1. Introduction
Following the GPI04, subsequent research has introduced a series of enhanced empirical indices that incorporate various environmental variables and their functional forms (Emanuel 2010; Murakami and Wang 2010; Tippett et al. 2011; McGauley and Nolan 2011; Wang and Murakami 2020). Additionally, other investigations have focused on fitting these indices to specific time scales and basins (Bruyère et al. 2012; Wang and Moon 2017; Moon et al. 2018). Both basin-dependent and global genesis indices have their own positive aspects. Basin-dependent indices focus on capturing local characteristics, improving performance within specific regions, while global indices provide a broader view for conducting comprehensive analyses across multiple models on a global scale. The choice of different GPIs should depend on the specific research goals and the level of detail required in the analysis.
In this paper, we focus on two limitations in the existing GPIs. First, existing GPIs have not thoroughly investigated or compared the potential variations in the relationship between TC and their surrounding environments across different basins and time scales. Second, existing GPIs only take into account environmental variables at the grid point where TC genesis occurs, neglecting the influence of nonlocal seed activity, which could greatly affect the effectiveness of GPIs.
It has been widely accepted that all TCs develop from preexisting semiorganized disturbances (Dunn 1951), also known as “seeds,” which can travel in a relatively steady state over distances of 1000–1500 miles (Riehl 1951) before either dissipating or developing into TCs. Recent research has provided insight into TC genesis by focusing separately on seed genesis and the transition from seeds to TCs (Vecchi et al. 2019; Hsieh et al. 2020; Emanuel 2022; Hsieh et al. 2022). Hsieh et al. (2020) expressed the TC genesis rate as the product of seed genesis rate ns and the probability of a seed forming a TC P2. This framework does not model the spatial distribution of TC genesis as accurately as it models the hemispheric-integral TC frequency. We hypothesize that the lower skill in spatial representation is primarily due to the fact that the seed-to-TC processes (as expressed by the parameters ns and P2) occur at different locations from the TC genesis point and are thus controlled by conditions along the full seed-to-TC trajectories.
To address the first limitation, we apply forward sequential feature selection (SFS) described in Ditchek et al. (2016) to select important environmental predictors objectively from 20 candidate variables on different time scales (seasonal and interannual) and in individual basins [the eastern North Pacific (ENP) and the North Atlantic (NA)]. Based on the Poisson regression methodology described in Tippett et al. (2011), we then establish the mathematical relationship between selected variables and observed distributions of TC genesis points to construct “local-GPIs.” This approach contrasts with the more subjective derivation employed in the construction of existing GPIs (Emanuel and Nolan 2004; Emanuel 2010). By comparing the accuracy of local-GPIs and GPI04, we can assess how much the performance accuracy of empirical indices can be improved by constructing them based on specific basins and time scales using an objective variable selection method. This should reveal the portion of GPI04’s bias that is due to the first limitation. Moreover, by comparing the GPI04 predictors to those objectively selected local-GPIs based on different basins and time scales, we can answer the following questions: 1) Within a single basin, are the major large-scale factors affecting seasonal and interannual variability in TC genesis the same? 2) Across different basins, how does the relationship between the large-scale environment and TC genesis differ? We are not aware of any studies that quantitatively answer these questions.
To tackle the second limitation, the nonlocal environmental conditions affecting seed activity along the seed-to-TC trajectory will be added to the index. In the absence of a complete theory of seed genesis, we need to identify which environmental factors are most important for controlling or defining seed genesis activity. Thus, in the first step, we construct seed genesis indices (seed-GPIs) using the same methods that were used in building local-GPIs for TC genesis. From the objective variable selection results, we can learn what environmental factors are the most important in representing seed activity and whether the key environmental factors vary across basins and time scales. In the next step, we develop trajectory-based GPIs (traj-GPIs) linking the TC genesis likelihood to environmental conditions along the seed-to-TC trajectory. This is achieved by weighting the nonlocal environmental conditions in adjacent grids according to the trajectory density of the seeds, merging them into the local environment of the TC genesis grid, and then applying the same processes of variable selection and regression as used in local-GPIs. Metrics connected with the variable selection will show how the environmental variables representing seed activity improve the index. This will also be done by directly comparing the performance of local-GPIs and traj-GPIs. The observations to be used are monthly averages of environmental conditions and genesis. When used as operational diagnosis, the indices would be valid as long as the environment is stationary.
The structure of this paper is as follows. In section 2, we outline the methodology and data used in developing local genesis indices for both TC genesis (local-GPIs) and seed genesis (seed-GPIs). The results for both indices in the ENP and NA are presented in sections 3 and 4, highlighting the most significant local environmental variables influencing TC and seed genesis. In section 5, we introduce the method to incorporate nonlocal environmental conditions when developing the traj-GPIs. We also describe the processes of building and applying the filter kernels and discuss the important nonlocal environments affecting the TC genesis selected in traj-GPIs. The paper concludes with a summary and discussion in section 6 of how this work advances our understanding of TC genesis and mentions future work.
2. Data and methods
Here, we conduct a systematic study of indices that model storm distributions considering only local environmental factors. For both TC genesis and seed genesis, we will examine their statistical relationship with local environmental conditions in the ENP and the NA. The statistical models are trained for two outcomes: 1) reproducing the seasonal variation of monthly mean storm frequency (seasonal-cycle GPIs) and 2) capturing the interannual variations of storm frequency (interannual GPIs). Thus, eight GPIs are generated, as detailed in Table 1.
Observed genesis counts for TCs and seeds (as defined in the text) and the sample sizes of various subsets based on different basins and time scales.
a. Tropical cyclone and seeds genesis points
TC observations are extracted from NOAA’s International Best Track Archive for Climate Stewardship (IBTrACS) data (Knapp et al. 2010, 2018) version 4 release 0 (v04r00). We restrict our analysis to the satellite era and only consider TC data from 1980 to 2018. The TC genesis locations are defined as the first positions of those storms that reached tropical storm intensity (34 kt or 17.5 m s−1). A resolution of 2.5° × 2.5° is used for analyzing the spatial distribution of TCs. In constructing the seasonal-cycle GPIs, the mean seasonal cycle of TC genesis frequency is calculated by summing monthly frequency over 39 years for each of the 12 months of the year. For the interannual GPIs, the interannual variability of TC genesis frequency is restricted to the TC season from June to October (JJASO) for each year of the 39-yr period, which accounts for over 93% of all TC genesis events and over 90% of all hurricane [64 kt (1 kt ≈ 0.51 m s−1) and higher] genesis events for both basins.
Genesis points of seeds were obtained from the global Tropical Cloud Cluster (TCC) dataset by Hennon et al. (2011), hereinafter referred to as the TCC dataset. To our knowledge, this is the first publicly available long-term dataset of TCCs, enabling the studies of seed activity over longer time scales (Teng et al. 2014; Zhao et al. 2019; Teng et al. 2019). TCCs were objectively tracked in Gridded Satellite (GridSat) infrared (IR) data with 8-km spatial resolution (Knapp et al. 2011) based on the characteristics of atmospheric convection, including intensity, size, shape, and persistence. The resolution for seeds is also reduced to 2.5° × 2.5° in our analysis. The TCCs labeled as “developing” are those formed into TCs (34 kt and higher), identified by comparing their locations with the TC genesis locations recorded in the IBTrACS dataset. The raw data are available in 3-hourly time increments and cover global TCC activity only from 1982 to 2018, resulting in the sample size of seeds being 2 years smaller than that of TCs. The seasonal cycle and interannual variability of seed are calculated in the same way as for TC. A full description of the identification and tracking algorithms can be found in Hennon et al. (2011).
b. Environmental parameters
We create a comprehensive pool of 20 environmental variables and select the most relevant ones based on an objective method, as opposed to prescribing which variables enter the index (as in GPI04). The candidate variables (Table 2) are chosen based on dynamic and thermodynamic parameters that have been associated with TC genesis in existing GPI studies (listed in the last column in Table 2), including all parameters in GPI04 (Emanuel and Nolan 2004), listed as “GPI04.” The candidates can be grouped into five categories: 1) vorticity, 2) humidity, 3) heating, 4) upward motion, and 5) shear (a detailed explanation of some special candidate variables is given in appendix A). All of the environmental variables used in the present study are obtained from the fifth generation of European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (ERA5; Copernicus 2017; Hersbach and Dee 2016; Hersbach et al. 2020). The base data are in hourly intervals and with a spatial resolution of 31 km (0.25° × 0.25°), which is coarsened to 2.5° × 2.5° following Tippett et al. (2011), a size small enough to resolve the spatial pattern of genesis, while large enough to contain sufficient storms in each grid. These data are then averaged over the period June–October (JJASO) to represent the interannual variability for constructing the interannual GPIs, while monthly averages are used for the analysis of the seasonal cycle in constructing the seasonal-cycle GPIs. All candidate variables are normalized by their individual means and standard deviations.
Candidate variables for the genesis potential index.
c. Regression methods
We use the Poisson regression methodology described by Tippett et al. (2011) to estimate the statistical relationship between storm genesis and the environmental variables. Poisson regression is a generalized linear model typically used for the modeling count data, such as the number of cyclones in a given region and time period (Solow and Nicholls 1990; Elsner and Schmertmann 1993; McDonnell and Holbrook 2004; Tippett et al. 2011; Ditchek et al. 2016; Zhang et al. 2018). The idea is to represent the expected value of the Poisson distribution as a function of a set of predictors. The Poisson distribution restricts the possible outcomes to nonnegative integers, as required for TC genesis frequency (Elsner and Schmertmann 1993).
The regression process is repeated for each of the scenarios in Table 1. Each scenario has a different sample size (the fifth column in Table 1). For example, the seasonal-cycle seed-GPI (Fig. 1, top left) and interannual seed-GPI (Fig. 1, bottom left) both have 420 spatial grids in the ENP but have different time-dimension sizes of 12 months and 37 years, corresponding to 5040 and 15 540 data points, respectively.
Schematic illustration of ENP seed-GPI construction using Poisson regression. (top left) Seasonal-cycle seed-GPI; (bottom left) interannual seed-GPI.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
Not all candidate variables in Table 2 remain in the statistical model, Eq. (2). Forward SFS objectively chooses the variables to include. In SFS, candidate variables are added to a constant-only model [μi = exp(b0)] sequentially by selecting the variable that minimizes a specific error criterion at each step until the addition of extra variables does not reduce the error criterion significantly. The error criterion is the Akaike information criterion (AIC) (Akaike 1973). AIC represents a trade-off between goodness of fit and simplicity of a model, rewarding increased likelihood across a dataset and penalizing overfitting. A model with lower AIC is considered better. While there is some subjectivity in our original candidate variables list, the SFS process allows us to develop a statistical model that is objective, understandable, and reproducible.
During the regression and variable selection process, tenfold cross validation is used to avoid overfitting and selection bias (Cawley and Talbot 2010). This starts with randomly dividing the data into 10 subsamples of approximately equal size. A single subsample is retained for validation and nine subsamples are used as training data. The cross-validation process is then repeated 10 times, with each of the 10 subsamples used exactly once as the validation data. The variables are selected in each training based on the decrease in AIC calculated using the training set, and the AIC using the testing set of the corresponding model is also calculated for validation. We also calculate the evolution of the deviance D in each training, a goodness-of-fit measure analogous to the sum of squared errors (McCullagh and Nelder 1989), with lower D corresponding to a more accurate model. The data division step is repeated 10 times so that 100 sets of AICs and deviances are calculated for both training and testing sets.
Applying the 10-time tenfold cross validation to the SFS algorithm results in 100 sequences of models with an increasing number of variables. For the seed-GPIs in the ENP as an example, we show the mean cross-validated AIC (Fig. 2, blue right y axis) and deviance D (Fig. 2, red left y axis) calculated by the training set and testing set. The result shows similar trends with an increase in the number of environmental parameters. The AIC and D decrease as more variables are added, until adding more variables after an inflection point does not significantly reduce AIC or D. We only include the objectively selected variables before the abrupt change in slop, namely, the first two (three) variables in the ENP seasonal-cycle seed-GPIs (interannual seed-GPIs), the combination of which should make the model both simple and accurate. The variation in AIC and D in the construction of other GPIs can be seen in appendix C. The corresponding number of variables selected by the SFS algorithm and their corresponding coefficients in all GPIs listed in Table 1 will be identified later when results are presented.
The variation of AIC and deviance as a function of the number of environmental parameters in the construction of the (a),(b) seasonal-cycle seed-GPIs and (c),(d) interannual seed-GPIs in the ENP: 10 iterations of tenfold cross-validated deviance D (red, left y axis), and AIC (blue, right y axis) as a function of the number of environmental predictors calculated from (top) the training sets and (bottom) the testing sets. The mean (dots) and ±1 standard deviation (error bars) are shown. As indicated by the orange dashed lines, both D and AIC decrease little when the number of predictors is increased from two to three (from three to four) in the left (right) panels, supporting the decision to use two (three) environmental predictors in the seasonal-cycle seed-GPIs (interannual seed-GPIs).
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
3. The local indices for TC genesis: local-GPIs
In this section, we show the results of constructing basin- and time-scale-specific GPIs for TC genesis (local-GPIs). These are functions of local environmental variables and thus only address the first limitation mentioned in the introduction.
a. Key environmental variables for TC genesis
By applying the SFS algorithm to the candidate variables (Table 2), the environmental variables that can significantly improve the performance of the local-GPIs as well as the associated regression coefficients (numbers and color shades, Fig. 3a) are identified. The associations between each of the selected variables and the response are statistically significant with p values under 0.01. In the ENP, the first two variables selected in both the seasonal-cycle GPI and the interannual GPI (in the order determined by the SFS) fall into the 1) wind shear and 2) vorticity categories, respectively, while in the NA, the first two selected variables based on the two different time scales belong to the 1) heating and 2) vorticity categories. The result indicates that TC genesis is influenced by similar environmental factors over different time scales. However, the specific environmental factors governing TC genesis differ across different basins. Although the physical mechanism of TC genesis is consistent globally, the environmental variables that influence TC genesis in each basin generate diverse signals driven by distinct atmospheric oscillations. This leads to a discrepancy in the selection of controlling variables for TC genesis in different basins.
Variables selected by the SFS algorithm and their Poisson regression coefficients in different (a) local-GPIs and (b) seed-GPIs based on Eq. (2). In each line, the variables are listed in the order chosen by the SFS procedure; the number and the color shading in each grid represent the magnitude of the coefficient. The resulting GPI is in the form of a log-linear equation
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
The coefficients represent the response of the log of the local-GPI to a one-standard-deviation shift in each variable. They are all positive except those of the wind shear, indicating that in the ENP (the NA), the environment with lower wind shear (more favorable heating condition) and higher vorticity is more likely to generate TCs. Among the four local-GPIs, the clipped vorticity (clipped η; Tippett et al. 2011) has the coefficient with the largest absolute value. However, the most important independent factor in each local-GPI is the first variable chosen by the SFS procedure. The magnitudes of the coefficients are not used as the basis for evaluating this importance because there may be correlation between the selected variables (Davis et al. 2008). Correlations can make the coefficients highly sensitive to the mixture of variables in the index; for example, the inclusion of a new variable may cause the largest coefficient to drop significantly, hiding the most important variable. Moreover, the SFS algorithm is designed such that the order of selection is crucial; the clipped η is selected only after the most impactful variable is already in the model. It supplements the information provided by the first variable and cannot replace its importance or uniqueness. In summary, in the ENP, the most important local environmental variable controlling TC genesis is wind shear, while in the NA, a favorable heating condition is most critical for TC genesis.
b. Diagnostic accuracy comparison: Local-GPI and GPI04
Each GPI formula follows a log-linear format:
1) Spatial variability
For the comparison of spatial accuracy, we consider the time integral value. Because the variables selected by the SFS algorithm are similar for the two different time scales (Fig. 3a), we will only display the interannual local-GPIs (Figs. 4c,f for two basins) to illustrate the spatial patterns.
Spatial distribution of the total number of (a),(d) observed, (b),(e) GPI04-diagnosed, and (c),(f) local-GPI-diagnosed TC genesis points in JJASO in the ENP and the NA. Each 2.5° × 2.5° grid is shaded according to the number of TCs that occur over 39 years in the given 5-month season. The numbers at the top-right corner of (b), (c) (e), and (f) indicate the pattern correlation coefficients between each GPI and observed TC genesis frequency in the corresponding basin. The regions enclosed by dashed lines are explained in the text.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
The spatial distribution diagnosed by GPI04 is shown in Figs. 4b and 4e and the local-GPIs in Figs. 4c and 4f. Overall, the global-based GPI04 does not give an accurate representation of the spatial distribution of TC genesis in the ENP (r = 0.54) or NA (r = 0.53). In the ENP, the GPI04 clearly underestimates the TC genesis density in the main development region (MDR; 110°–100°W, 10°–15°N, blue dashed box in Fig. 4a), while it overestimates the genesis density along the Pacific coast of Mexico (purple dashed boxes in Fig. 4b) and the intertropical convergence zone (ITCZ) in the central Pacific (west of 140°W, red dashed boxes in Fig. 4b), mainly due to the high potential intensity Vpi in GPI04 in this region (not shown). In the NA, TC genesis points are diagnosed to be concentrated along the ITCZ (5°–12°N, red dashed boxes in Fig. 4e) by GPI04, but the observed TC density maximum is poleward of the ITCZ in the MDR (9°–16°N, blue dashed box in Fig. 4d) (van Hengstum et al. 2016). GPI04 also overestimates the TC genesis densities in the Caribbean Sea (purple dashed box in Fig. 4e).
Because of the objectivity of variable selection and the flexibility of basin-specific training, we might expect some improvements in the performance of local-GPI (Figs. 4c,f) over GPI04. However, using local-GPIs, we find only a slight improvement in spatial distributions in the ENP (r increases from 0.54 to 0.58) and in the NA (r increases from 0.53 to 0.59). There is still the issue of overestimating TC genesis off the Pacific coast of Mexico (red dashed box in Fig. 4c) and in the Caribbean Sea (purple dashed box in Fig. 4f). The fit is improved primarily by the use of clipped absolute vorticity (clipped η) in the local-GPIs, which results in more realistic spatial distributions with fewer TC genesis events near the equator (south of 10°N). Indices based on clipped η respond more strongly to near-equatorial latitudinal gradients in absolute vorticity η without producing excessive TC genesis events at high latitudes (Tippett et al. 2011).
2) Temporal variability
Compared with GPI04 (Figs. 5c and 6c), the basin-specific local-GPIs (Figs. 5d and 6d) do not significantly enhance the performance of the TC seasonal cycle. Because GPI04 and seasonal local-GPIs are all designed to capture the seasonal cycle of TC genesis, they all exhibit high accuracy in reproducing the seasonal TC variability in the ENP and NA, with r values exceeding 0.96. To understand the seasonal cycles of GPI04 and local-GPIs, we calculate the partial contribution of each variable to the basin-integrated seasonal cycle of the index. Specifically, we take the seasonal-cycle values of only one parameter in the index with other parameters set to their climatology and then normalize this seasonal cycle using the observed total TC number and calculate the anomalous value. The slight improvement observed in the local-GPIs primarily arises from the incorporation of clipped absolute vorticity (clipped η). Clipped η is in good agreement with the seasonal cycle of TC genesis (r = 0.90 for the ENP and 0.91 for the NA); however, there is no clear correlation between seasonal time series and the absolute vorticity η in GPI04. The clipped absolute vorticity, by definition, remains almost constant north of 16°, indicating that the seasonal cycle of TC genesis is predominantly influenced by vorticity near the equator rather than vorticity at higher latitudes.
Partial contributions of each individual environmental variable in (a) GPI04 and (b) interannual local-GPI to the interannual variability of TC genesis number in the ENP by year, and in (c) GPI04 and (d) seasonal-cycle local-GPI to the seasonal cycle of TC genesis number in the ENP by month. Partial contributions are calculated by taking the interannual values (seasonal-cycle values) of only one parameter in GPI with all other parameters fixed to their climatology, summing over the space, and forming the anomaly over the 39 years (12 months). The number at the top-right corner of each panel indicates the coefficient between each GPI and observed TC genesis frequency over the ENP.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
As in Fig. 5, but for the North Atlantic basin.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
Because GPI04 is not specifically designed to simulate interannual variability, its ability to replicate such variability is limited (Figs. 5a and 6a). There is a clear difference in the interannual accuracy based on GPI04—and therefore the large-scale control of TC genesis—between the ENP (r = 0.56) and the NA (r = 0.76). The absolute vorticity term (|105η|3/2) and midlevel relative humidity [(r700/50)3] term are almost uncorrelated with the interannual genesis variation in the ENP (r values are close to 0) but have a statistically significant correlation with that in the NA. This again highlights the interbasin differences in the relationship between environmental conditions and TC genesis which are not considered by GPI04. However, despite being specifically designed to replicate interannual variability, the interannual local-GPIs do not yield a substantial improvement in reproducing such variability when compared to GPI04, with r values even dropping slightly (Figs. 5b and 6b). Recall that the parameters chosen by the SFS method are the best trade-offs between spatial distribution and interannual variation, which may sacrifice the accuracy of interannual performance for better spatial matching. It is evident from the above results that the relationship between the large-scale environments and the interannual variation of TC genesis is not as tight as it is for its seasonal cycle. The seasonal cycle of environmental conditions exhibits a marked shift from relatively dry, cold, and high wind shear wintertime conditions to wet, warm, and low wind shear summertime conditions in the main TC genesis area (Gray 1979). In contrast, the interannual variability of JJASO-mean environmental conditions is much smaller and exhibits a more complex spatial pattern. The relatively flat interannual variability is easily affected by small processes and errors in reanalysis and thus is not sufficient to accurately describe the TC variation. Also, previous studies suggest that the likelihood of TC genesis does not increase without bound with more favorable environmental conditions. Genesis may depend on whether a certain environmental factor reaches a required threshold (Tippett et al. 2011), beyond which this environmental condition will no longer be decisive in TC genesis. During a single year, environmental conditions fluctuate within and beyond the thresholds required for TC genesis, allowing good representation of the seasonal cycle. It is possible, however, that the JJASO means of environmental variables exceed their thresholds and no longer contribute to the probability of TC genesis on the interannual time scale.
In summary, local vertical wind shear and local heating conditions are the most important factors controlling TC genesis in the ENP and NA, respectively. To some extent, local-GPI solves the first limitation of GPI04, and it is the best index we can build based with the SFS method and Poisson regression if only local environmental variables are used. However, the basin- and time-scale-specific local-GPIs are not able to give a significant improvement in diagnosing the spatiotemporal distribution of TC genesis. Therefore, most of the interannual bias in GPI04 is likely due to the limitation of not including nonlocal factors in the TC genesis index. As discussed in the introduction, seed activity is one of the most important nonlocal factors influencing TC genesis. We address this systematically in section 5 but begin by developing GPIs for seeds (seed-GPIs) in the next section.
4. The local indices for seed genesis: seed-GPIs
Since seeds may be frequent enough to contaminate the climate mean state, environmental variables selected in seed-GPIs may be directly related to the definition of seed itself. Caution needs to be taken in interpreting the statistical association between seed genesis and time-mean environments.
a. Key environmental variables for seed genesis
According to the SFS algorithm, the most significant variables selected in all seed-GPI are associated with upward motion, regardless of the basin or time scale. However, the primary factors influencing TC genesis differ across various basins. This disparity arises because the seeds can be frequent enough to contaminate the climate mean state, so the environmental variables selected in seed-GPIs may be directly linked to the definition of seeds themselves. The spatial distribution of seeds aligns with the pattern of large-scale upward motion, which corresponds to the globally unified definition of convective seeds in the TCC dataset. This result also agrees with Hsieh et al. (2020), who incorporated middle troposphere pressure velocity ω in the expression for the initial stage of seed genesis.
b. Diagnostic accuracy of seed-GPIs
The seasonal cycle of the seed genesis in each basin is well reproduced by the seasonal-cycle seed-GPI, with r values of 0.97 for the ENP and 0.98 for the NA. The seasonal seed variation is generally consistent with that of upward motion, which even reproduces the double peak of seed genesis number in the NA (Fig. 7d, purple line), although the first peak is diagnosed 1 month later in May.
As in Fig. 5, but for partial contribution of each individual environmental variable in the interannual seed-GPI to the overall interannual variability of the seed genesis number in the (a) eastern North Pacific and (b) the North Atlantic by year, and in the seasonal-cycle seed-GPI to the overall seasonal cycle of the seed genesis number in the (c) eastern North Pacific and (d) the North Atlantic by month.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
The accuracy of interannual representation for seed genesis is lower compared to seasonal-cycle diagnosis, as the environmental conditions during summertime tend to remain relatively stable from year to year, in contrast to the larger changes in monthly mean environments between winter and summer. When compared to local-GPIs (Figs. 5b,d and 6b,d), which are used for interannual TC diagnosis, seed-GPIs better represent the interannual variability of seed frequency (Figs. 7a,b), with correlations of r = 0.82 in the ENP and 0.77 in the NA.
Time integrals of the seed-GPIs closely reproduce the pattern and magnitude of the observed seed genesis (Fig. 8, r = 0.94 for the ENP and 0.89 for the NA). The seed-GPI for the ENP has a maximum along the ITCZ (Fig. 8b), as in the observations (Fig. 8a). The seed-GPI is also able to reproduce detailed features in the NA (Fig. 8c), such as the maxima near northern Africa and over the Caribbean Sea (Fig. 8d). Overall, the seed-GPIs successfully describe seed genesis frequency based on just 2–4 selected variables (Fig. 3b).
As in Fig. 4, but for the spatial distribution of the total number of (a),(c) observed and (b),(d) interannual seed-GPI-diagnosed seed genesis points in JJASO in the ENP and the NA.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
In conclusion, it is possible to accurately reproduce the spatiotemporal distribution of seed genesis using only local environmental conditions. The frequency of seed genesis in both basins is closely related to local upward motion, a result that fits the definition and theoretical understanding of a seed.
5. The trajectory-based genesis potential index for TC: traj-GPI
The remainder of this work aims to construct GPIs of TC genesis that contain nonlocal information about seed activity and environmental conditions on seed-to-TC trajectories. The method is described first, followed by details of the regression and finally the performance.
a. Method of constructing spatial filtering kernels
In addition to environmental variables on the grid where TC genesis occurs, variables from multiple adjacent grids are needed in the traj-GPI index. To prepare the variable selection and regression process, we combine values of each of the 20 candidate environmental variables at TC genesis grid points with those at neighboring grid points by applying spatial filter kernels. Figure 9 is a flowchart of the construction of the ENP filter kernels (see appendix B for a similar flowchart in the NA). Each step is discussed in detail below.
Flowchart of spatial filter kernels construction in the eastern North Pacific. (a) Seed genesis locations observed in the TCC dataset from 1982 to 2018 relative to their corresponding future TC genesis locations; (b)–(d) density maps of the displacement vectors (Δlon, Δlat) belonging to the three clusters in (a), respectively. Note that (b)–(d) show all points recorded along seed trajectories to the TC genesis point; (e)–(g) normalized center 5 × 7 grids of (b)–(d) will be used as spatial filter kernels for the three clusters in the ENP. In (a), one-dimensional distributions of the zonally and meridionally integrated seed count are shown at the right and top of the scatterplot.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
The filter kernel is a small matrix of weights for integrating the environmental fields. The resolution of the matrix is 2.5°. The sum of the elements is unity. We construct the kernels based on the scatter of displacement vectors (Δlon, Δlat) of every seed position relative to their TC genesis points recorded in the observation from 1982 to 2018 (Fig. 9a). The resolution of the scatter is 0.25°. The physical principle is that, if seeds go through a series of grids before developing into TCs in the center grid, the environments in those grids should influence the future TC genesis. The grids that are passed through by more seeds should be weighted more heavily in the filter kernel, as they may be on the preferred paths of the seeds to reach the center grid.
Seeds can move in different directions within the same basin depending on the direction of mean low-level steering winds. It is therefore helpful to identify several subbasins where seeds show a similar pattern of movement and construct separate filter kernels for each subbasin. We perform k-means clustering to identify different groupings of seeds according to a set of chosen features related to the seed-to-TC trajectory. Specifically, we consider two key features: 1) The relative locations, denoted as (Δlon, Δlat), which represent the displacement of seed genesis points in relation to the corresponding TC genesis points. These relative locations are depicted by the dots in Fig. 9a; 2) the absolute locations of seed genesis points (illustrated by the dots in Fig. 11). The two statistics are normalized. If groups of seeds exhibit the two features in significantly different ways, a robust separation into distinct “clusters” is feasible. The first feature categorizes seeds according to their moving direction before becoming TCs, and the second feature ensures that seeds in the same cluster are geographically close to each other.
The number of clusters K is a free parameter that is chosen a priori. An optimal value of K is determined using AIC and Bayesian information criteria (BIC; Akaike 1973). Similar to AIC, BIC measures the quality of a statistical model, with a lower value representing a better model. The AIC and BIC change as a function of the number of clusters (Fig. 10); K = 3 (K = 2), indicated by the orange dashed line, is used for the remaining analysis in the ENP (in the NA). Beyond this number, the decrease in AIC becomes notably smaller and the BIC begins to increase, suggesting that little information is gained by further increasing K. We divided the entire ENP basin (NA basin) into K = 3 (K = 2) subbasins based on where the different clusters of seeds are gathered (different subbasins are framed by rectangles with different colors in Fig. 11). Within each subbasin, seeds follow a similar path in terms of distance and direction before developing into TCs. Attempts were made to adjust the boundary by 2.5° and 5°. However, these adjustments had minimal impact on variable selection in the traj-GPI, with only slight changes observed in deviance and AIC values. Consequently, the boundary location leading to the lowest deviance value was chosen.
Akaike information criterion (AIC) and Bayesian information criterion (BIC) asymptotes in (a) the eastern North Pacific and (b) the North Atlantic; K = 3 (K = 2) is chosen for the analysis in the ENP (the NA). Error bars of 2σ capturing the stochastic seed are shown.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
Schematic diagram of applying filter kernels to 20 candidate variables in (a) the eastern North Pacific and (b) the North Atlantic. Different kernels are convolved with environmental variables in different subbasins framed by rectangles. The dots in different colors show the genesis locations of seeds in different clusters. The vectors show the summertime-mean low-level wind (averaged between 1000 and 850 hPa), and the color shading shows the magnitude of meridional wind (blue: northerly wind; red: southerly wind).
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
This clustering analysis is conducted using data spanning from 1982 to 2018. Figure 11a illustrates the relative seed locations (Δlon, Δlat) in various clusters, represented by different colors. Additionally, the curves on the top and right depict the univariate kernel density estimation (KDE) over longitude and latitude, respectively. In the ENP, the three clusters are 1) the east equatorward cluster (purple dots), 2) the east poleward cluster (blue dots), and 3) the west cluster (black dots). The relative seed locations (Δlon, Δlat) in the east equatorward cluster show a north-sector preference (purple dots in Fig. 9a), hence equatorward seed movement, which can be explained by the equatorward low-level steering wind in subbasin 1 (the blue shade in subbasin 1 in Fig. 11a indicates the equatorward low-level wind). The displacements (Δlon, Δlat) in the east poleward cluster (blue dots in Fig. 9a) show a south-sector preference, hence poleward seed trajectory, which is driven by poleward low-level winds in subbasin 2 (the red shade in subbasin 2 in Fig. 11a indicates the poleward low-level winds). There is relatively little displacement in the west cluster (black dots in Fig. 9a) since seeds in subbasin 3 are mainly located where the northerly and southerly low-level winds converge (subbasin 3 in Fig. 11a). Interestingly, the seed displacements (Δlon, Δlat) for each of the three clusters closely approximate a two-dimensional Gaussian distribution, respectively. We crop the central 5 × 7 grid (12.5° × 17.5°) latitude–longitude matrices in Figs. 9b–d where most seeds are included, and normalize each cropped map so that the sum equals unity. We thereby arrive at the final format of filter kernels (Figs. 9e–g). In the NA, a similar method is applied to construct two filter kernels for 1) the west cluster and 2) the east cluster (see appendix B for results in the NA).
The k-means cluster analysis was also applied to three different 20-yr periods: 1982–2001, 1986–2005, and 1991–2010. Remarkably, consistent clustering patterns and relative locations were observed across these periods (not shown). This suggests a stable nature of low-level steering winds throughout these time frames, indicating their potential use as a reliable proxy for the seed-to-TC trajectory.
To incorporate the nonlocal environment into each grid, we convolve the filter kernel with all candidate environmental fields in the corresponding subbasin (Fig. 11). Convolution produces the “filtered environmental fields,” i.e., the weighted averages of the environments over the central grid (where TC genesis occurs) and adjacent grids. The value of a filtered environmental variable contains information from the neighboring grids according to the prevailing path of the seeds and the frequency at which those grids are transited by the seeds.
b. Key nonlocal environmental variables selected in traj-GPIs
The construction of nonlocal GPIs for TC genesis is based on the same variable selection and regression methods as in the local-GPIs. The only difference is that the 20 environmental fields are prefiltered as just described. The resulting traj-GPIs represent the likelihood of TC genesis based on both local and nonlocal effects.
The resulting traj-GPIs are shown in Fig. 12, and a total of four variables are kept in each basin- and time-scale-specific traj-GPI based on the downtrends of the mean cross-validated D and AIC (Fig. 13). The categories of selected variable combinations vary across basins but do not vary much between the two time scales, a pattern also observed when constructing local-GPIs (section 3).
As in Fig. 3, but for variables selected by the SFS algorithm and their Poisson regression coefficients in traj-GPI.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
As in Fig. 2, but for the variable selection process in the construction of the interannual traj-GPIs in (a),(b) the ENP and (c),(d) the NA: 10 iterations of tenfold cross-validated deviance (red solid lines, left y axis) and AIC (blue solid lines, right y axis) as a function of the number of environmental predictors calculated from (top) the training sets and (bottom) the testing sets. For comparison, dashed lines show the corresponding trends of local-GPIs based on unfiltered environmental fields.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
In the local-GPIs, wind shear (heating condition) has been identified as the most important local environmental variable for TC genesis in the ENP (the NA). This finding is consistent with its continued selection as a significant factor in the nonlocal traj-GPIs, demonstrating that the traj-GPIs effectively capture local information about TC activity. Furthermore, the inclusion of variables related to upward motion in the traj-GPIs (ω700 in the ENP, Precip in the NA), which are absent in the local-GPIs, indicates their ability to capture nonlocal information about seed activity. It is worth noting that although upward motion was not chosen in the local-GPIs, previous studies (Wang and Murakami 2020) have shown its potential to improve the representation of interannual variations in TC genesis frequency. However, its importance in this context is not as pronounced as it is for the nonlocal traj-GPIs. By considering both local and nonlocal upward motion, traj-GPIs are able to enhance the representation of both interannual variations and spatial distribution in the ENP and NA, offering a comprehensive understanding of the factors influencing TC genesis.
The overall performance of the local-GPIs and the traj-GPIs can be compared using AIC, with a smaller value representing a better and simpler model. In the ENP, the AIC values of interannual traj-GPIs are consistently lower than those of interannual local-GPIs as the number of parameters increases (Figs. 13a,b), suggesting that including nonlocal environmental information in the index better represents the spatiotemporal distribution of TC genesis. In the NA, however, AIC values of interannual traj-GPIs are not significantly reduced from the interannual local-GPIs (Figs. 13c,d). We will explain this by examining the spatial distribution and interannual variation of traj-GPIs for both basins, respectively. Similar conclusions apply to the performance of seasonal-cycle GPIs.
c. Diagnostic accuracy comparison: Traj-GPI and local-GPI
Here, we evaluate whether including the nonlocal effects in the index can significantly improve the performance by comparing the performance of traj-GPI and local-GPI. We start with the interannual diagnosis.
By examining the 39-yr integrated values of the interannual traj-GPI, we can see a significant increase in the spatial performance of traj-GPIs over the original local-GPI across the entire ENP (Fig. 14b). The basinwide r value increases from 0.58 to 0.93. This improvement can be attributed to the inclusion of seed activity information. The regions with the highest TC genesis density in the east Pacific MDR are now accurately diagnosed (blue dashed box). As mentioned in section 5, seeds from east part of ENP (subbasins 1 and 2 in Fig. 11a) tend to move toward the east Pacific MDR and form TCs, thus making the MDR the place with the highest TC genesis density. Furthermore, there is a significant reduction in the previously observed overestimation of TC density along the Pacific coast of Mexico (purple dashed box). The limited occurrence of TCs along this coast, despite favorable conditions, can be attributed to that seeds originating from land need to travel a considerable distance over the ocean to acquire the necessary energy for TC genesis.
Spatial distribution of the total number of the observed and interannual traj-GPI-diagnosed TC genesis points in JJASO in (a),(b) the ENP and (c),(d) the NA. Partial contributions of each individual environmental variable in the interannual traj-GPI to the overall interannual variability of the TC genesis number in (e) the ENP and (f) the NA by year. The coefficients between observed TC genesis frequency and each traj-GPI are presented in red, while the coefficients between observations and local-GPI are indicated in black for comparison.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
The traj-GPI also improves the spatial accuracy of TC genesis in the NA, especially the overestimation of TC frequency in the Caribbean Sea by GPI04 and the local-GPI (purple dashed boxes in Fig. 14e). However, the improvement in spatial accuracy of traj-GPI is not as pronounced in the NA as in the ENP, with basinwide r only increasing from 0.59 to 0.61. In the ENP, seeds are heavily concentrated in the narrow ITCZ region (Fig. 8a), reflecting very sharp gradients of the environmental factors controlling seed genesis in and near the ITCZ. There is a large difference between the filtered environmental fields with seed influence and the original fields. However, in the NA, seed genesis and the environmental conditions that control it are more evenly distributed, especially in the western part of the basin (Fig. 8c). Therefore, the seed influence in the filter kernels does not produce significant differences from the local-GPI.
The diagnosis of interannual TC variability using traj-GPIs in both basins is only slightly improved by the inclusion of information on seed activity. Compared to the local-GPIs, the r value of the observed and diagnosed time series increases from 0.52 to 0.59 in the ENP (from 0.74 to 0.83 in the NA). It is important to note that the filter kernels only affect the spatial distributions of large-scale environmental variables. The temporal variations of environmental variables are not affected by the spatial filtering since they are spatially integrated values.
In summary, the combination of seed activity driven mainly by upward motion and maturation controlled mainly by wind shear (ENP) or heating conditions (NA) is now captured. Compared to local-GPIs based solely on local environmental variables, traj-GPIs more accurately model the spatiotemporal patterns of TC genesis, although the improvement varies by basin and time scale.
6. Summary and conclusions
Empirical genesis indices like GPI04 (Emanuel and Nolan 2004) provide a simple way to model how TC genesis likelihood varies over time and space according to the local environmental conditions. In this study, we point to two possible limitations of existing GPIs that can degrade performance.
First, the current GPIs have not conducted a comprehensive exploration or comparison of the possible variations in the TC–environment relationship across different basins and time scales. An objective variable selection method is desirable, especially one that can be trained on individual basins and different time scales. Second, we suggest incorporating the environmental conditions along the entire trajectory from seed to TC in the GPIs to effectively model the probability of TC genesis.
The first limitation can be addressed by constructing time-scale- and basin-specific GPIs for TC genesis (local-GPIs) using the Poisson regression. We focused on two different basins (the eastern North Pacific and the North Atlantic) and two different time scales (the seasonal cycle and the interannual variability) and used the forward sequential feature selection (SFS) algorithm to objectively choose the variables included in each local-GPI from 20 candidates (Table 2). The variables selected for local-GPIs vary by basin but do not change significantly across time scales (Fig. 3a). In the ENP, the most influential environmental variable controlling the TC genesis is wind shear, while in the NA, favorable heating conditions are the most critical. The performance of local-GPIs in diagnosing the spatial TC distribution is only slightly better (Figs. 4c,f) over GPI04 (Figs. 5 and 6), primarily due to the inclusion of clipped vorticity (clipped η) which shifts high values of TC genesis density north of the ITCZ and narrows the spatial distribution latitudinally. However, other spatial mismatches still exist in basin-specific local-GPIs, indicating that TC genesis must be strongly affected by other important factors besides the local environmental conditions.
This led us to explore nonlocal control over the TC genesis. The first goal is to identify which environmental variables control the important nonlocal factor, i.e., the genesis of TC seeds (Vecchi et al. 2019). We start by constructing seed-GPIs using the same approach as for local-GPIs for TC genesis. In contrast to local-GPIs, we have good results for the basinwide spatial distribution (Figs. 8b,d), seasonal cycle (in the NA, Figs. 7c,d), and interannual variation (Figs. 7a,b). Apparently, seed genesis is more directly and to a greater extent controlled by the climatological mean state than TC genesis. Regardless of the basin and time scale, upward motion is the most important variable for seed genesis (Fig. 3b). This is a reasonable result, as upward motion is conducive to deep convection, which clusters to form seeds (Hennon et al. 2011). It is also consistent with the idea in Hsieh et al. (2020) that seed genesis can be concisely represented as a function of midtropospheric pressure velocity ω.
Finally, we proposed a nonlocal GPI for TC genesis based on the climatological mean state along the seed-to-TC trajectory (traj-GPI). To the environmental conditions at the TC genesis grid in the local-GPIs, we add the environmental conditions at multiple adjacent grids weighted by the seed-to-TC trajectory density to form the traj-GPIs. If a neighboring grid is transited by more seeds, the weight of the environmental information there will be greater in the genesis potential at the central grid. We used the same variable selection and regression methods as when building local-GPIs to construct traj-GPIs with weighted averages of the environmental conditions at adjacent grids and the central grid. As with the local-GPIs, the most important local environmental variables to TC genesis, the wind shear in the ENP and the heating condition in the NA, remain selected in the traj-GPIs. What is new compared to the local-GPIs is the addition of variables in the upward motion category that control the seed activity. The combination of seed activity driven mainly by upward motion and seed maturation controlled mainly by wind shear (ENP) or heating conditions (NA) is successfully captured in the traj-GPIs. Compared with local-GPIs based on the original environmental variables, traj-GPIs better explained the spatiotemporal distribution of TC genesis points in both basins. The temporal performance shows less improvement than spatial performance because spatial filter kernels only affect the spatial distribution of environmental fields, but not their temporal variations which are integrated over space.
Several issues deserve further investigation. First, the primary objective for constructing traj-GPIs is to identify crucial local and nonlocal environmental variables for TC genesis, rather than for predictive purposes. During the variable selection process, apart from random sampling, we also explored dividing the data into training and testing sets based on time order. This division allowed us to evaluate whether the established relationships, derived from specific time periods, could be generalized and applied to new time frames. The findings reveal that while the selected variables and their weights in traj-GPI remained stable over time, the predictive capability of the index can vary. This indicates that utilizing traj-GPIs for predictions under climate change scenarios can introduce uncertainties. Additionally, it is important to acknowledge that biases stemming from the constraints of the regression method and the selection of candidate environments are significant factors to consider. However, these aspects are outside the scope of the current study; we strongly believe that further investigation into these issues is essential and should be pursued. Third, constructing spatial kernels based on seed trajectories poses challenges for some coarse resolution GCMs. In such cases, we propose using the direction of the low-level steering wind as a substitute for the seed-to-TC trajectory.
Although our methodology is statistical, the results can be expected to yield dynamical insight into mechanisms for genesis, as with any existing GPI. The proposed approach of identifying important environmental factors and constructing nonlocal GPIs can be applied to diverse reanalysis products, model outputs, and different definitions of “seeds.” It would be interesting to compare results obtained using different seed definitions or datasets, as this can contribute to a better understanding of the robustness and variability of the findings. As a proposed next step, we intend to provide a physical perspective (e.g., the role of the Hadley circulation) linking upward motion (governing seed activity) and vertical wind shear (governing TC activity) to better understand the link of seed activity and TC activity.
Acknowledgments.
We thank Jie Chen, Kun Gao, and Tsung-Lin Hsieh for the discussion and feedback on earlier versions of this manuscript. This report was prepared by Lingwei Meng under Award NA18OAR4320123 from the National Oceanic and Atmospheric Administration, U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the author(s) and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the U.S. Department of Commerce.
Data availability statement.
Reanalysis data used in this study are openly available from the Copernicus Climate Change Service (C3S) Climate Data Store at https://doi.org/10.24381/cds.6860a573 as cited in Hersbach et al. (2020). The results contain modified Copernicus Climate Change Service information 2020. Neither the European Commission nor ECMWF is responsible for any use that may be made of the Copernicus information or data it contains. The tropical cyclone data were downloaded from NOAA’s International Best Track Archive for Climate Stewardship (IBTrACS) data at https://doi.org/10.25921/82ty-9e16 as cited in Knapp et al. (2010, 2018). The tropical cloud cluster data are included in Hennon et al. (2011). The diagnostic routines are available from the first author.
APPENDIX A
Candidate Variables
a. Vpi
b. Clipped η
Tippett et al. (2011) found that the probability of TC genesis does not increase further with absolute vorticity η once it reaches a threshold value (about 3.7 × 10−5 s−1), leading to the construction of “clipped” absolute vorticity (clipped η) which is the quantity minimum (η, 3.7 × 10−5 s−1). The index based on clipped absolute vorticity is more sensitive to near-equatorial latitudinal gradients in absolute vorticity without overestimating TC genesis frequency at high latitudes.
c. Shear
d. Land
APPENDIX B
Flowchart of Spatial Filter Kernels Construction in the North Atlantic
As indicated in Fig. 10b, the entire North Atlantic can be divided into K = 2 subbasins, and in each subbasin, seeds will follow a similar path before forming TCs. The two clusters are 1) the west cluster, the cluster of seeds gathered in the west part of the NA (the subbasin 1, purple dots in Fig. 11b); their relative location shows a little east-sector preference; and the 2) the east cluster, the cluster of seeds gathers in the east (the subbasin 2, black dots in Fig. 11b); their relative location shows a more obvious southeast-sector preference. The construction of the filter kernels for these two clusters in the NA is similar to that in the ENP (as described in section 5a). The spatial filter kernels for the NA are shown in Figs. B1d and B1e.
As in Fig. 9, but for the construction of spatial filter kernels in the North Atlantic. The filter kernels for the two clusters in the NA (west cluster and east cluster) are shown in (d) and (e).
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
APPENDIX C
Variation in AIC and Deviance with the Number of Environmental Parameters
The variation in AIC and D in the construction of other GPIs can be seen in this appendix (Figs. C1–C3).
As in Fig. 2, but for the construction of the local-GPIs: (a),(b) seasonal-cycle local-GPIs and (c),(d) interannual local-GPIs in the ENP and (e),(f) seasonal-cycle local-GPIs and (g),(h) interannual local-GPIs in the NA. The number of variables decided to use in each GPI is indicated by the orange dashed lines and the text in each panel.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
As in Fig. 2, but for the construction of the seed-GPIs: (a),(b) seasonal-cycle seed-GPIs and (c),(d) interannual seed-GPIs in the ENP and (e),(f) seasonal-cycle seed-GPIs and (g),(h) interannual seed-GPIs in the NA.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
As in Fig. 2, but for the construction of the traj-GPIs: (a),(b) seasonal-cycle traj-GPIs and (c),(d) interannual traj-GPIs in the ENP and (e),(f) seasonal-cycle traj-GPIs and (g),(h) interannual traj-GPIs in the NA.
Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-23-0025.1
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