## 1. Introduction

Integrated kinetic energy (IKE) is a recently developed metric that is designed to approximate the damage potential of landfalling tropical cyclones (Powell and Reinhold 2007). As its name suggests, IKE is defined as a summation of the kinetic energy within the near-surface wind field of a tropical cyclone (TC). By integrating energy across a large portion of a storm’s wind field, IKE considers the overall structure of a TC. This is in stark contrast to many other existing hurricane metrics, which often quantify only a wind or pressure extreme at a single point within a TC. Intensity metrics such as maximum sustained wind speeds (VMAX) are undoubtedly useful for assessing the maximum potential damage caused by the winds in a TC (e.g., Emanuel 2005; Bell et al. 2000), but they do not paint a complete picture of storm damage potential.

In the decade following the landfall of Hurricane Wilma, no major hurricanes (VMAX > 96 kt; 1 kt = 0.5144 m s^{−1}) have made landfall in the United States. This drought is thought to be a rather rare event, (Hall and Hereid 2015), depending on the metric that is used to classify major hurricanes (Hart et al. 2016). Despite this perceived quiet period of significant U.S. hurricane activity, there has been no shortage of damaging storms that have made landfall in the past decade. According to initial estimates from the National Hurricane Center,^{1} Hurricanes Ike (AL092008), Irene (AL092011), and Sandy (AL182012) each caused more than $15 million (U.S. dollars) in losses across the United States during the major hurricane drought despite each storm’s somewhat weak landfall intensity. This disconnect between VMAX and damage often occurs because storm size and structure must also be considered to properly evaluate storm surge potential (e.g., Irish et al. 2008). Since Sandy, Irene, and Ike were such large storms, they were able to produce higher storm surge and damage totals than otherwise would be expected by storms of similar intensities. For this reason, it is likely that the IKE metric could add value to existing intensity metrics, by anticipating the higher damage potential of larger landfalling TCs (Powell and Reinhold 2007), especially considering that Ike, Sandy, and Irene all ranked very highly in terms of IKE relative to other storms in the historical record (Kozar and Misra 2014, hereafter KM14).

Despite the potential advantages of IKE, the concept of forecasting the energy metric in real time is still in its infancy. Currently, operational forecasters have little to no guidance to predict IKE. Recently, KM14 explored whether or not it is feasible to fill that void with a simple statistical model in a proof-of-concept exercise. The resulting statistical model from that study was named the Statistical Prediction of Integrated Kinetic Energy (SPIKE), and it used linear regression to predict changes of IKE from a series of environmental predictors. Despite its simplicity, SPIKE was ultimately capable of outperforming a persistence forecast in a perfect-prognostic mode, indicating that statistical–dynamical forecasts of IKE might be possible in the future.

Building upon those results, the focus of this study is to further evaluate the operational potential of IKE forecasts using a more sophisticated statistical–dynamical scheme in a hindcast mode. Despite the successes of the proof-of-concept SPIKE model from KM14, linear regression is suboptimal for statistical weather prediction because the earth system is quite complex and contains several nonlinear signals. As such, the fixed linear regression coefficients in the SPIKE model will never be able to fully process the complex changing relationships between the environment and IKE variability within a TC. Therefore, a second-generation version of SPIKE is developed in this work by utilizing a more complex and nonlinear statistical framework in lieu of linear regression. More specifically, SPIKE, version 2 (SPIKE2), utilizes a series of artificial neural networks (ANNs) to predict IKE tendency from a similar series of environmental input parameters. Ultimately, these networks are capable of learning and anticipating complex patterns in the environment, and as a result they are better suited to model a nonlinear system.

Furthermore, SPIKE2’s evaluation will be moved from a perfect-prognostic space previously used in the initial KM14 work to a hindcast space. Obviously, in an operational setting, a statistical–dynamical forecast scheme must contend with imperfect input parameters that contain forecast errors of increasing magnitude with increasing lead time. Therefore, by running SPIKE2 in a hindcast mode with model data from the National Oceanic and Atmospheric Administration (NOAA) Second-Generation Global Ensemble Reforecast archive (Hamill et al. 2013), we are able to more comprehensively measure the potential performance of the IKE prediction models.

The next section discusses the historical and reforecast data that are used to calibrate and evaluate the SPIKE2 neural network system. In the subsequent sections, the discussion shifts toward the methodology and procedures for creating, calibrating, and evaluating the neural networks used in SPIKE2. Finally, the calibration and hindcast performance of SPIKE2 is compared against various persistence and climatology benchmarks for Atlantic TCs between 1990 and 2011 in the penultimate section, which is followed by the conclusions.

## 2. Historical and model reforecast data

Similar to KM14, a historical record of IKE in North Atlantic TCs is used to train and validate the SPIKE2 neural networks. This historical record covers the 1990–2011 hurricane seasons, and includes over 5000 (in total) 6-hourly fixes from nearly 300 individual storms (Misra et al. 2013). Since gridded wind fields are not available for all of these cases, the IKE values contained in this record are all estimated from operational wind radii and intensity metrics in the extended best track dataset (Demuth et al. 2006) using a series of equations from Powell and Reinhold (2007) and Misra et al. (2013). The mean value of IKE across all of the storm fixes included in our historical archive is 35 terajoules (TJ), with a standard deviation of 43 TJ. The distribution of observed IKE values takes a somewhat lognormal shape with a long tail toward higher values (KM14). As such, although most storms never reach 50 TJ of IKE, Hurricane Sandy likely had more than 400 TJ of IKE before it made landfall in New Jersey in 2012.

It should also be noted that past works have documented significant uncertainty within the historical record of wind radii that fluctuates depending on the data platforms that are available when analyzing each storm (e.g., Knaff et al. 2014; Landsea and Franklin 2013). Therefore, our historical IKE record, which again is based on the operation wind radii, likely inherits many of the same uncertainties found in the extended best track dataset.

Unlike KM14, SPIKE2’s environmental input variables are drawn from a historical model reforecast database during the same 1990–2011 interval. The second-generation Global Ensemble Reforecast archive (Hamill et al. 2013) is selected as the source for this model data because it includes model runs dating back multiple decades using a static February 2012 operational configuration (version 9.0.1) of the National Centers for Environmental Prediction (NCEP) Global Ensemble Forecast System (GEFS).

These archived GEFS reforecasts include forecasts out to 16 days beyond the initialization time. The first 8 days of the forecast are run at T254 horizontal resolution (~50 km) with 42 vertical levels. The latter half of the forecast is run at a lower T190 resolution (~70 km), with the same 42 vertical layers. Each of the reforecast runs is initialized once daily at 0000 UTC, as opposed to the 6-hourly approach for generating operational GEFS forecasts. The initial conditions for the reforecast dataset are produced from the Climate Forecast System Reanalysis (CFSR; Saha et al. 2010) prior to February 2011 and operational Gridpoint Statistical Interpolation analysis system after that time.

The reforecast archive includes 98 output fields from initial time out to *F* + 384 h for each of the daily GEFS reforecasts.^{2} The archived data are stored at 3-hourly intervals for the first 72 h of the forecast and then at 6-hourly intervals after that time. Each meteorological field is bilinearly interpolated down to a somewhat coarse 1° resolution global grid. In addition to the 1° datasets, a smaller selection of 28 fields is also stored in the GEFS’s higher-resolution native Gaussian grid (~0.5°). However, the higher-resolution fields are all single-level variables, primarily near the surface. As a result of this limitation, mid- and upper-atmospheric dynamic and thermodynamic fields (winds, temperatures, humidity, etc.) are only available in the 1° grids. Therefore, to maximize consistency, we use only the 1° data to examine the dynamical and thermodynamical processes that relate to IKE variability for our SPIKE2 neural network system.

As its name suggests, the GEFS archive does not just include a single deterministic forecast. In fact, the reforecast dataset comprises 11 ensemble members (1 control run, and 10 perturbation runs) compared to 21 ensemble members in the operational GEFS. For the purposes of this work, only the control run in the GEFS reforecast set is considered, but future works can and should clearly expand upon these results to produce probabilistic forecasts that resolve the uncertainty in the model’s initial environment.

This GEFS reforecast dataset includes some noteworthy biases with regards to resolving TCs that will be addressed here. Obviously, the GEFS reforecasts will include position and intensity errors, and the reforecasted environment is expected to be imperfect as well, with all errors increasing as lead time increases. For reference, Galarneau and Hamill (2015) analyzed track errors in the GEFS reforecast archive for TCs in the Gulf of Mexico between 1985 and 2010 and found average positions errors to be 100 km with a lead time of 24 h, 250 km with a lead time of 72 h, and 400 km with a 120-h forecast interval. Typically, these track reforecasts in the Gulf of Mexico were found to have a left and slow bias relative to the storms motion.

Furthermore, Galarneau and Hamill (2015) also indicated that the GEFS reforecasts had a significant and consistent low-intensity bias. This does not come as a surprise and leads us to the potentially most concerning issue for using the GEFS reforecast database in this study. Simply put, the 1° horizontal resolution data taken from the model will not be sufficient to properly resolve the wind field of a TC. As a result, intensities will be underestimated, and wind fields may be too broad. In fact, the GEFS reforecasts may fail to generate a TC vortex altogether in some extreme scenarios.

However, since we are not trying to predict IKE directly from the model’s wind field, but instead by relating environmental parameters to IKE variability, this low-resolution data might still be sufficient, albeit less than ideal. By using the lower-resolution GEFS reforecast data, we can estimate the lower bounds of skill for a real-time version of SPIKE2. Furthermore, the static model configuration provided by the GEFS reforecast dataset (Hamill et al. 2013) allows us to focus on the performance of the SPIKE2 predictive scheme independent of changes in the underlying dynamical model’s configuration and performance.

## 3. Selection of input parameters for statistical–dynamical prediction

Before the neural networks can be constructed, we must first establish which environmental and storm-specific input parameters will be taken from the GEFS control reforecasts to produce predictions of IKE variability. The initial SPIKE model built in KM14 utilized a series of 14 predictors that contained a significant linear relationship with IKE variability, many of which were taken directly from the Statistical Hurricane Intensity Prediction Scheme (SHIPS) developmental dataset (DeMaria and Kaplan 1999). These input parameters included various environmental predictors (both dynamical and thermodynamical), storm-specific parameters (e.g., position, minimum pressure), and persistence values of IKE based on known relationships between IKE and the environment (e.g., Maclay et al. 2008; Musgrave et al. 2012). However, since these parameters were selected based on their linear relationships with IKE, it is necessary to reselect predictors to highlight the nonlinearities in the storm-environment system that hopefully can be captured by the more sophisticated neural network scheme utilized here for SPIKE2.

As was done in KM14, the goal in selecting these parameters should be to target physical processes that govern variability with a TC’s structure and ultimately the IKE index. Therefore, we started with a large pool of predictors, including both predictors used in the linear model that had clear and justifiable relationships with IKE as well as control variables such as day of month. Properly tuning a nonlinear complex neural network is a bit more difficult than tuning a linear regression model as there are more weights and neurons than there are coefficients in a linear regression model. Nonetheless, as we constructed the neural network we removed predictors if network performance over the testing sample increased by subtracting the predictor. As such, each of the control parameters and a few of the other environmental predictors with weaker ties to IKE were not selected for the final version.

Ultimately, we settled on 18 input parameters for SPIKE2, each of which is related to targeted relationships between the environment and IKE, in order to maximize the neural networks potential predictive power. The specific predictors are listed in Table 1. From this point forward, each predictor will be referred to by its abbreviation in the table. This predictor list is very similar to those used in the linear SPIKE model, but does include a total of four additional predictors. As such, we acknowledge that a few of these predictors could be removed, and the performance of SPIKE2 would likely not change by an appreciable margin. However, removal of any of the predictors did not seem to improve validation performance, suggesting that the predictors were not setting the model back via overfitting. Therefore, we felt that by including some of these extra predictors, the neural network may have a better chance to resolve some of the nonlinear signals between the environment and TCs if we were careful to limit the number of neurons in the ANN, thus minimizing the chances of overfitting.

Variables used in the SPIKE2 neural networks. These input parameters are obtained from GEFS reforecasts and analyses, NOAA OISSTs, and the historical record. Many of these predictors are inspired by the parameters contained in the SHIPS developmental dataset.

Nonetheless, we ran a series of perturbation tests and case studies to ensure that each individual variable had some physical relationship that could explain how it is affecting projections of IKE from SPIKE2. For brevity, the remaining discussion in this section is meant to highlight the physical relationships that can explain how each of the individual predictors affect IKE variability, followed by a short explanation about how the predictors are directly calculated from the model fields.

Predictors such as D200, VORT, SHTD, and SHRD are designed to represent the certain dynamical features (upper-level divergence, low-level vorticity, weak easterly shear) that are favorable for TC development. These predictors were some of the more significant predictors in the linear regression SPIKE model, and their impact over SPIKE2’s IKE projections remains strong. Meanwhile, SST, T150, and VMPI are meant to be tied to thermodynamical properties that govern the maximum intensity of the storm, the height of the tropopause, and how far a storm has to go before it reaches said maximum intensity (e.g., Emanuel 1988; Bister and Emanuel 1998). RHLO and RHMD capture well-known relationships between moisture and TC development. MSLP, PENV, and VMAX are storm-specific parameters that give some information about the TC’s intensity and breadth at the validation time, wherein a more intense storm or a larger storm with all else being equal will have higher wind speeds and more IKE. LAT, LON, SDAY, and PDAY obviously give information about the storm’s position and time. These can be useful for identifying climatological tendencies across the basin. Finally, predictors such as PIKE and dIKE12 give information about persistence (i.e., how much IKE the storm had previously, and was it gaining or losing IKE previously) that can be useful for predicting future trends in certain instances.

However, as alluded to in the opening section, the signals between IKE and these predictors are quite complex. Unlike traditional storm development, which has a somewhat straightforward relationship with some of these predictors (i.e., the combination of low shear and high SSTs typical translates to a stronger storm all else being equal), IKE is also tied to storm size and the many different processes that govern it. For instance, many storms tend to expand as they move poleward and interact with other baroclinic features or through extratropical transition (e.g., Evans and Hart 2008). As such, recurving TCs often gain IKE in midlatitude environments that would traditionally be considered nonfavorable for development (Maclay et al. 2008).

Considering that extratropical transition occurs in just under half of all Atlantic TCs (Hart and Evans 2001), our prediction scheme must be calibrated to anticipate the correct IKE tendencies from these complex signals. As a result, the nonlinear equations within the ANNs will also use predictors such as LAT, SHRD, T150, RHLO, and SST to determine whether or not a storm is likely to expand in size (and also in IKE) from baroclinic forcings. Encouragingly, some simple case studies revealed that a hypothetical storm in the midlatitudes (high LAT), late in its life cycle (high SDAY) will actually gain IKE as expected in a more baroclinic environment with lower SSTs and higher SHRD. However, if the storm is under a similar environment in the deep tropics or if shear and SSTs are too prohibitive in the midlatitudes, the neural networks will correctly identify that the storm is more likely to decay. Ultimately, by considering both baroclinic influences and traditional developmental mechanisms from this wide-ranging predictor base through a nonlinear system of equations, the ANNs should be able to improve upon the results of KM14.

The majority of the predictors discussed above (LAT, LON, MSLP, VORT, D200, etc.) are calculated directly from the corresponding TC signature within 3D atmospheric fields from the GEFS’s control run. However, it should be noted that the GEFS dataset by itself is insufficient to calculate all 18 of the input parameters. For instance, some of the input parameters require information about the ocean surface (VMPI, SST), time and date of year (SDAY, PDAY), and past values of IKE (PIKE, dIKE12). Therefore, to obtain hindcasts for each of the input parameters the GEFS reforecast dataset will be supplemented with a number of other datasets. Daily 1° NOAA Optimum Interpolation SST (OISST; Reynolds et al. 2007) is used to estimate observed ocean surface conditions. The historical IKE record (derived from the extended best track dataset) is used to produce the persistence parameters, and finally the NHC best track dataset is used to get the time information for each storm fix.

Once the input parameters are calculated from the GEFS control run for all forecast hours between initial time and *T* + 72 h, each parameter is normalized by its sample within the GEFS control run for all storms between 1990 and 2011. Normalizing the input parameters offers the benefit of filtering out some of the systematic biases in the GEFS, which in turn should enhance the performance of the operational IKE prediction schemes.

## 4. Setup of artificial neural network for SPIKE2

With the predictors and data sources now established, this section details how the artificial neural networks are constructed, calibrated, and then run in a hindcast mode. As highlighted earlier, ANNs are chosen for SPIKE2 because of their ability to resolve and adapt to changing nonlinear signals in a certain system (e.g., Kriesel 2007). Thanks in part to their versatility, ANNs have been used in meteorology over the past several years to complete a wide array of tasks. A nonexhaustive list of tasks that ANNs have been used for includes evaluating uncertainty in hurricane wind analyses (DiNapoli et al. 2012), processing remotely sensed data (e.g., Atkinson and Tatnall 1997), classifying circulation patterns (e.g., Cawley and Dorling 1996), predicting troposphere ozone levels (e.g., Abdul-Wahab and Al-Alawi 2002), forecasting wind speeds (Cao et al. 2012), forecasting precipitation and flooding (e.g., Hapuarachchi et al. 2011), and predicting the strength of the Indian monsoon on a seasonal scale (Shukla et al. 2011). A more detailed summary of earlier ANN applications in meteorology can be found in a review by Gardner and Dorling (1998).

### a. Network hierarchy and algorithms

The SPIKE2 prediction scheme will be built using a system of multiple two-layer feed-forward ANNs. Our two-layer feed-forward networks’ hierarchy includes a hidden layer with 20 artificial neurons and an output layer with a single neuron that will ultimately produce the desired results from the input parameters. The 20 neurons were chosen for the hidden layer to maximize predictive skill based on the results of an exhaustive search test, in which we found that this number of neurons corresponded to the best validation performance over the test subsample. By testing model performance with a wide varying number of neurons in this exhaustive search, we were able to find the approximate point at which ANN complexity is small enough to minimize the chance of overfitting, without compromising its ability to recognize and generalize the nonlinear signals in the TC-environment system.

In our case, the output of the neural networks will be IKE tendency for a given forecast hour, or in other words the difference between IKE at validation time and IKE at initialization time. Meanwhile, the 18 normalized parameters discussed in section 3 are selected as the input parameters of the neural network. As such, the goal of each ANN is to produce an estimate of IKE tendency from environmental and storm-specific values within a model solution.

Ultimately, each of these ANNs within the SPIKE2 scheme are trained using a shared learning algorithm, wherein the networks are calibrated using a set of input parameters and known target (IKE tendency). The weights of the network’s neurons are designed to adapt from a somewhat random initial value to a more optimal value, as the error function reaches a minimum. More specifically, the learning algorithm uses a Levenberg–Marquardt backpropagation algorithm (Marquardt 1963) to find this error minimum. This specific algorithm is designed to solve nonlinear least squared problems and is typically thought to be an efficient and stable method for converging at an optimal solution in neural network learning (e.g., Hagan and Menhaj 1994).

### b. Training, validation, and test samples for calibration

To avoid overfitting and to promote generalization in the above supervised learning algorithm, the historical input and target output data series that are used to construct the ANNs will be randomly split into three subsets. The first subset of data, named the training sample, comprises 70% of the input and target series. As its name suggests, the training sample is used to train the network by establishing the optimal weights within the neurons. The validation sample is a smaller subset, comprising 15% of the historical input and target series. This subset is ultimately used to determine when the neural network can stop learning based on the network’s ability to generalize effectively. As such, the learning algorithm searches for the point at which the neural network has the least amount of error over the validation subset during calibration. Finally, the third subset of input and target data is called the testing sample. This test sample is not used in the calibration of the model in any way. Instead it simply provides a more accurate measurement of out-of-sample network performance during calibration.

It should be noted that the three subset samples used in calibration are not entirely independent from one another because of storm-based serial correlation The general population of calibration data for any given forecast hour contains multiple target IKE tendency values from long-lived storms, but will not ever contain multiple sets of predictors from the same model run. Furthermore, each GEFS run that predictors are taken from is separated by at least 24 h from the next closest analysis, as the GEFS is only initialized once daily in the NCEP reforecast dataset. As KM14 showed, past IKE change did not have significant ties to future IKE tendency beyond the first 24 h. Therefore, storm-based serial correlation between subsequent target IKE tendency values for each forecast hour should be somewhat limited. Nonetheless, these three subsets are only used in the calibration of individual neural networks. Once the weights are established with analyses as detailed in section 5, the evaluation exercises done in section 6, will use out-of-sample hindcast data to drive the neural network in an effort to best simulate how the models may perform in real time.

### c. Neural network random variability

Inevitably, the methodology used to construct the neural networks introduces random variability into each individual ANN. Specifically, random variability is first introduced when the general population of input and target parameters from 1990 to 2011 is randomly split into the three separate subset samples. Additional random variability is introduced to the neural networks because the weights within the neurons are initialized somewhat randomly before arriving at their optimal weights. Ultimately, the random variability makes it all but impossible for two ANNs to be exactly identical to one another, even if they are calibrated on the exact same input and target output datasets. Each neural network weighs connections in the nonlinear system somewhat differently, and as a result, some of the networks will seem more accurate in certain situations but less accurate in other situations. Therefore, it is insufficient to base SPIKE2 off a forecast from just a single neural network.

Instead, SPIKE2 will utilize a system of 100 individual neural networks to make its prediction of IKE tendency for each of the forecast intervals (i.e., a separate system of neural networks for each forecast interval). As shown by the schematic of SPIKE2 (Fig. 1), the system of neural networks will produce 100 separate independent predictions of IKE tendency from a single set of input parameters. A deterministic forecast of IKE tendency from SPIKE2 will be taken from the median of these 100 individual predictions. Using the median from a large sample of ANNs helps to minimize the random variability present in a single neural network’s forecast, thus allowing SPIKE2 to focus on the true skill of the neural networks. The overall skill of this deterministic forecast will be discussed at length in section 6.

Not covered in this paper for brevity, but in development, are a series of probabilistic products that take each of the 100 individual ANNs within SPIKE2 into account, rather than just the median estimation. These SPIKE2 probabilistic products are used to evaluate the uncertainty within the ANN statistical scheme from a single set of input parameters, as each ANN within SPIKE2 has a slightly different set of weights. Probability of exceedance, uncertainty ranges, and error bars, are just a small sample of some of the probabilistic utilities that are possible with SPIKE2 before even considering the idea of forcing the model with a wide array of input parameters from multiple forecast models or ensembles.

### d. Comparison of neural network and linear model performance

Once again, the primary goal for developing SPIKE2 is to create a statistical dynamical model that is capable of predicting IKE in a hindcast mode, which would mark a significant step toward moving to real-time operational forecasts of IKE. This objective differs from the goals of the linear regression version of SPIKE in KM14. That previous linear regression model was designed in a perfect prognostic mode to prove that IKE prediction is possible when given accurate environmental predictors. As a result of the different objectives and the different running environments (perfect prognostic vs hindcast), SPIKE and SPIKE2 are trained on two completely different sets of predictors, even if the storms in the calibration and evaluation sample are identical.

Therefore, to explicitly show the advantages of the new neural network methodology over the previously used linear regression model, we also created a perfect prognostic version of our neural networks using the same data source (developmental SHIPS; DeMaria and Kaplan 1999) that was used for calibration and evaluation of the SPIKE model in KM14. Unsurprisingly, the added flexibility provided by the nonlinear equations allowed the neural network to outperform the linear regression model across a 1995–2011 comparison period. For instance, SPIKE2 has a mean absolute error of 12.6 TJ over its training sample at its longest 72-h forecast interval. In contrast, the linear regression model has a comparable mean absolute error of 14 TJ at a much shorter 24-h forecast interval. As such, we will progress onward out of the perfect prognostic space, and begin to calibrate the neural networks with predictors from numerical analyses in section 5, in an effort to prepare the neural networks for evaluation with hindcast predictors in section 6.

## 5. Calibration of neural networks using GEFS analyses

Ultimately, to establish the weights of the ANNs, we calibrate the entire system with targets of IKE tendency taken from our historical record and normalized input parameters taken from the control 0-h analyses (F00) at validation time within the GEFS reforecast archive. These analyses represent the best estimation for observed environmental conditions within the model data’s 1° resolution. As such, the SPIKE2 system will be calibrated to accept reforecast input parameters from the same coarse resolution when it is ultimately evaluated in a hindcast mode. It is important to note that the F00 analyses do not include forecast errors. Therefore, only the persistence IKE predictor will change with advancing forecast hour as the persistence IKE value becomes further removed from the validation time.

Since the GEFS reforecast runs are only initialized at 0000 UTC, the maximum sample size for the F00 calibration dataset is the 1377 storm fixes that occur at 0000 UTC between 1990 and 2011. However, SPIKE2 requires persistence parameters of varying forecast lead times. Therefore, the sample size of the calibration dataset will decrease with increasing forecast hour because short-lived storms will not have longer-term persistence values. For comparison purposes, there are 1097 fixes at a forecast hour of 12 h and 614 fixes for the 72-h forecast interval.

The performance of SPIKE2’s deterministic forecast for the analysis-based calibration dataset is shown in Table 2. Similar to SHIPS (DeMaria and Kaplan 1994) and SPIKE (KM14), the explained variance for the targeted tendency value increases with increasing forecast hour. The correlation between IKE tendency predictions from SPIKE2 with GEFS F00 data and the observed historical dataset was *r* = 0.73 at a 12-h forecast window compared to *r* = 0.91 at 72 h. This seemingly counterintuitive result can be explained by considering that the magnitude of IKE tendencies increases with growing forecast hour, such that random fluctuations and observational biases are less impactful at longer forecast hours. Furthermore, forecast errors are not present in any of the GEFS F00 input parameters such that the input parameters are no less accurate at 72 h than they are at 12 h.

Performance of SPIKE2’s deterministic forecast when evaluated with the calibration input set from the GEFS F00 analyses. Sample size indicates the amount of storm fixes that were included at each forecast hour, *R*_{tendency} measures the correlation between the observed IKE tendency value and the predicted IKE tendency value from SPIKE2’s output, and *R*_{IKE} measures the correlation between the observed IKE value at validation time and the predicted IKE value calculated by adding SPIKE’s tendency prediction to the existing persistence value from initialization time. Mean error is simply the mean absolute difference between the predictions from SPIKE2 and the observed IKE values.

In addition to predicting IKE tendency, SPIKE2 can also predict the actual value of IKE at the validation time by adding its IKE tendency prediction to the persistence IKE value from the model’s initialization time. KM14 found that the IKE metric was somewhat resistant to change because it considers the energy across a storm’s entire wind field. As a result, it is unsurprising that SPIKE2 performs better at predicting IKE than it does at predicting IKE tendency because it can use the decent performance of a persistence IKE forecast to its advantage, especially in short forecast windows. At a 12-h forecast window, SPIKE2 explains 91% of the observed variance (*r* = 0.95) when using the GEFS F00 input parameters. That performance does not degrade sharply, as the explained variance remains near 80% (*r* = 0.90) at a longer 72-h window.

While these high correlations are promising, they are somewhat meaningless if similar performance can be achieved by simply using a persistence forecast. Encouragingly, the SPIKE2 calibration model has a lower 72-h forecast error than does a much shorter 24-h persistence forecast. To provide another metric for comparison, we have evaluated the mean-squared error (MSE) from SPIKE2 over its calibration dataset against a persistence forecast at each corresponding forecast hour in Fig. 2. Overall, SPIKE2 has lower MSE than does persistence by a fair margin (45% at a 12-h forecast window, climbing up to 82% by 72 h). The improvements over persistence are statistically significant at a *p* = 0.025 level for all forecast intervals based on a two-sample bootstrapping test.

Evaluation of SPIKE2 skill in a calibration mode with GEFS analyses relative to a persistence forecast. Calibration skill is measured as a percent reduction of MSE for the model’s deterministic predictions from 1990 to 2011, with respect to similar MSE calculations for a persistence forecast at various forecast hours. A reduction of MSE is plotted as a positive percentage, indicating that the model has outperformed persistence. SPIKE2 has significantly lower MSE than persistence at the *p* = 0.05 level for all forecast hours. For reference, the reproduced results of the linear regression version of SPIKE as detailed in KM14 are also shown.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Evaluation of SPIKE2 skill in a calibration mode with GEFS analyses relative to a persistence forecast. Calibration skill is measured as a percent reduction of MSE for the model’s deterministic predictions from 1990 to 2011, with respect to similar MSE calculations for a persistence forecast at various forecast hours. A reduction of MSE is plotted as a positive percentage, indicating that the model has outperformed persistence. SPIKE2 has significantly lower MSE than persistence at the *p* = 0.05 level for all forecast hours. For reference, the reproduced results of the linear regression version of SPIKE as detailed in KM14 are also shown.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Evaluation of SPIKE2 skill in a calibration mode with GEFS analyses relative to a persistence forecast. Calibration skill is measured as a percent reduction of MSE for the model’s deterministic predictions from 1990 to 2011, with respect to similar MSE calculations for a persistence forecast at various forecast hours. A reduction of MSE is plotted as a positive percentage, indicating that the model has outperformed persistence. SPIKE2 has significantly lower MSE than persistence at the *p* = 0.05 level for all forecast hours. For reference, the reproduced results of the linear regression version of SPIKE as detailed in KM14 are also shown.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Also shown in Fig. 2 are the reproduced results from the original linear version of SPIKE detailed in KM14. These results are also calculated over the model’s calibration interval, 1990–2011, but as noted earlier, the two models used predictors from entirely different datasets making this comparison uneven. Nonetheless, the calibration statistics indicate that the linear SPIKE model simply cannot measure up to the neural networks in SPIKE2, although both models offer substantial improvement over persistence. Like the results of section 4d, this evidence continues to support our hypothesis that the neural networks will be superior to simple linear regression because it can account for the nonlinearities in the TC-environment system.

Although these initial calibration results appear to be encouraging, it should once again be noted that the hindcast version of SPIKE2 discussed in the following section will use imperfectly reforecasted input parameters from the GEFS control runs. As such, it would be unfair to expect SPIKE2’s hindcasts in the following section to achieve these high performance benchmarks. Instead, the performance metrics shown in Table 2 can be viewed as the maximum potential skill that can be obtained by SPIKE2. The intent of these performance benchmarks is to determine how the model will degrade when forecast errors are introduced to the model input fields. Nonetheless, the exercise proved useful by identifying a set of weights within the artificial neurons that can be used to produce hindcasts of IKE from the GEFS reforecasts.

## 6. Performance of SPIKE2 hindcasts using GEFS reforecasts

In this section, we will adapt the SPIKE2 ANN system to run in a hindcast mode with the GEFS reforecast control run from 1990 to 2011. As just discussed, the network will retain the same neuron weights that were calibrated in the previous exercise with GEFS control analyses. However, unlike the calibration exercises the neural networks will be given imperfect input parameters from the GEFS reforecast control run at various lead times out to 72 h. This will enable us to determine how forecast errors affect SPIKE2’s ability to predict IKE. We can understand from this analysis of predictive skill whether or not SPIKE2 might offer skillful operational support in a real-time environment.

Much like the last section, we will evaluate the deterministic forecast from SPIKE2 using the target IKE tendency and IKE values as the historical baseline. Statistics such as correlations and mean absolute errors will be used to detect the magnitude of performance deterioration relative to the maximum potential performance levels obtained in the calibration exercise. As was done in the earlier calibration exercises and in KM14, SPIKE2’s deterministic skill will be evaluated relative to simple persistence forecasts. However, in addition, a new more challenging benchmark will also be introduced by way of a simple statistical model that considers climatology and other nonforecast parameters.

Such a benchmark model would follow in the footsteps of the Statistical Hurricane Intensity Forecast model (SHIFOR), which uses seven known parameters at initialization time to set the baseline performance for operational intensity forecasts (Jarvinen and Neumann 1979; Knaff et al. 2003). The exact parameters of SHIFOR include the following: Julian day, initial storm intensity, previous 12-h intensity change, initial latitude, initial longitude, initial zonal component of storm motion, and initial meridional component of storm motion. These SHIFOR climatology and persistence predictors are somewhat relevant to IKE tendency as well. Therefore, an IKE statistical persistence model named the Benchmark of Integrated Kinetic Energy (BIKE) is created to predict IKE tendency in a simple linear regression model using the same seven input parameters, with two exceptions. First, the 12-h intensity change parameter will be switched out for a 12-h IKE change parameter. Second, the initial or persistence value of IKE will be added as an eighth predictor. This BIKE regression model is trained using all 0000 UTC storm fixes from 1990 to 2011, such that its calibration fit will be compared to the GEFS–SPIKE2 hindcasts at lead times of 24, 48, and 72 h for the same 1990–2011 interval.

Case studies act as a good first step to evaluate the SPIKE2 hindcasts relative to their assortment of benchmarks in an effort to see how IKE forecasts might perform during significant landfalling events. To that end, Fig. 3 contains a plot of SPIKE2 hindcasts shown against historical values of IKE just prior to landfall for Hurricanes Floyd (AL081999), Katrina (AL122005), Ike (AL112008), and Irene (AL122011). Each of these four storms gained considerable IKE as they approached land, and as a result, a persistence forecast would have greatly underestimated the storm’s destructive potential at landfall. BIKE proves to be a more challenging benchmark for SPIKE2 in these four case studies, as it arguably outperforms SPIKE2 for Hurricane Floyd. Nonetheless, the SPIKE2 hindcasts outperform BIKE in most other cases. The SPIKE2 hindcasts for Irene were particularly impressive as the green curve representing the hindcast remains very close to the black line representing the observations throughout the 72-h forecast period. One final item of note, in nearly each case, the SPIKE2 hindcast using reforecasted predictors performs worse than the SPIKE2 calibration model using predictors from analyses. This result is expected, as it suggests that the performance of the ANNs will degrade with the introduction of forecast errors in the series of input predictors.

72-h runs of SPIKE2 plotted against benchmarks and the observed IKE values (black lines) for notable hurricanes immediately prior to their landfalls. SPIKE2 hindcasts utilize GEFS reforecasted predictors from a run initialized at the time specified in each legend. SPIKE2 calibration runs utilize analyzed predictors from the GEFS archive valid at each forecast time. The benchmark model is initialized at the same time as the SPIKE2 hindcast run for direct comparison purposes. Not explicitly shown is a persistence forecast that would be represented by a horizontal line stretching from the first observed IKE value on the left through the entire 72-h period.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

72-h runs of SPIKE2 plotted against benchmarks and the observed IKE values (black lines) for notable hurricanes immediately prior to their landfalls. SPIKE2 hindcasts utilize GEFS reforecasted predictors from a run initialized at the time specified in each legend. SPIKE2 calibration runs utilize analyzed predictors from the GEFS archive valid at each forecast time. The benchmark model is initialized at the same time as the SPIKE2 hindcast run for direct comparison purposes. Not explicitly shown is a persistence forecast that would be represented by a horizontal line stretching from the first observed IKE value on the left through the entire 72-h period.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

72-h runs of SPIKE2 plotted against benchmarks and the observed IKE values (black lines) for notable hurricanes immediately prior to their landfalls. SPIKE2 hindcasts utilize GEFS reforecasted predictors from a run initialized at the time specified in each legend. SPIKE2 calibration runs utilize analyzed predictors from the GEFS archive valid at each forecast time. The benchmark model is initialized at the same time as the SPIKE2 hindcast run for direct comparison purposes. Not explicitly shown is a persistence forecast that would be represented by a horizontal line stretching from the first observed IKE value on the left through the entire 72-h period.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Moving to a more general perspective, mean error and correlation statistics are shown on a line plot in Fig. 4 for all of the storm fixes within the 1990–2011 evaluation sample. The SPIKE2 hindcasts are capable of explaining more than 80% of the variance in the historical IKE record with a day of lead time, and mean absolute errors are approximately 12 TJ in the same 24-h forecast window. As lead time increases, hindcast performance expectedly decays, but the model is still capable of explaining 70% of the historical IKE variance at 48 h and 62% at 72 h, with errors of 16.6 and 20.7 TJ at those times, respectively.

Performance statistics for SPIKE2. (a) Correlation and (b) mean absolute error values, in units of TJ, are shown between the historical record and SPIKE2 in various modes or a persistence forecast. The correlation value that is shown in these plots is for IKE, not IKE tendency. Calibration statistics are identical to those in Table 2, and are used as a maximum potential reference point to determine the degradation of skill when forecast error is introduced to the model in the hindcast runs via the input parameters from the GEFS reforecast. Also shown for reference is the performance of a persistence forecast and the statistical climatological and persistence model, BIKE.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Performance statistics for SPIKE2. (a) Correlation and (b) mean absolute error values, in units of TJ, are shown between the historical record and SPIKE2 in various modes or a persistence forecast. The correlation value that is shown in these plots is for IKE, not IKE tendency. Calibration statistics are identical to those in Table 2, and are used as a maximum potential reference point to determine the degradation of skill when forecast error is introduced to the model in the hindcast runs via the input parameters from the GEFS reforecast. Also shown for reference is the performance of a persistence forecast and the statistical climatological and persistence model, BIKE.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Performance statistics for SPIKE2. (a) Correlation and (b) mean absolute error values, in units of TJ, are shown between the historical record and SPIKE2 in various modes or a persistence forecast. The correlation value that is shown in these plots is for IKE, not IKE tendency. Calibration statistics are identical to those in Table 2, and are used as a maximum potential reference point to determine the degradation of skill when forecast error is introduced to the model in the hindcast runs via the input parameters from the GEFS reforecast. Also shown for reference is the performance of a persistence forecast and the statistical climatological and persistence model, BIKE.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

The performance of the hindcast easily exceeds the performance benchmark set forth by a persistence forecast. For instance, a 72-h SPIKE2 hindcast has comparable error on average to a half-as-long 36-h persistence forecast. Mean-squared error statistics paint a similar picture (Fig. 5), as the SPIKE2 model offers a 60% reduction of MSE at 24-h relative to persistence. This reduction in MSE relative to persistence holds steady as forecast hour increases, fluctuating between 50% and 70% between 24 and 72 h of lead time. The lack of a trend with advancing forecast hour in this MSE reduction metric (outside of the first 12–24 h, where persistence forecasts excel) is likely attributed to a balance between rapidly increasing persistence error (lowering the benchmark), and increasing forecast errors in the input data holding back the SPIKE2 scheme (decreased hindcast performance). The significance of these improvements is once again tested with a two-sample bootstrapping exercise. Results indicate that the SPIKE2 hindcasts are significantly better than persistence at the *p* = 0.05 level for all forecast windows greater than 12 h, and at the *p* = 0.01 level for all forecast windows greater than or equal to 48 h.

Evaluation of SPIKE2 skill relative to a persistence forecast. Performance is once again measured as a percent reduction of MSE for the model's deterministic predictions from 1990 to 2011, with respect to similar MSE calculations for a persistence forecast at various forecast hours. The calibration skill is reproduced from Fig. 2 and is shown alongside the skill of the SPIKE2 hindcasts with reforecasted input parameters relative to persistence. The hindcast model is significantly better than persistence at the *p* = 0.05 level for all forecast hours greater than or equal to 24 h in length.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Evaluation of SPIKE2 skill relative to a persistence forecast. Performance is once again measured as a percent reduction of MSE for the model's deterministic predictions from 1990 to 2011, with respect to similar MSE calculations for a persistence forecast at various forecast hours. The calibration skill is reproduced from Fig. 2 and is shown alongside the skill of the SPIKE2 hindcasts with reforecasted input parameters relative to persistence. The hindcast model is significantly better than persistence at the *p* = 0.05 level for all forecast hours greater than or equal to 24 h in length.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Evaluation of SPIKE2 skill relative to a persistence forecast. Performance is once again measured as a percent reduction of MSE for the model's deterministic predictions from 1990 to 2011, with respect to similar MSE calculations for a persistence forecast at various forecast hours. The calibration skill is reproduced from Fig. 2 and is shown alongside the skill of the SPIKE2 hindcasts with reforecasted input parameters relative to persistence. The hindcast model is significantly better than persistence at the *p* = 0.05 level for all forecast hours greater than or equal to 24 h in length.

Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-16-0030.1

Unsurprisingly, the BIKE model is indeed a tougher benchmark than just a simple persistence forecasts as noted by both the correlation and mean error metrics. For instance, BIKE has a 12% lower mean absolute error at 72 h than does persistence. Nonetheless, the SPIKE2 hindcasts still clearly outperform BIKE. For instance, BIKE’s mean absolute errors are more than 30% higher than the SPIKE2 hindcast errors at all three of the shown forecast windows.

On the other hand, the performance of the hindcasts falls short of the higher performance levels found during the calibration exercises. Again, this result was expected because statistical–dynamical prediction schemes are only as accurate as the input data going into the statistical model. In this case, the GEFS reforecasts include forecast errors that were not present in the analyses, which results in this degradation of performance. Furthermore, a lesser decrease in performance should also be expected just by running the ANNs on a dataset that they were not calibrated with (i.e., the GEFS F00 analyses).

Nonetheless, the drop in performance from the calibration tests to the hindcast tests is not a hindrance. Mean errors only increased by less than 15% and correlations only decreased by less than 7% inside the shorter 12- and 24-h windows. Growing inaccuracies in the GEFS input variables, led to a more dramatic decrease in performance at larger longer forecast windows relative to the maximum potential performance level in the calibration exercises. However, once again, these hindcasts are still convincingly skillful relative to a persistence forecast. In fact, the hindcast performance metrics (green curve) are much closer to the potential performance metrics in the calibration runs (blue curve) than they are to the persistence performance benchmarks (red curves).

## 7. Conclusions and outlook for future operational development

Despite the promise of the hindcast results presented above, there is still some work left to be done to adapt this model for operational use. For example, the neural networks would likely need be recalibrated to receive operationally predicted input parameters from a desired model in real time unless the targeted model is similar to the GEFS reforecast data used here (such as the operational GEFS). If recalibration is needed, a sufficiently long historical database will once again be needed to normalize the predictors and set the neuron weights. This limitation is one of the primary reasons for using the control run in the available GEFS reforecast database. Despite its coarse 1° resolution, the GEFS archive contained a long record of data from a static version of the same model. Unfortunately, few operational models have long archives of forecasts or hindcasts that are readily available. Therefore, adapting SPIKE2 to be used with a higher-resolution operational model or model ensembles is dependent upon securing an archive for the desired model. As such, adapting SPIKE2 to the rest of the GEFS ensemble members at a similar resolution or to archived model data stored in The Observing System Research and Predictability Experiment (THORPEX) Interactive Grand Global Ensemble (TIGGE) archive would be easier to accomplish than would be adapting SPIKE2 to work with predictors from the Hurricane Weather Research and Forecasting (HWRF) Model.

In addition to calibration, future work must focus on determining whether or not SPIKE2 forecasts can be made in a timely manner. SPIKE2’s products almost certainly cannot be issued instantaneously at initialization time. Although the neural networks themselves can be run fairly quickly, an operational version of SPIKE2 still requires dynamically forecasted input parameters, and unfortunately, the output from most modern dynamical models is not available until a few hours after their initialization time. Therefore, statistical-dynamical models dependent upon forecast model data, such as model output statistics (MOS) and in the future SPIKE2, cannot come out until the dynamical models’ run time concludes. As a result, a 72-h SPIKE2 forecast using GFS or GEFS data would be already a few hours into its forecast period by the time it was issued, thus shortening it to a 66–70-h forecast depending on the actual issuance time.

A large delay between issuance and initialization would be detrimental to the usefulness of SPIKE2 because most operational forecasters are required to issue their forecasts at regular intervals. To alleviate this concern some operational statistical–dynamical models, are run in a so-called early cycle mode, wherein each product uses environmental predictors from the previous dynamical model run, which is typically initialized 6 h earlier (i.e., the 0000 UTC forecast uses dynamical predictors from an 1800 UTC model). Adapting this early cycle approach to SPIKE2 will ensure that its IKE forecasts are in advance of each forecast advisory. Consequently, SPIKE2’s dynamical predictors in an early cycle mode would be several hours old before SPIKE2 is even issued. As such, the need to forecast the input parameters an additional few hours into the future would likely result in a slight degradation to model skill. Considering that SPIKE2 hindcasts outperform much shorter persistence forecasts, this may not have a substantial effect on performance. Nonetheless, it will be necessary to compare the pros and cons of running SPIKE2 in this early cycle mode against attempting to develop an interpolator to issue SPIKE2 forecasts in a “late cycle” mode as real-time development of SPIKE2 continues.

Nonetheless, the results presented in the earlier sections serve as a proof of concept, suggesting that SPIKE2 could be a viable product in an operational setting once these hurdles are cleared. In calibration exercises, the deterministic scheme is capable of explaining the majority of variance in the historical IKE archive, and offers a significant improvement over persistence. Importantly, the addition of imperfectly predicted input parameters from a coarse GEFS control run archive did not cause SPIKE2’s performance to drop off severely. Instead, SPIKE2 hindcasts from 1990 to 2011 still exhibit significant skill over any known IKE persistence or climatology metrics, despite the inclusion of forecast errors from a rather coarse-resolution GEFS dataset. Not to mention, the briefly discussed SPIKE2 probabilistic products add value to the deterministic IKE forecasts by offering a quantitative estimate of uncertainty in the statistical neural network scheme.

With the inclusion of input parameters from a higher-resolution dataset that is capable of better resolving some of the storm specific predictors, it may be possible to improve SPIKE2’s skill even further. Nonetheless, if even the level of performance by SPIKE2 with the GEFS reforecast data can be maintained when adapting SPIKE2 for operations, it would surpass the ability of any known guidance specifically targeted for deterministic IKE prediction.

## Acknowledgments

Thanks to Robert Hart, Phillip Sura, Allan Clarke, Ming Ye, and Mark Bourassa for their helpful comments and feedback. This work was graciously supported by grants from NOAA (NA12OAR4310078, NA10OAR4310215, and NA10OAR4320143). Finally, we greatly appreciate the constructive comments and suggestions given to us by two anonymous reviewers during the submission process.

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