A Poisson Regression Model of Tropical Cyclogenesis for the Australian Southwest Pacific Ocean Region

Transcription

1 440 WEATHER AND FORECASTING VOLUME 19 A Poisson Regression Model of Tropical Cyclogenesis for the Australian Southwest Pacific Ocean Region KATRINA A. MCDONNELL AND NEIL J. HOLBROOK Division of Environmental and Life Sciences, Macquarie University, Sydney, New South Wales, Australia (Manuscript received 3 October 2002, in final form 4 August 2003) ABSTRACT This paper seeks to address some of the limitations in previous statistical forecast models of tropical cyclogenesis through the development of a series of Poisson regression models on a 2 latitude 5 longitude spatial grid and a monthly grid in time. The Gray parameters [low-level relative vorticity, vertical wind shear parameter, ocean thermal energy, (saturated) equivalent potential temperature gradient, and middle-troposphere humidity] were analyzed as potential predictors of tropical cyclogenesis for the Australian southwest Pacific Ocean region. Various predictor lead times of up to 5 months were investigated, with the most significant Poisson regression models being cross validated, and the skill of their hindcasts evaluated. The Poisson regression model incorporating a combination of saturated equivalent potential temperature gradients at various leads was found to be the most skillful in hindcasting the temporal (phase and amplitude) variability of tropical cyclogenesis for the Australian southwest Pacific region, with a correlation coefficient between the observed and cross-validated hindcast time series of 0.54 (significant at the 99% level), and a rootmean-square error 26% better than climatology. Models using the thermal (ocean thermal energy, saturated equivalent potential temperature gradient, and middle-troposphere humidity) and all (thermal plus low-level relative vorticity and vertical wind shear parameter) predictor variables showed the most skill in hindcasting the spatial distribution of cyclogenesis in this region. The model hindcast skill in predicting individual tropical cyclone occurrences and nonoccurrences was also examined. The all-gray parameter Poisson regression model was found to correctly hindcast up to 72.6% of cyclogenesis events and nearly 70% of nonoccurrences in the Australian southwest Pacific region. The model design enabled the investigation of tropical cyclogenesis on subregional/subseasonal scales, with promising model hindcast skill evident. The results presented herein suggest that useful and more detailed forecasts may be possible in the future in addition to those currently provided at the basin-wide and seasonal scales. 1. Introduction Most previous and existing statistical models of tropical cyclogenesis have been limited in their spatial and temporal prediction skill. For example, the models used by Gray (1984) and Gray et al. (1992) for the Atlantic Ocean, Nicholls (1992) for the Australian southwest Pacific Ocean region, and Chan et al. (1998) for the northwest Pacific Ocean regions each provide basinwide seasonal forecasts. They each incorporate largescale predictors without spatial dependence, such as the Southern Oscillation index (SOI), quasi-biennial oscillation (QBO) index, and African rainfall. Nevertheless, it has been well established by the work of Gray (1975, 1979), McBride (1981), Watterson et al. (1995), and others that, on climatological time scales, the spatial and temporal distributions of tropical cyclogenesis are Corresponding author address: Dr. Neil J. Holbrook, Dept. of Physical Geography, Division of Environmental and Life Sciences, Macquarie University, Sydney NSW 2109, Australia. Neil.Holbrook@mq.edu.au well accounted for by the various components of the Gray parameters. This study seeks to investigate previously identified relationships between the Gray physical parameters and tropical cyclogenesis by developing an appropriate statistical (Poisson regression) model, and to explore the potential of the Gray parameters for forecasts at smaller spatial (subbasin) and temporal (subseasonal) grid scales (for both the predictor variables and tropical cyclone occurrences) in the Australian southwest Pacific region. The advantage of the Gray parameters over the more traditional large-scale predictors, such as SOI, basinwide pressure, etc., is that through the point-wise nature of the predictors, they offer the potential for forecasting not only the number of tropical cyclones, but also in what part of the season and in what location they may develop. Gray (1975) identified six physical parameters: (i) low-level relative vorticity, (ii) local or planetary vorticity (Coriolis parameter), (iii) inverse of the vertical shear of the horizontal wind between the lower and upper troposphere, (iv) ocean thermal energy due to 2004 American Meteorological Society

2 APRIL 2004 MCDONNELL AND HOLBROOK 441 temperatures above 26 C to a depth of 60 m, (v) vertical gradient of equivalent potential temperature between the surface and 500 mb (hpa), and (vi) middle-troposphere relative humidity. When combined, it has been shown that these parameters broadly identify the geographical and seasonal distribution of tropical cyclogenesis in each of the major ocean basins. This combination of parameters, known as the seasonal genesis parameter (SGP), has also been used as a diagnostic tool for analyzing general circulation model climatologies (e.g., Ryan et al. 1992; Watterson et al. 1995; Royer et al. 1998). Ward (1995) also examined individual tropical cyclogenesis events in the southwest Pacific and produced a hurricane index derived from sea surface temperature and two of the Gray dynamic parameters, relative vorticity and vertical wind shear. Ward found that when the index was above a certain fixed threshold, most of the disturbances studied developed into tropical cyclones. DeMaria et al. (2001) have developed an experimental real-time daily genesis index for the tropical Atlantic utilizing parameters similar to Gray s. The genesis parameter combines the scaled 5-day running mean vertical shear and two thermodynamic variables, vertical instability and midlevel moisture. Jagger et al. (2002) recently developed a new space time count model for seasonal hurricane prediction in the rth Atlantic using Poisson processes. The hurricane data were divided spatially into grid boxes and annual counts in time, with one of the three main model forms developed including instantaneous local and autoregressive coupling between the grids. Their study ties together cyclogenesis and tracking of hurricane strength tropical cyclones, with their hurricane counts being analyzed to also pass between neighboring boxes. This builds upon earlier research work in the Atlantic Ocean predicting hurricane tracks based on climate conditions in advance of the season (e.g., Lehmiller et al. 1997; Elsner et al. 2000; Jagger et al. 2001). In the southwest Pacific Ocean region, small island states are particularly at risk to potential loss of life and infrastructure due to tropical cyclones, and have a limited capacity to mitigate and adapt to future climate change (McCarthy et al. 2001). Any possible change to the location of cyclogenesis and/or tracks under climate change could have serious consequences on these island states. The Australian southwest Pacific region is considered to be the most difficult to forecast tropical cyclone tracks in, due to the highly variable tracks in this region with up to 35% of tracks being characterized as erratic (Holland 1993; Elsberry 1995). Compared to the Atlantic region, there has been far less research into the development of seasonal tropical cyclone forecast models in the Australian southwest Pacific region. Hence, our understanding of this region is not as well advanced. Being able to better forecast the point of tropical cyclogenesis is an important first step toward our ability to properly track tropical cyclones, which most benefits risk management strategies. The present study involves the development of a series of statistical models of tropical cyclogenesis for the Australian southwest Pacific Ocean region on a 2 latitude 5 longitude spatial grid and monthly grid in time. Links between tropical cyclogenesis and the Gray (1975) parameters, at monthly leads of up to 5 months, are investigated. The statistical forecast models developed in this study use the Poisson regression. The advantage of Poisson regression over linear regression is that it is more applicable for modeling the occurrence of rare, discrete events such as the occurrences of tropical cyclones (Wilks 1995). Solow and Nicholls (1990), for example, successfully constructed a statistical model of the relationship between tropical cyclone frequency in the Australian region and the SOI, using Poisson regression. Elsner and Schmertmann (1993) showed that a nonlinear Poisson model provides a large increase in skill over linear statistical models in their study of seasonal numbers of intense tropical cyclones in the Atlantic. The hindcast skill, and forecasting potential, of a series of Poisson regression models developed in our study are investigated using correlation, root-mean-square errors, spatial distribution maps, skill scores, and cross validation. Based on these analyses, we determine whether the Gray parameters are useful forecast variables on a monthly scale, and whether forecasting on smaller spatial and temporal scales is likely to provide useful additional information for improving the skill of forecasting tropical cyclogenesis in the Australian southwest Pacific region. This paper is structured as follows. The data are described in section 2. The modeling methods and analysis techniques are explained in section 3. Section 4 describes the Poisson regression models. The results are summarized and discussed in section 5. Finally, section 6 provides some concluding remarks. 2. Data a. Tropical cyclone track data Tropical cyclone track data for the Australian southwest Pacific Ocean region (6 20 S, E) were supplied by the Tropical Cyclone Warning Services, Bureau of Meteorology, Australia (J. Gill 1997, personal communication). The dataset includes the date, position (degrees of latitude and longitude), and storm central pressure (hpa) at 6-h intervals. It has been reported that tropical cyclone data in this region are most reliable since 1960 (Holland 1981). For the present study, we develop our model using data between 1960/61 and 1992/93. This provides an expected 33 yr of reliable tropical cyclone observations for the tropical cyclone season, which extends from vember to May in this region.

3 442 WEATHER AND FORECASTING VOLUME 19 Tropical cyclone genesis is identified at the time and location where the wind speed first exceeds 34 kt (17.5 ms 1 ) (e.g., Bureau of Meteorology 1978). For historical Australian region tropical cyclones, only tropical cyclone central pressure readings are available. These pressure readings were first converted to wind speeds using a nonlinear form of the tropical cyclone minimum sea level pressure maximum sustained wind relationship (Atkinson and Holliday 1977) modified for local conditions. The Bureau of Meteorology divided the Australian region into three subregions, namely, Perth ( E), Darwin ( E), and Brisbane ( E). Empirical value tables were used to convert central pressures to maximum wind speeds appropriate to each region (Love and Murphy 1985; J. Gill 1997, personal communication). This is essentially an objective technique for determining the tropical cyclogenesis point, sufficient for the accuracies required by the spatial and temporal constraints of the gridded dataset used in this study. Cyclone genesis occurrence points were binned into monthly cells in time and 2 latitude 5 longitude boxes in space. The tropical cyclone track data contained some errors and missing values. The quality control of these data resulted in the loss of only about 3% of the tropical cyclone observations. Problems included the occasional unavailability of central pressure readings and a single erroneous genesis position reading, which was located in the middle of the Australian continent. It was noted that a small number of tropical cyclones in the dataset had their first wind speed reading (in time) greater than 34 kt as the genesis point. This coincides with either the recording of the cyclone already part way into its life cycle, or an underresolved rapidly intensifying cyclone. If these isolated initial occurrences of cyclonic wind speed were much greater than 34 kt, we considered them to be potentially misleading to use as genesis points. Hence, we chose to remove those tropical cyclones with the first recorded wind speed greater than a threshold value of 45 kt. During the 1960/ /93 seasons, 239 tropical cyclones formed in the Australian southwest Pacific Ocean region. Figure 1a shows the climatological monthly distribution of tropical cyclogenesis occurrences in this region during the months of vember May (tropical cyclone season for the region) over the 33-yr period. The largest number of cyclones occurred during January, with a total of 65 in the 33-yr period, an average of almost 2 during January in any 1 yr. The minimum number of cyclones formed in the region within the tropical cyclone season was seven, occurring in both vember and May, an average of about one cyclone during these months every 5 yr. The mean annual number of tropical cyclones formed in the region during this period was 7.2, with a standard deviation of 2.7. Figure 1b shows the time series of the annual number of tropical cyclone genesis occurrences in the region during the 1960/ /93 period. The FIG. 1. (a) Distribution by month of tropical cyclone genesis events, (b) annual number of tropical cyclone genesis events, and (c) locations of origin of tropical cyclones formed in the Australian southwest Pacific Ocean region (6 20 S, E) during the v May season, 1960/ / /84 season had the maximum number of events with 15, while a minimum of 3 events was shared by both the 1987/88 and 1991/92 seasons. All tropical cyclogenesis locations in the region from 1960/61 to 1992/93 are shown in Fig. 1c. The majority of these occur within the latitude belt from 10 to 20 S, with many forming close to the Australian coast. As demonstrated by McBride and Keenan (1982), genesis is clustered around three locations: off the northwest coast of Australia (approximately 122 E), in the Gulf of Carpentaria (central portion of the region), and in the Coral Sea (eastern portion of the region). b. Upper-ocean thermal data The upper-ocean temperature data used to calculate the available ocean thermal energy were obtained from the Data Support Section of the National Center for Atmospheric Research (NCAR). These data are from W. White s ocean temperature climatology from 1950 to 1993 (information online at ds260.0/). This gridded product was produced by W. White and T. Walker from the Scripps Institution of Oceanography (University of California, San Diego). The data consist of uniformly gridded (2 latitude 5 longitude) temperatures generated from expendable bathythermograph (XBT) data obtained between 1950 and 1993, and which have been gridded at 11 vertical levels from the surface to 400-m depth. Due to the data sparsity of subsurface information

4 APRIL 2004 MCDONNELL AND HOLBROOK 443 collected in the real ocean, the original ocean thermal energy dataset covers only about 75% of the Australian southwest Pacific Ocean region. Since ocean temperature is essentially a continuous variable (both spatially and temporally) that is relatively smoothly varying in time and space at the large scale, linear interpolation was used to fill in the data gaps. c. Atmospheric data The atmospheric data required to calculate four of the five Gray atmospheric parameters were obtained from the National Centers for Environmental Prediction (NCEP) NCAR reanalysis project. The reanalysis data are maintained and distributed by the Data Support Section of NCAR. The resolution of the reanalysis spectral model is T62 (equivalent to a horizontal resolution of about 210 km) with 28 vertical sigma levels (Kalnay et al. 1996). The data used in the present study are the NCEP NCAR reanalysis monthly mean subsets from 1960 to 1993, archived in World Meteorological Organization (WMO) gridded binary (GRIB) format (information online at The coordinate system for these data is a pressure level stack (17 levels) on a 2.5 latitude 2.5 longitude grid. Fields used to calculate the Gray parameters include relative vorticity at 925 mb (low-level relative vorticity), u (eastward) and (northward) components of the wind at 925 and 200 mb (to calculate the vertical wind shear), and relative humidity at 700 and 500 mb (to calculate the middle-troposphere humidity). For computational convenience, the vertical gradient of equivalent potential temperature used by Gray was replaced by the vertical gradient of saturated equivalent potential temperature (calculated using the temperature fields at 1000 and 500 mb), which is a more appropriate measure of the potential for cumulonimbus convection from a lapse rate stability viewpoint. This has the virtue of separating out the temperature lapse rate information in this predictor, and the moisture field information into the midtropospheric relative humidity predictor. The calculated monthly Gray parameters were averaged into 2 latitude 5 longitude boxes. 3. Modeling methods and analysis techniques a. Poisson regression Regression analysis is an effective way to model expected outcomes based on information from several variables (or predictors; e.g., Gray et al. 1992, 1993, 1994; Landsea et al. 1994; Nicholls 1999). However, linearity conditions should be satisfied in order to use linear regression analysis. These assumptions break down where the observed data are a relatively small number of counts from a large sample size. This is the case with tropical cyclones where, for example, in the Australian southwest Pacific region (6 20 S, E), there were only 239 tropical cyclogenesis events between 1960/61 and 1992/93 within a sample size of [33 yr 7 months (tropical cyclone season) 78 spatial ocean grid points] possible outcomes. The Poisson distribution is often used to model the occurrence of rare, discrete events, such as tornado counts, the occurrences of droughts, or cold spells (e.g., Wilks 1995). The Poisson distribution also restricts the possible outcomes to non-negative integers, making it ideal for modeling tropical cyclone occurrences (Elsner and Schmertmann 1993). In a Poisson model, the probability distribution, that is, the probability of occurrence of exactly y tropical cyclones, is given by y exp i i Pr(Yi y) y!, y 0,1,2,...,, (1) where i is 0 j j X exp ij, i (2) and where X ij is the data value for predictor j on observation i and j is the corresponding Poisson regression coefficient for predictor j. It can be shown theoretically that the expected number of tropical cyclones E(Y) Var(Y), when Y has a Poisson distribution (Kleinbaum et al. 1988). The general method of fitting a Poisson regression model is to use the Poisson model formulation to derive a maximum likelihood function. This means that for a given (vector of Poisson regression coefficients), is calculated for each set of predictors, and the likelihood of the observed number of tropical cyclones is estimated [Eq. (1)]. The that is then used for the forecast (or hindcast) is the one that maximizes the product of the probabilities [in Eq. (1)] over all time (Elsner and Schmertmann 1993). The solution to the maximum likelihood equations must generally be obtained through an iterative procedure. The iterative procedure used here is iteratively reweighted least squares (Kleinbaum et al. 1988). A Matlab script (Smyth 2002), employing such a method, was used to generate the Poisson regression models developed in this study. The significance of the regression coefficients was tested using a z score, calculated by dividing the regression coefficient by its standard error. The null hypothesis (H 0 ) is that a regression coefficient i 0. The z scores are test statistics for testing whether the coefficient does equal zero (Christensen 1990). A significance level is chosen, typically 0.05, and the critical value for the z score (z crit ) using a two-tailed test is determined from a table of z scores. For a 0.05 significance level, and in the present case, z crit would be equal to 1.96 (from a z-score table). If the absolute value of the z score for the regression coefficient is greater than or equal to z crit, H 0 can be rejected. In this case, the regression coefficient is not equal to zero, and is significant.

5 444 WEATHER AND FORECASTING VOLUME 19 The deviance, D, can be thought of as a measure of the residual variation about (or deviation from) the fitted model (Kleinbaum et al. 1988), being an overall summary of the difference between the fitted values and observations. The larger the difference between the observed (y i ) and predicted ( ˆ i) values, the larger D will be. The tropical cyclone data are sparse, being essentially binary (0s and 1s), and so the deviance, and hence the Pearson chi square statistic, is uninformative as a goodness-of-fit measure of the model (McCullagh and Nelder 1989). The deviance, however, can be used to compare models, for example, models using low-level relative vorticity with different lead times as predictor variables, where the model with the lowest deviance would be considered to have the best fit. b. Cross validation Cross validation is a resampling technique, where the available data are repeatedly divided into development and verification (prediction) subsets (Wilks 1995). It attempts, with a limited data sample, to simulate actual forecasts and provide an accurate estimate of the true predictive skill of the model or algorithms (Hess and Elsner 1994). Cross validation involves developing a separate prediction rule for each observation (or subset of observations) in the dataset based only on the remaining observations, with the resulting predictions being termed hindcasts (Elsner and Schmertmann 1994). Each hindcast is made with a different set of regression coefficients. If hindcast skill is acceptable, then the entire class of models (in this case, the Poisson regression models) is accepted as a useful forecast algorithm, rather than only relying on a single model (i.e., a single set of coefficients) (Hess et al. 1995). The method used is called K-fold cross validation, with the data split into K roughly equal-sized subsets (Efron and Tibshirani 1993). For the kth subset, the model is developed using the other K-1 data subsets, and then the fitted model is used to predict the kth data subset. The above process is undertaken for k 1, 2,...,K, hence resulting in every data point being treated as independent data (Wilks 1995). Threefold cross validation was used in this study with the dataset divided into three equal subsets. One of the three subsets is used each time as the verification (prediction) set and the other two subsets are put together to form a developmental (training) set. The cross-validation procedure was performed 3 times so that each year of observations fell in the verification subset once (within each year there are 546 separate observations, i.e., 7 months 78 spatial ocean grid points). Since the data quality is better in the later years of the observing period, rather than developing the model over the period , say, and attempting to predict cyclone numbers from 1982 to 1992, any biases due to record length are minimized in the model development procedure by instead hindcasting every third year in the series. Hence, for the first validation, the developmental subset consists of 1961, 1962, 1964, 1965, 1967,..., 1992, and the verification subset is 1960, 1963, 1966,..., For the second validation, the model is developed over the years 1960, 1962, 1963, 1965,...,1992, and the years 1961, 1964,..., 1991 are hindcast. Finally, the developmental subset, 1960, 1961, 1963, 1964,..., 1991, and the verification subset, 1962, 1965, 1968,..., 1992, are used in the third validation. Hence each observation within each year is independently hindcast, simulating an actual forecast. In this way, the maximum likelihood estimate of the regression parameters is independent of the years for which the predictions are required (Elsner and Schmertmann 1993). The cross-validation procedure produces a larger error estimate than that given by an in-sample hindcast, but it is more accurate, as cross validation mimics actual forecast situations (Elsner and Schmertmann 1994). c. Forecast verification In this study, Poisson regression has been undertaken for each Gray parameter as a predictor variable of tropical cyclogenesis for a number of different lead times. The predictor variables are relative vorticity, vertical shear parameter, ocean heat storage, saturated equivalent potential temperature gradient, and relative humidity. The data were standardized by subtracting the overall mean and dividing by the overall standard deviation (in time and space), to provide an equal weighting for all of the variables used as predictors. Two categories of lead times were used. First, leads before the season began, that is, June October, and second, leads ranging from 1 to 5 months prior to the month being hindcast, which takes into account changes in the parameter values throughout the tropical cyclone season. These leads essentially represent an absolute value, or threshold, of the predictor above or below that which is required for tropical cyclogenesis to most likely occur. Lead differences, for example, June October, August October, 5-month lead 1-month lead, etc., were also used as predictors. As an example, a 3 1-month lead would mean that for the prediction of April cyclogenesis, the January March gradient of the predictor (difference between these month values) is used. For May cyclogenesis, the February April gradient is used, and so on. These differences, or gradients, reflect the rate of change of a predictor over time. A rapid (slow) increase (decrease) in a predictor over a number of months may provide a better estimate of the conditions required for tropical cyclogenesis (e.g., Gray et al. 1992; Nicholls 1992). In the present study, the regression includes both spatial and temporal information about the predictor variables in the models: here we include month, cyclone year (i.e., the tropical cyclone season), and latitude and

6 APRIL 2004 MCDONNELL AND HOLBROOK 445 TABLE 1. Poisson regression model predictor variable leads used in the development of models of tropical cyclogenesis for the Australian southwest Pacific region. Poisson regression model predictor variable leads 1-month lead 2-month lead 3-month lead 4-month lead 5-month lead 5 1-month gradient lead 5 3-month gradient lead 3 1-month gradient lead 2 1-month gradient lead 3 2-month gradient lead 4 3-month gradient lead 5 4-month gradient lead Jun lead Jul lead Aug lead Sep lead Oct lead Jun Oct gradient lead Jun Aug gradient lead Aug Oct gradient lead Sep Oct gradient lead Aug Sep gradient lead Jul Aug gradient lead Jun Jul gradient lead longitude (i.e., position in horizontal space) in our analyses. The geographical and seasonal information, on its own, provides a background estimate for the climatological forecast (hindcast) (see the appendix). In some ways, these seasonal forecasts reproduce the work of Gray (1975). Here, however, we use the more appropriate Poisson regression model and also take account of variations in time of the physical parameters. A total of 24 different lead time models were produced in our analyses of the benefits of each predictor variable in Poisson models of tropical cyclogenesis for the Australian southwest Pacific region (see Table 1). For model parameter (predictor variable) significances at 95% level (assessed using the z score), those particular models were singled out for further investigation. Of these, the model with the lowest deviance for each predictor variable was cross validated, and the resulting independent hindcasts were analyzed both temporally and spatially in order to assess the model s ability to predict future events. These hindcasts were further evaluated using analysis techniques including (a) comparisons between the rootmean-square error (rmse) of the hindcasts and the rmse for persistence and climatology forecasts, (b) skill scores, and (c) two-way contingency tables (see the appendix for details). Two remaining methods used to analyze the prediction skill of the Poisson regression models are by examination of the time series (correlation coefficients to indicate the predictability of the phase of the variations) and spatial maps. The tropical cyclones that formed during a cyclone year across the region are determined, giving a total seasonal number for that year and region. The cross-validated hindcasts are integrated in the same way and are compared to the observed seasonal numbers. The correlation between the two time series (hindcast and observed) represents how well the phase is captured by the model, while the rmse compares the amplitudes of the observed and predicted cyclone occurrences. Spatial prediction skill is investigated by comparison of the spatial maps showing the total number of observed tropical cyclones formed in each 2 latitude 5 longitude grid box in the region against the model cross-validated hindcast numbers during the overall time period. The five lead times with the lowest deviance for each predictor variable (parameter), along with the spatial temporal predictors, were combined into a single model and the Poisson regression analysis was again performed. Any parameters showing (i) little significance in combination with each other, (ii) incorrect regression coefficient signs, or (iii) between-predictor correlations greater than 0.9 (collinearity) were removed from the model. The remaining predictor variables (considered here to be useful) were used to produce the final Poisson regression model appropriate to this study. This model was also further cross validated and analyzed in the same way as described previously. 4. Poisson regression models Table 2 summarizes the main results for various predictor model representations showing the lead times used in both simple and combination Poisson regression models for each of the Gray parameters when used as predictors of tropical cyclogenesis for the Australian southwest Pacific Ocean region. The skill of the models in hindcasting the phase and amplitude of the temporal variability of the observed seasonal tropical cyclones is included. As can be seen, the saturated equivalent potential temperature gradient combination, dynamic, thermal, and all-gray parameter Poisson regression models showed significant skill over and above climatology. These four models will now be discussed in further detail. a. Saturated equivalent potential temperature gradient For the Poisson regression models using the saturated equivalent potential temperature gradient (at various leads) as a predictor, 19 out of 24 were significant at the 95% level. The 1-month lead model had the lowest deviance. The temporal relationship between the observed seasonal cyclogenesis occurrences and cross-validated hindcasts from the simple (single predictor) model is poor with a correlation coefficient of 0.12 (insignificant) and an rmse of 2.66, demonstrating that the simple oneparameter model fails to represent the phase variations in the observations. The cross-validated hindcast spatial

7 446 WEATHER AND FORECASTING VOLUME 19 TABLE 2. Summary of the main results from the Poisson regression model analyses using each of the various predictors of tropical cyclone genesis for the Australian southwest Pacific Ocean region. Details include the correlation coefficient between the observed seasonal cyclone numbers and the cross-validated model hindcasts, with the significance levels given in boldface if they are significant at greater than the 99% (98% marked with an asterisk) level and in italics if they are significant at greater than the 90% level; and the rmse of the seasonal cross-validated model hindcasts, and the percent improvement over the climatological rmse of 2.64, if any. Model Correlation coef Rmse Relative vorticity simple model (4-month lead) Relative vorticity combination model (3-, 4-, and 5-month and Jun Aug gradient leads) Shear parameter simple model (1-month lead) Shear parameter combination model (5-month, Jun, 2 1- and 3 1-month gradient leads) Ocean thermal energy simple model (5 1-month gradient lead) Ocean thermal energy combination model (5 1- and 2 1-month leads, and Sep Oct gradient leads) Saturated equivalent potential temperature gradient simple model (1-month lead) Saturated equivalent potential temperature gradient combined model (Sep and 5 1-month gradient leads) Relative humidity simple model (4-month lead) Relative humidity combination model (3-, 4-, and 5-month and Aug leads) Dynamic parameters model Thermal parameters model All-Gray parameters model * 0.49* Improvement over climatology 26% 2% 26% 21% skill is nevertheless reasonable. The observed (spatial) occurrence maximum in the Gulf of Carpentaria is at least partly resolved, while twin maxima off the northwest coast of Australia were also represented by the model (not shown). While these maxima were found to be farther south and smaller in area than observed, their magnitudes are comparable. The five Poisson regression models of tropical cyclogenesis with the lowest deviances incorporate the saturated equivalent potential temperature gradient at leads of 1 month, 2 months, 5 1 month gradient, September, and October as predictors. The October saturated equivalent potential temperature gradient contributed insignificantly when used in combination. Hence, this predictor was removed from the model with little change to the deviance. The 1-month lead predictor was also removed as its regression coefficient became negative (instead of the expected positive; i.e., an increase in the saturated equivalent potential temperature gradient leads to an increase in tropical cyclogenesis). Following this reduction, the 2-month lead predictor became insignificant and was removed from the model. The final combination model incorporates two saturated equivalent potential temperature gradient leads that include September and the 5 1-month gradient as predictors. Using this combination model, the correlation coefficient between the observed seasonal cyclogenesis occurrences and the cross-validated hindcasts is 0.54 (significant at the 99% level). The rmse of 2.27 corresponds to a skill score of 0.26, or a 26% improvement over climatology (i.e., the number of tropical cyclones predicted using only month, latitude, and longitude as predictors in a Poisson regression model). Figure 2a shows quite good phase correspondence between the observed occurrences of cyclogenesis and the model cross-validated hindcasts during a number of seasons, although the sharp peaks are not captured. The spatial skill is improved over the simple saturated equivalent potential temperature gradient model (see Fig. 2c) with a doubling in the amplitude of the maximum in the Gulf of Carpentaria, and a more realistic northward extension of the maximum off the northwest coast of Australia. The Coral Sea maximum is not represented by this model. b. Dynamic parameters Gray s (1975) dynamic parameters include the shear parameter, low-level relative vorticity, and planetary vorticity (Coriolis parameter). Since the Coriolis parameter approximates a simple linear function of latitude only at low latitudes, and since latitude is already included as a variable in each of the Poisson models, the Coriolis parameter was not included further. The relative vorticity and shear parameter leads calculated for their respective combination regression models (see Table 2) are included here in combination to produce a dynamic predictor Poisson regression model of tropical cyclogenesis. The predictor variables in this dynamic combination model are the 2 1- and 3 1-month gradient shear parameters, and June and 5-month lead shear parameters; the 3-, 4-, and 5-month lead relative vorticities, and the June August gradient lead relative vorticity. It was found that, in combination, only the June and 5-month lead shear parameters have z scores significant at the 95% level.

8 APRIL 2004 MCDONNELL AND HOLBROOK 447 FIG. 2. (a) The actual (solid line), cross-validated hindcasts (dashed line), and Poisson-modeled (dotted line) seasonal numbers of Australian southwest Pacific tropical cyclones formed during the 33-yr period 1960/ /93. (b) Total number of tropical cyclones formed in the Australian southwest Pacific region in each 2 latitude 5 longitude grid box, between 1960/61 and 1992/93. (c) Crossvalidated hindcasts of the total number of tropical cyclones formed. Predictors used are a combination of saturated equivalent potential temperatures at leads of Sep and the 5 1-month lead gradient. Month, cyclone year, latitude, and longitude are also included as predictors. While relative vorticity and the shear parameter in the individual Poisson regression models provided very low correlation coefficients between the cross-validated hindcasts and the observed seasonal cyclogenesis numbers when used as predictor variables on their own, when used in combination the correlation coefficient increased slightly to 0.21, although this is still low and insignificant. Some phase agreement can be seen between the dynamic parameter model results and the observations (see Fig. 3a), although the relationship is weak. The rmse of the cross-validated seasonal hindcasts is 2.61, which is 2% better than the climatological rmse (2.64). The modeled maximum in the Gulf of Carpentaria is about half of the observed magnitude (see Fig. 4b), while the maximum off the northwest coast of Australia is not captured. FIG. 3. (a), (b) The actual (solid line), cross-validated hindcast (dashed line), and Poisson modeled (dotted line) seasonal numbers of Australian southwest Pacific tropical cyclones formed during the 33-yr period 1960/ /93. Predictors used are (a) a combination of dynamic parameters (shear parameter and low-level relative vorticity) and (b) a combination of thermal parameters (saturated equivalent potential temperature gradient and ocean thermal energy). Month, cyclone year, latitude, and longitude are also included as predictors. c. Thermal parameters Gray s (1975) thermal parameters include the middletroposphere relative humidity, ocean thermal energy, and in this study, saturated equivalent potential temperature gradient. As with the dynamic parameter analysis, the significant leads calculated for the individual predictor variable models (see Table 2) are again considered here. These variables include the saturated equivalent potential temperature gradient at leads of September, and the 5 1-month gradient; relative humidity at 3-, 4-, and 5-month and August leads; and ocean thermal energy at 5 1-month, 2 1-month, and September October gradient leads. In combination, the relative humidity lead predictors have incorrect regression coefficient signs (i.e., a decrease, instead of the expected increase, in relative humidity leads to an increase in tropical cyclogenesis), and reduced the model skill in hindcasting cyclogenesis. Hence, they were removed from the combination model. Figure 3b shows the relationship between the Poisson regression model cross-validated hindcasts and the observed cyclogenesis occurrences; the correlation coefficient between these two time series being 0.51 (significant at the 98% level). This correlation coefficient is nevertheless slightly lower than the 0.54 recorded for the saturated equivalent potential temperature gradient combination model. The rmse of the cross-validated seasonal hindcasts of tropical cyclogenesis occurrences was 2.28, which is again a 26% improvement over climatology (2.64), and approximately the same as for the saturated equivalent potential temperature combination model (2.27). Hence the amplitude of the phase variations is quite well represented using the thermal parameter model. Two of the three maxima, those in the Gulf of Carpentaria and off the northwest coast of Australia (eastern Indian Ocean), are both reasonably well defined (see Fig. 4c). The Coral Sea structure is again not resolved.

9 448 WEATHER AND FORECASTING VOLUME 19 FIG. 4. (a) Total number of tropical cyclones formed in the Australian southwest Pacific region in each 2 lat 5 lon grid box, between 1960/61 and 1992/93. (b) Cross-validated hindcasts of the total number of tropical cyclones formed using a combination of dynamic parameters (shear parameter and low-level relative vorticity) as predictors. (c) Cross-validated hindcasts of the total number of tropical cyclones formed using a combination of thermal parameters (saturated equivalent potential temperature gradient and ocean thermal energy) as predictors. Month, cyclone year, latitude, and longitude are also included as predictors in (b) and (c). d. All-Gray parameters The combination of all of Gray s (1975) parameters was taken from the parameters and lead times used for the dynamic and thermal predictors in the previous Poisson regression models. Of these, the 3- and 4-month lead relative vorticity, and the 2 1-month gradient ocean thermal energy, were found to be insignificant (z scores lower than 1) and were removed from the set of predictors used in the final Poisson regression model. This dynamic and thermodynamic model of tropical cyclogenesis occurrences in the Australian southwest Pacific region consists of the following predictors: the shear parameter (June, 2 1- and 3 1-month gradients, and 5- month leads); low-level relative vorticity (5-month and June August gradient leads); saturated equivalent potential temperature gradient (September and 5 1-month gradient leads); and ocean thermal energy (5 1-month and September October gradient leads). The correlation coefficient between the observed seasonal cyclogenesis occurrences and the Poisson regression model cross-validated hindcasts is 0.49 (significant at the 98% level). This is, nevertheless, slightly lower than for the simpler thermal parameter Poisson model (0.51). While the rmse for these model results is a 21% improvement over climatology, the rmse is higher than for both the thermal parameter and saturated equivalent FIG. 5. (a) The actual (solid line), cross-validated hindcast (dashed line), and Poisson-modeled (dotted line) seasonal numbers of Australian southwest Pacific tropical cyclones formed during the 33-yr period 1960/ /93. (b) Total number of tropical cyclones formed in the Australian southwest Pacific region in each 2 lat 5 lon grid box, between 1960/61 and 1992/93. (c) Cross-validated hindcasts of the total number of tropical cyclones formed. Predictors used are a combination of dynamic and thermodynamic Gray parameters. Month, cyclone year, latitude, and longitude are also included as predictors. potential temperature gradient combination models. As can be seen in Fig. 5a, there is good visual agreement in the phase variability between the modeled and observed occurrences overall. However, the agreement for , 1974, , and 1991 is poor. Figure 5c shows the spatial distribution of the crossvalidated Poisson regression model hindcasts of tropical cyclones formed during the 33-yr period. The maximum in the Gulf of Carpentaria, and to a lesser extent the maximum off the northwest coast of Australia, are hindcast reasonably well by the model in both structure and amplitude. However, the structure of the maximum in the Coral Sea is again not represented. e. Subregion and subseason forecasts The design of the suite of Poisson regression models developed in this study permits the forecast potential and skill to be examined at various spatial and temporal scales. For each Poisson regression model analysis, a hindcast was produced for every possible observation event in space and time within the Australian southwest Pacific region, that is, within every 2 latitude 5 longitude cell in the region bounded by 6 20 S, E, and for every month in the tropical cyclone sea-

10 APRIL 2004 MCDONNELL AND HOLBROOK 449 FIG. 6. (left) The actual (solid line), cross-validated hindcast (dashed line), and Poisson modeled (dotted line) seasonal numbers of Australian/southwest Pacific tropical cyclones formed in the eastern subregion ( E), during the 33-yr period 1960/ /93. (right) The actual (solid line), cross-validated hindcast (dashed line), and Poisson modeled (dotted line) midseason numbers of Australian southwest Pacific tropical cyclones formed across the entire region, during the 33-yr period 1960/ /93. Predictors used are (top) saturated equivalent potential temperature gradient, (middle) a combination of thermal parameters (saturated equivalent potential temperature gradient and ocean thermal energy), and (bottom) a combination of thermodynamic and dynamic parameters (low-level relative vorticity, shear parameter, saturated equivalent potential temperature gradient, and ocean thermal energy). Independent tropical cyclone observations (exes, ) between 1993/94 and 1995/96 are also forecast (open circles, ) for the model using the combination saturated equivalent potential temperature gradient predictors. Month, cyclone year, latitude, and longitude are also included as predictors. son (vember May) from 1960/61 to 1992/93 (a total of possible observation events). Hence, the cross-validated hindcasts generated by each model can be summed across different subregions and subseasons, and evaluated against the corresponding observations, to determine the potential Poisson regression model hindcast skill at the finer scales. Independent years from 1993/94 to 1995/96 are also forecast using the saturated equivalent potential temperature gradient combination model. Here, our investigation is extended by further dividing the Australian southwest Pacific region into three subregions (based on the Bureau of Meteorology forecast regions): the western ( E), northern ( E), and eastern ( E) regions. The tropical cyclone season is also divided further here into three subseasons: early (vember December), middle (January March), and late (April May) tropical cyclone season. Correlation coefficients and rmses were again calculated and used to evaluate the Poisson regression model cross-validated hindcasts for each subregion against the observed numbers of tropical cyclones formed for the (i) total seasonal number of tropical cyclones formed in each of the three subregions, (ii) early, middle, and late tropical cyclone season numbers over the entire region, (iii) early season numbers in each of the three subregions, (iv) midseason numbers in each of the three subregions, and (v) late season numbers in each of the three subregions. Figure 6 shows the phase and amplitude relationships between the observed and cross-validated Poisson regression model hindcast tropical cyclogenesis occur-

11 450 WEATHER AND FORECASTING VOLUME 19 TABLE 3. Statistical summary of the Poisson regression model hindcasts for the defined seasons regions. For example, entire eastern identifies the seasonal number of tropical cyclones in the eastern region, while mid entire refers to the midseason tropical cyclone numbers over the entire region. The correlation coefficient and the rmse between the observed and cross-validated hindcasts are shown, together with the percentage improvement over climatology. Values of the correlation coefficient are given in boldface if they are significant at greater than the 99% (98% marked with an asterisk) level, and in italics if they are significant at greater than the 95% level. Season region Correlation coef Rmse Combination EPT Entire western Entire northern Entire eastern Early entire Mid entire Late entire Thermal Gray Entire western Entire northern Entire eastern Early entire Mid entire Late entire All Gray Entire western Entire northern Entire eastern Early entire Mid entire Late entire Late eastern * Improvement over climatology 11% 19% 9% 24% 47% 4% 26% 8% 29% 55% 23% 7% 25% 61% 29% rences for the eastern subregion and for midseason using combination saturated equivalent potential temperature gradient, thermal Gray, and all-gray predictors. In all six combinations of season and region the phase and amplitude are reasonably well captured overall, with the hindcasts matching the amplitude of several of the observed peaks. While the hindcasts for midseason and the entire region are out of phase with observed occurrences in the early 1970s, the hindcasts for the entire season eastern region are in phase with observed for the same time period. The hindcasts are back in phase with observed for the second half of the 1970s and the 1980s. The three models produce reasonable hindcasts across the 33 yr at these subregion and subseason scales. Due to the unavailability of upper-ocean thermal data after 1993, the three extratropical cyclone seasons from 1993/94 to 1995/96 can only be forecast using the combination saturated equivalent potential temperature gradient Poisson regression model. The 1993/94 season was forecast using the model coefficients already developed. The 1994/95 season was then forecast with the model created using the 1960/ /94 data, and similarly the 1960/ /95 data were used to forecast the 1995/96 season. The observed and forecast tropical cyclone numbers for the midseason entire region and entire season eastern region combinations during these three seasons are shown in the top panels of Fig. 6. For the entire season eastern region, the model does a good job, forecasting the correct number of tropical cyclones for each year, except for 1995, where two were observed but three were predicted. Two out of the three years are also predicted reasonably in the midseason entire region pair, the exception being the 1993/94 season, where only just over half the tropical cyclones observed were predicted. From our analyses, we found that the Poisson regression models using either a combination saturated equivalent potential temperature gradient, or thermodynamic (thermal Gray) or dynamic plus thermodynamic (all Gray) predictors are able to hindcast tropical cyclogenesis in the afore-described subseasons subregions with the most consistent and largest improvements over the climatology (i.e., the number of tropical cyclones predicted using only month, latitude, and longitude as predictors in the Poisson regression model). Correlation coefficients and rmses between the observed and cross-validated hindcasts of tropical cyclogenesis, together with the percentage improvement over climatology, are presented in Table 3. The time variability of the seasonal number of tropical cyclones formed in the eastern region, as well as the midseason tropical cyclogenesis numbers across the entire region, are hindcast with the most skill. Early season cyclogenesis occurrences, together with those cyclones formed in the northern region, are the most poorly hindcast by the various Poisson regression models. This is perhaps not surprising as this subseason and subregion typically contain the smallest number of tropical cyclogenesis occurrences. Even though the hindcasts of late season cyclogenesis show the greatest improvements over the climatology, only the all-gray model hindcasts are correlated with the observed occurrences at better than the 95% significance level. It is noted in this case that the climatology rmse is actually higher than when simply using the average number of tropical cyclone occurrences for the late season. Also note that although the spatial structure of the eastern region tropical cyclones is poorly represented by the models, the phase and amplitude are nevertheless well represented. 5. Summary and discussion Forecast contingency tables (see the appendix for details) can be used to compare individual predictions of tropical cyclone occurrences and nonoccurrences against observations of same. The vertical wind shear combination model (details in Table 2) hindcasts individual tropical cyclogenesis occurrences well, with 70% being hindcast correctly (see Table 4a). This model also correctly hindcasts 66% of nonoccurrences. a. Dynamic parameter model McBride (1981) investigated the differences between developing and nondeveloping systems in relation to