θ.

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As in Fig. 6, but for translation speed.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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As in Fig. 8, but for a bimodal joint PDF.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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In the joint PDF of the translation direction, a downward-sloping (upward-sloping) ridge line means that the translation direction tends to converge toward (diverge from) the mean value. Figures 8 and 9 show that in general the translation direction tends to converge toward the mean value at low latitudes and to diverge from the mean value at mid- or high latitudes. The shape of the unimodal joint PDF in Fig. 8 indicates that the joint PDF is almost flat in the θ direction at latitudes from 18° to 33°N. Therefore, the change in translation direction is similar to the result obtained from a random walk simulation, where changes in direction (clockwise and counterclockwise) occur with almost the same probability. However, the variance of the rate of change depends on θ. If the previous direction is considerably different from the average, the magnitude of the rate of change is likely to be large. On the other hand, the bimodal model shows a joint PDF similar to that of the unimodal model, but at low latitudes there is a cluster around the mean value (180°) and another cluster 90° clockwise with respect to the previous cluster (Fig. 9). Also, the difference in R_c decreases at high latitudes. The R_c values of the two clusters are almost the same at 25°N, whereas the clockwise cluster is dominant around 30°N. That is, the translation direction of the bimodal model exhibits two convergence points around 18°–27°N. The clockwise cluster shows a tendency to diverge from the mean value around 30°–33°N, and both clusters show a tendency to diverge from the mean values at higher latitudes. These features of the joint PDF of the translation direction match the observational data because TC tracks in NWP are divided into two main categories. One type of track passes straight from low latitudes through the Philippines and toward the South China Sea, and the other type of track curves sharply along the way and goes toward Japan (Fig. 10).

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Tracks of observed TCs in IBTrACS (1950–2000).

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Figures 11 and 12 show the joint PDF of the translation speed V (km h⁻¹) and its rate of change ΔV (km h⁻²) at latitudes from 6° to 45°N and longitudes from 135° to 138°E. Based on our preliminary experiments, we used a logarithmic distribution for V during PCA and subsequent calculations. It is estimated that the rate of change ΔV of the translation speed is similar to the result of a random walk because the ΔV of observational data is distributed around 0. However, the rate of change of the translation speed is positive when V is small at low latitudes. As TCs move from low toward higher latitudes, the mean translation speed increases, but the mean rate of change ΔV remains around 0 km h⁻² at latitudes below 25°N. The mean of ΔV is positive and becomes large around latitudes from 25° to 30°N as the translation speed increases. After that, the tendency is reversed, and the mean of ΔV becomes slightly negative around 43°N as the translation speed increases. The variance of ΔV becomes larger above 24°N, and it is particularly large around 37°N. These features of the joint PDF of the translation speed indicate that some TCs accelerate suddenly at midlatitudes under the influence of subtropical westerly winds and then, having lost their strength, decelerate at high latitudes. The bimodal joint PDF of V–ΔV is not significantly different from the unimodal distribution at low latitudes. However, the bimodal joint PDF reaches a cluster peak around 0 km h⁻², and another cluster reaches another peak around the region of positive ΔV. Therefore mean V of the latter cluster is larger than that of the former cluster. However, another cluster is observed around negative ΔV above 43°N.

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As in Fig. 6, but for translation speed.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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As in Fig. 11, but for a bimodal joint PDF.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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The choice of distribution and number of clusters is rather arbitrary, and careful verification is necessary. Based on the above analysis and verification of basic TC statistical distributions, we present the proposed GSTCM in the next section.

4. Results and discussion

a. Model verification

Using numerical analysis, the eight models shown in Table 1 are constructed and compared here. To evaluate the reproducibility of the development process of TC, simulation results based on GSTCM and observational data (IBTrACS) were classified by several areas (denoted as areas 1–4) in NWP (Fig. 13). The reproducibility of TC parameters was compared for each area. Although GSTCM model parameters were automatically decided by PCA and cluster analysis, the cyclolysis ratio had to be calibrated to agree with observation data in this study. This calibration was performed on the result of NWP mainly. The verifications of the GSTCM in NAT and SP were also conducted and their results are shown in this section.

Table 1.

Combination of joint PDF models.

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Areas used for verification of the GSTCM.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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The results of comparing TC parameters such as average of minimum SLP, minimum of minimum SLP, average translation direction, and average translation speed for each model are shown in Figs. 14–17. Here, average and minimum values were calculated from the parameter values of individual TCs at the time steps while each TC was passing through the respective areas. In these figures, histograms denote observational data and lines indicate simulation results based on a given PDF. The observational dataset covered 59 yr (from 1950 to 2008), and simulation results were an ensemble mean of 50 groups of 50 yr of data each (for a total of 2500 yr) to obtain statistical convergence.

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Comparison of the PDFs of the average of minimum SLP in areas 1–4 (histograms: observation; lines: simulation).

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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As in Fig. 14, but for the minimum of minimum SLP.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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As in Fig. 14, but for the average translation direction.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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As in Fig. 14, but for the average translation speed.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Figure 14 shows the results for the average of minimum SLP, where the bimodal model (Bp, Bps, Bpd, and Bpsd, shown in blue) provide higher reproducibility than the unimodal model (U, Bs, Bd, and Bsd, shown in green and red). The PDF of the average of minimum SLP shows a peak around 1000 hPa, and 70% of the samples occupy the region above 985 hPa in area 1 (lowest latitude). The simulation based on the unimodal model overestimates the PDF of weak TCs (80% of the samples with a minimum SLP above 985 hPa). On the other hand, the simulation based on the bimodal model yields highly reproducible results in comparison with the unimodal model. Furthermore, the PDF of strong TCs with a minimum SLP below 970 hPa is underestimated by the unimodal model, however, the results of the bimodal model show close agreement with observational data. In area 2 (midlatitudes), weak TCs with a minimum SLP above 985 hPa are reduced to about 55%, while strong TCs (with a pressure between 925 and 970 hPa) increase. In this area, the unimodal model yields an overestimation for weak TCs (the minimum SLP of 65% of TCs is above 985 hPa) and an underestimation for strong TCs, although the bimodal model result is in close agreement with the observational data. In area 3, there are fewer highly intense TCs with a minimum SLP between 925 and 940 hPa and a larger number of strong TCs with a minimum SLP between 955 and 970 hPa because TCs that have developed in area 2 weaken in area 3. In the unimodal model, the reproducibility of these changes is low, and its PDF is almost the same as that in area 2. On the other hand, the results of the bimodal model agree with the PDF of observational data, although they are slightly overestimated around 925 hPa and underestimated between 955 and 970 hPa. This trend is similar to that at much higher latitudes in area 4, indicating that the reproducibility of the bimodal model is higher than that of the unimodal model. The accuracy of the simulated PDF of the average of minimum SLP depends heavily on the modeling of the joint PDF of the minimum SLP.

The superiority of the bimodal model becomes clear from a comparison of the temporal minimum of minimum SLP PDF (Fig. 15). The reproducing of the minimum value of SLP in its lifetime with a GSTCM is more difficult than that of the average value because the minimum value is an instantaneous value. Indeed, the results obtained with the unimodal model (U, Bs, Bd, and Bsd, shown in green and red) are overestimated around 985 hPa and underestimated between 895 and 955 hPa. Consequently, the reproducibility is lower than in the case of the average of minimum SLP. On the other hand, the bimodal model (Bp, Bps, Bpd, and Bpsd, shown in blue) achieves high reproducibility even for the temporal minimum of minimum SLP. The important point is that improving the accuracy of approximation of joint PDFs contributes toward the high reproducibility of the minimum SLP, as clear from the case of the bimodal model. For example, on average, weak TCs (>985 hPa) become weaker when using a GSTCM with the unimodal joint PDF of minimum SLP. In addition, the low reproducibility of the early stages of TCs at low latitudes results in difficulty of reproduction of TC development at mid- and high latitudes. On the other hand, although the rate of change might be small, weak TCs become stronger on the average when using a GSTCM with the bimodal joint PDF for minimum SLP, showing that the joint PDF of stronger TCs is in close agreement with that of historical observational data at mid- and high latitudes.

Next, the performance of each model is examined with respect to the average translation direction (Fig. 16). The translation direction is defined in nautical coordinates to increase in clockwise direction, and 0° is defined as north. The reproducibility of results obtained with the bimodal model (Bd, Bpd, Bsd, and Bpsd, shown in blue) is higher than that of results obtained with the unimodal model (U, Bp, Bs, and Bps, shown in green and red). At low latitudes (area 1), major TCs propagate in the direction of 270° (westward), and their distribution is slightly biased to the north. This trend is reproduced relatively well by all models, but the bimodal model achieves a slightly higher reproducibility than the unimodal model for peak values. At midlatitudes (area 2), the distribution of TCs moves in clockwise direction, and the peak of the PDF is located around 300°. The simulation results show a similar distribution, but they are slightly biased to counterclockwise. A comparison between the two models indicates that the bimodal model achieves higher reproducibility in terms of translation direction than the unimodal model. In areas 3 and 4, the PDF moves in clockwise direction, and its peak is located around 45° (northeastward). There is no notable difference in PDF of observational data between areas 3 and 4. Although the PDFs in the simulation results of the unimodal model also shift in clockwise direction, the magnitude of the shift is smaller than that of observation results. However, a GSTCM with the bimodal model achieves higher reproducibility than that with the unimodal model.

The bimodal model (Bs, Bps, Bsd, and Bpsd, shown in blue) does not show an improvement in terms of reproducibility of the translation speed (Fig. 17). The PDF of the observational data shows a sharp distribution around the peak (15 km h⁻¹) in area 1. The PDF of the translation speed shifts toward higher values as the latitude increases, after which the slope becomes gentler. The difference in PDF of observation between area 3 and 4 is small and the simulation results show a similar PDF distribution in comparison with observational data. However, in area 1, the peak of the PDF is higher than that of observational data, and the ratio of slow TCs is reduced in the bimodal model. Furthermore, the reproducibility of the transition in translation speed distribution following the shift toward the poles is not so good in both the unimodal and the bimodal model, and the simulated TCs tend to be faster than those in observational data. Therefore, the peak of the PDF is shifted to 20 km h⁻¹, and slow TCs (with a speed between 5 and 15 km h⁻¹) are underestimated, whereas fast TCs (with a speed above 20 km h⁻¹) are overestimated.

In this way, the bimodal joint PDF of minimum SLP improves reproducibility of mean and minimum of minimum SLP. But it does not play a major role to reproducibility of translation direction and speed. Similarly, the bimodal joint PDF of translation direction improves reproducibility of translation direction, while it does not influence reproducibility of minimum SLP and translation speed. Then, the bimodal joint PDF of translation speed does not improve reproducibility of translation speed, but it does not influence reproducibility of minimum SLP and translation direction. As a result, each joint PDF of one TC parameter is independent of reproducibility of other TC parameter.

As mentioned above, although qualitative verification was carried out for the PDFs of TC parameters, a quantitative comparison is performed by using the error index K defined in the following equation:

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Here, p_s and p_o are the simulated and observed PDFs of certain TC parameters, and i is the bin of the histogram. Therefore, K = 0 means a perfect match, and a large value of K indicates low reproducibility. Figure 18 shows a cobweb chart of the K values of TC parameters for each PDF model. First, it is obvious that in all models the K values of the average and minimum of minimum SLP are larger at higher latitudes. This means that the accumulation of error starts at lower latitudes. The bimodal model is superior to the unimodal model with respect to the minimum SLP, as expected. The K value obtained with the bimodal model is smaller than that of the unimodal model at all latitudes, and the reproducibility is always high. Second, in a similar manner, the K value of the average translation direction tends to become large at higher latitudes, although K in area 3 is larger than that in area 4 in the case of the unimodal model. Conversely, the K value with respect to the latitude is constant in the bimodal model. Finally, in areas 3 and 4, the difference in K values between different models is small in the case of average translation speed. However, it is clear that the K value for the joint PDF of the translation speed is larger in the bimodal model than in the unimodal model, and its reproducibility is low already in areas 1 and 2. Similarly to the qualitative verification above, the superiority of the bimodal joint PDF model was confirmed with respect to minimum SLP and translation direction, although not in the case of translation speed.

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Comparison of model performance by cobweb charts of the individual error index K.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Figure 19 shows the average error index K averaged over areas 1–4. The error index K averaged over each TC parameter is also shown in Fig. 19. The most appropriate model in terms of reproducibility can be considered to be Bpd, although most of the other bimodal models show better performance than that of the unimodal model.

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Comprehensive comparison of model performance by a cobweb chart of the average error index K.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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A comparison of the PDFs of TC parameters at representative areas in NAT and SP are shown in Fig. 20. Improvement in the GSTCM by the bimodal joint PDF was observed in the case of minimum SLP and forward direction, particularly in SP. However, the reproducibility of a GSTCM with the bimodal joint PDF for translation speed showed little difference in comparison to that with the unimodal joint PDF. These trends were similar to the results in NWP. In the case of NAT, the number of medium-strength TCs with a minimum SLP of about 970–985 hPa is overestimated. However, the reproducibility of stronger TCs (<955 hPa) was relatively high. Furthermore, the translation direction in GSTCM results is biased to the northwest direction in NAT.

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Comparison of the PDFs of average TC parameters for areas 5 and 6 (histograms: observation; lines: simulation).

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Figure 21 presents the TC tracks simulated with the Bpd model. The quality and diversity of simulated tracks are close to those of observational data shown in Fig. 10, and general characteristics, such as that the direction of TCs is likely to recurve around 25° north and south, are well reproduced.

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The TC tracks obtained through simulation with the Bpd model for a 50-year period.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Finally, we examine the reproducibility of the TC frequency, which is required for planning disaster prevention measures. Figure 22 shows a comparison between observation and simulation results obtained with the Bpd model with respect to the mean annual TC frequency at each grid cell (cell size: 5° × 5°; unit: number per year per grid). The mean annual TC frequency is highest in NWP, exceeding 5 events in the area around the northeast of the Philippine Sea and the South China Sea. The mean annual TC frequency is between 1.5 and 2 around the west coast of Mexico (NEP), the northeast coast of Florida (NA), the 12°S band in SIO, and north Australia (SP). The simulation results provide a close reproduction of this distribution pattern of mean annual TC frequency. However, areas that are frequently hit by TCs according to the simulations spread slightly northward, whereas the frequency is slightly reduced at latitudes above 30°N. As a result, the frequency of TCs in the simulation is lower than that derived from observational data near the Philippine Islands. On the other hand, it should be noted that the observation period in IBTrACS is limited to about 50 years, and the validity of the mean TC frequency at each grid might be uncertain. In addition, as shown in Fig. 1, there is a long-term variation in the annual number of cyclogenesis events, which also affects the annual TC frequency. For example, it has been shown that the frequencies of TC in the eastern North Pacific in about last 30 yr were lower than before 1980. However, in our STCM, the continuity of annual TC frequency is not able to be examined from simulation results because it has no physical meaning. Thus, the annual TC frequency must be evaluated by considering its PDF in addition to its mean value.

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Comparison of mean annual TC frequency (number per year per grid; Δx = 5°).

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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In this regard, we examined the validation of the PDFs of annual TC frequency for areas 1–4. Figure 23 shows comparison of the PDFs of the annual TC frequency in each area. The simulation results were ensemble mean of 50 groups for 50 yr simulation to obtain statistical convergence. Each error bar shows standard deviation of bins between the groups. The variation in mean annual TC frequency in the groups is relatively large in comparison to the mean value. These large variances of PDF indicate that annual TC frequencies that were estimated in the target area are not stable values statistically. The peak position and distribution pattern of the PDF of the ensemble mean values agree with observational data. This result serves as evidence of the effectiveness of GSTCMs for quantitative prediction of frequency of TCs.

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Comparison of PDFs of annual TC frequency in areas 1–4 (histograms: observation; lines: simulation).

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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b. Further improvements and applications

Choosing appropriate cyclolysis criteria is important for improving GSTCMs because they strongly affect the outcome for the local TC frequency. As indicated in section 2, the cyclolysis criteria in this study are tentative, and the threshold of the cyclolysis probability distribution is arbitrary. It is assumed that TCs decay when the minimum SLP approaches the pressure level of the surrounding environment (1015 hPa), but this criterion does not apply to all observed TCs. Figure 24 shows the latitudinal distribution of final minimum SLP for historical observation data. At low latitudes, TCs are considered to disappear when the minimum SLP is relatively high (weak TC), and at mid- to high latitudes even TCs with low minimum SLP (strong TCs) tend to disappear. Another notable feature is that the minimum SLP at cyclolysis events shows a large variance at mid- to high latitudes. In IBTrACS data, the end of a TC includes two different events: one is cyclolysis and another is cyclone phase change. Therefore it is difficult to determine whether cyclolysis events occur by considering only minimum SLP. Additional criteria for cyclolysis events are therefore required to implement a GSTCM. For example, for their STCM, Rumpf et al. (2007) proposed a cyclolysis probability as an exponential function of maximum wind speed. However, this cyclolysis criterion is also arbitrary since results are sensitive to the PDFs of all TC parameters. Thus, we used a combination of cyclolysis criteria by considering the minimum SLP threshold and a cyclolysis probability estimated from actual observational data. The cyclolysis probability coefficient in this paper is taken as 0.4 to ensure that the TC frequency as shown in Fig. 23 corresponds to observational data. Figure 25b shows the cyclolysis probability distribution calculated with the Bpd model. This distribution pattern is almost the same as that of observational data, which is shown in Fig. 25a, and the spatial correlation coefficient between simulated and observed distributions was 0.86. In this study, we adopted an additional cyclolysis criterion whereby a cyclolysis event occurs if TC moves into an area where TCs have never arrived in the past. However, observational data are a subset of realistic data obtainable with GSTCMs. Therefore, this cyclolysis criterion is flexible. In such cases, appropriate statistics for TC parameters must be extrapolated from information from neighboring regions.

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Latitudinal distribution of the minimum SLP at the moment of disappearance of TCs.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Comparison of the cyclolysis probability distribution obtained (a) from observation and (b) through simulation.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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GSTCMs simulate the track, minimum SLP, translation direction, and translation speed of TCs. As indicated in the previous section, cluster analysis improves the accuracy with respect to minimum SLP and translation direction. However, the accuracy is reduced in the case of translation speed. The cluster analysis is not always proper method, for example, when the distribution of joint PDF of TC parameter has single peak. This can also be deduced from the fact that there are areas where separation into two clusters is impossible, even with cluster analysis. Another possible reason is the distribution characteristics of clusters of translation speed. For example, the two clusters are distributed parallel to the axis of the rate of change in the case of translation speed, although they are distributed parallel to the axis corresponding to the TC parameters at the previous time step in the case of minimum SLP and translation direction. Whereas in the unimodal model, the PDF of the rate of change shows a single peak in the TC development process, in the bimodal model the PDF of the rate of change shows two peaks for one value at the previous step. In other words, if each cluster represents the joint PDF of a different development stage, simulation with the bimodal model is likely to produce an incorrect development process in continuous-time calculation. This problem could be resolved in future work by adopting a selection algorithm that selects the joint PDF of one cluster by taking into account the continuity of the time series.

The developed model is useful for gaining knowledge to aid in disaster prevention and estimating the frequency of extreme TCs. Conventional extreme value statistics based on observational data yields considerable uncertainty because of the limited number of samples. However, extreme value analysis in combination with a GSTCM provides a more robust prediction for long-term estimation of extreme TCs. For example, Fig. 26 shows the approximated PDF and histogram of the minimum SLP depth of the TC with maximum strength for the year as prepared from data for TCs that hit Japan (129.5°–146.0°E, 31.0°–45.5°N). Figures 26a and 26b show observational and simulation data, respectively. The solid lines represent approximations based on generalized extreme value PDFs (GEV). The observation results are based on IBTrACS data (covering 59 years), whereas the simulation result is based on a GSTCM, which utilizes synthesized data corresponding to an observation period of 2500 yr. In Fig. 26b, error bars indicate the width of standard deviation calculated from 50 ensembles of 50 yr (a total of 2500 yr). There is little difference in peak location between Figs. 26a and 26b, but the feet of two PDFs are different. Observational data are scarce, and the estimation of appropriate PDFs from these data is prone to error. On the other hand, the GEV PDF is in close agreement with the simulation data. Although the two PDFs are similar, the simulation results show a wider tail of distribution than the observation data. The PDF estimated through simulation is slightly more asymmetric than that obtained from the observation data. The long tail of PDF obtained from the simulation does not automatically mean a low reproducibility of the GSTCM. In fact, if we consider that the uncertainty caused by the scarcity of observational data can be estimated from the error bars in this figure, the PDF of observational data is included in the range of μ ± σ. In other words, this information (long PDF tail) is essential and constitutes one of our research targets.

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Approximated PDF and histogram of the depth of the minimum SLP of TCs with the maximum strength for the year [histograms: (a) observations and (b) average of simulation results; lines: GEV distribution]. The horizontal error bars and filled circles are confidence intervals (μ ± σ) and average of the observation and simulation results, respectively. The vertical error bars in (b) are confidence intervals (μ ± σ) of each histogram bar estimated from 50 time trials.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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Finally, a comparison of extreme values calculated from observational data and GSTCM results based on conventional extreme value analysis is shown in Fig. 27. As we have seen in the previous Fig. 26, we can directly calculate extreme values from GSTCM results. That is, the histogram obtained with a GSTCM is expected to resemble the GEV PDF if the amount of simulation data is sufficient. However, here we attempted to estimate extreme values under the assumption of the GEV PDF. A direct comparison between extreme value analysis of observational data and GSTCM data is central to understanding the significance of the stochastic tropical cyclone model. Figure 27 shows the cumulative distribution function (CDF) of the minimum SLP depth of the TC with the maximum strength for the year calculated from data for TCs hitting Japan (the same as those in Fig. 26). The two lines indicate observation and simulation results, both of which are approximated by GEV. Extreme TCs estimated from observational data are weaker than those estimated through simulation. For example, the probability values for less than 40, 60, and 80 hPa are 0.03, 0.45, and 0.92 by observation and are 0.11, 0.49, and 0.83 by simulation, respectively. Here, the TC 100-yr return period is the value at CDF = 0.99. The 100-yr value estimated from observational data is 925 hPa, whereas the 100-yr value estimated with the GSTCM is 900 hPa.

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CDF of the depth of the minimum SLP of the TC with the maximum strength for the year.

Citation: Journal of Applied Meteorology and Climatology 53, 6; 10.1175/JAMC-D-13-08.1

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The most notable problem with the GSTCM is the limited number of observational data on a global scale. Specifically, quality data on minimum SLP are extremely scarce. Although many TCs have been observed in the northern Indian Ocean (NIO), the quality of data on minimum SLP is considerably lower than that for other basins. 65% of TC tracks in the northern Indian Ocean basin have no records or only one record of minimum SLP. One possible way to give some suggestion to this problem might be the indirect use of results from GCM simulations. The state-of-the-art GCM can be used for simulations spanning several decades with a high resolution of about 20 km, which is sufficient for the accurate calculation of TCs (Kusunoki et al. 2011). If joint PDFs estimated from a GCM can be utilized as observational data, the accuracy of the GSTCM might be improved. However, each GCM result has its own bias for reproducibility of the present climate conditions. Therefore, direct application to GSTCMs may be difficult. Caution must be exercised when applying GCM results to GSTCMs. In addition, long-term variations in the number of cyclogenesis events on a basin scale are disregarded in this study. Taking these variations into account might give climatological meaning to subsequent artificial TC data and might allow us to assess the periodicity of TC-related disasters. GCMs and GSTCMs can be used in parallel to obtain a quantitative evaluation of long-term changes in TC parameters under ideal conditions.

Furthermore, the shortage of simulation data makes it difficult to assess the impact of future changes in TC parameters due to global warming. Probably, a GSTCM would be an effective approach for assessment of damage brought by such changes in TCs, while additional assumptions are necessary. For example, GSTCMs can estimate a rough return period of simulation results of future climate conditions from the point of view of present climate conditions. Ishikawa et al. (2013) used a bogus method to perform impact assessment of the maximum possible TC strength by downscaling a global warming simulation with various tracks. However, that approach cannot provide information about how often maximum-strength TCs might hit a certain region. In this regard, GSTCMs can be used for assessing the frequency of such maximum-strength TCs.

The advantage of PCA is generalization of arbitrary variables. The basic approach is similar to standard methods of calculating the conditional probability density function, but we assume a correlation between a TC parameter and its rate of change as a function of location. PCA gives more general relations if two TC parameters are correlated. Additionally, PCA reveals a spatial transition in the relation between a TC parameter and its rate of change. That is, this model is useful not only for estimating the model coefficients necessary to reproduce the observational data (similarly to an MLR model), but also for obtaining synoptic knowledge about the TC development process through the approximation of joint PDFs. It is important to obtain a reasonable explanation of the physical meaning of the TC development process from the viewpoint of stochastic modeling. For example, we were able to recognize the need for cluster analysis from the joint PDF distribution pattern.

5. Conclusions

The findings of this study can be summarized as follows. We developed a GSTCM as a means for preparing numerous artificial TC data, which cannot be obtained from numerical weather prediction models. The GSTCM is sensitive to the approximations of joint PDFs of TC parameters and temporal correlations. Furthermore, newly introduced detailed approximations of joint PDFs based on cluster analysis improve the reproducibility of TC parameters. Although increasing the number of clusters might improve the approximation accuracy in the future, the examination of this relation is difficult with the current shortage of observational data.

The proposed temporal correlation PDF model is useful for the prediction of extreme TCs. In particular, one of its merits as compared with previous methods (e.g., the Markov chain model; Emanuel et al. 2006) is that it uses principal component analysis for approximation of joint PDFs from limited data. In the future, we may improve this method with additional information (such as numerical simulation data). In fact, intrinsic bias can be seen in the results of GCM, but the physical mechanism and basic relation between a TC parameter and its rate of change would be similar to those of actual phenomena. We expect that important information about joint PDFs can be obtained from analysis using GCM data, and we recognize the strong potential of this stochastic model with PCA and cluster analysis.