1. Introduction
The inherent uncertainty in tropical cyclone (TC) tracks (or position) has led to the desire to use ensemble-based estimates of the distribution of TC tracks as a forecasting tool. These ensemble-based estimates of the distribution of TC tracks can be used to predict the uncertainty in a TC track forecast as well as the probability that a TC may impact a particular region of the globe (Majumdar and Finocchio 2010; Dupont et al. 2011; Hamill et al. 2011). Another potential application of ensembles of TCs is that of targeting or adaptive sampling (Majumdar et al. 2011).
As with all forecasts, the quality of the forecasted track distribution depends on the accuracy of the forecasting system. Because all real-world forecasting systems are flawed this leads to errors in the prediction of the location and shape of the ensemble distributions. Errors in model formulation as well as the relatively coarse resolution of the model can lead to TC track distributions that have too little variance and a bias with respect to the true TC position. An example of a forecasting system with just such an underdispersive and biased TC track distribution is provided in Fig. 1. The ensemble forecasting system shown in this figure is the Navy Operational Global Atmospheric Prediction System (NOGAPS), which was operational for the latter part of the 2011 TC season as well as the entire 2012 TC season and includes 20 ensemble members (McLay et al. 2010). Figure 1 reveals that the typical bias in this ensemble forecasting system is in the tens of nautical miles and that the biases are distinctly different when calculated in the east–west versus north–south directions. In addition, the underdispersiveness in the ensemble’s variance (calculated with the bias removed from the error) of TC positions also depends on the direction in which the errors are calculated and at a forecast lead of 120 h the underestimate of the variance is quite small in the north–south direction and is about 75 n mi (1 n mi = 1.852 km) in the east–west direction. This underdispersiveness and persistent bias of the TC tracks is the feature of the ensemble distribution that we aim to control here.
Bias and variance of TC tracks. (a),(b) The ensemble mean track bias (n mi) in the north–south and east–west directions for the 2012 combined Atlantic and western Pacific basins as a function of forecast lead time. A positive bias in (a) refers to being too far north and in (b) being too far east. (c),(d) The track error variance about the mean and in the same directions as calculated against the best track data (black line) as well as the ensemble variance about the mean (gray line).
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
Recent work in ensemble forecasting has focused on validating the impact of general forms of stochastic forcing on TC forecasts (e.g., Snyder et al. 2011; Lang et al. 2012) and has shown some beneficial impact. We aim here to specifically address the basic issues of underdispersiveness and biases in ensemble-based TC track distributions through a stochastic parameterization that induces TCs to undergo Brownian motion. Because a characteristic of Brownian motion is an increasing ensemble position variance with time this allows for the inflation of forecasted distributions by user-defined amounts. The proper application of a stochastic parameterization, however, requires a choice of stochastic calculus. There exist two standard stochastic calculi that are commonly studied in the theory of stochastic differential equations (Kloeden and Platen 1991). The first is that of Itô (1951) and the second is that of Stratonovich (1966). The most important point about the choice of stochastic calculus is that each one implies a distinctly different algorithm is required to obtain a particular result. The algorithmic differences implied by the choice of stochastic calculus and their impact upon the structure and life cycle of TCs are the subject of this manuscript. In the course of this work we will show that the naïve implementation of a stochastic parameterization without properly accounting for the appropriate stochastic calculus will lead to undesirable results. In the cases presented here these undesirable results will manifest as overly intense TCs, which, depending on the strength of the forcing, could lead to problems with numerical stability and physical realism.
This manuscript is organized as follows. In section 2 we illustrate the basic idea on a simple, one-dimensional advection equation and then we follow with the details of the algorithm for operational NWP models. In section 3 we illustrate the broad potential that the stochastic parameterization of section 2 has on controlling the structure of a TC track ensemble distribution. Section 4 closes the manuscript with a brief summary of the main results and a discussion of possible improvements and applications.
2. Stochastic parameterization
This section will comprise three subsections. In the first we will present the theory of a stochastic parameterization that induces waves to undergo Brownian motion in a simple model setting and in the second we describe a few numerical methods appropriate for stochastic parameterizations of this type. In the third we will describe the algorithm we used in an operational numerical weather prediction (NWP) model.
a. Theory and simple example
Five example Brownian motion paths for the 100 n mi case of section 3b.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
Equation (2.6) is a stochastic differential equation (SDE) and as such the way the stochastic term is evaluated is critical to determining the solution of the equation. We will discuss two ways to evaluate the stochastic term. The first will be that of Stratonovich (1966) in which the function F in the stochastic term in (2.6) is evaluated at the center of each time step. The second way is that of Itô (1951) in which the function F in the stochastic term in (2.6) is evaluated at the beginning of each time step. A brief review of the differences between these two perspectives is provided below as applied to our stochastic parameterization. A more comprehensive review of the differences between the Itô and Stratonovich stochastic calculi may be found in Kloeden and Platen (1999) and a summary of the differences as it pertains to NWP can be found in Hodyss et al. (2013).
Note that the wave defined by (2.7) undergoes Brownian motion in its amplitude and/or phase. When the parameter σ is purely real the wave in (2.7) will undergo random variation in its amplitude, but when σ is purely imaginary the wave in (2.7) will undergo random variation in its phase. Whether σ is real or imaginary is determined by the state-dependent forcing function 
In other words, the evaluation at the beginning of each time step of a stochastic parameterization that induces waves to undergo Brownian motion will lead to two distinct differences away from (2.7), and therefore away from the desired result, if the Itô correction is not employed. The first is simply an increase in amplitude of the wave. The second results from the wavenumber dependence of the diffusion term, which implies an increase in the amplitude of higher wavenumbers such that a whiter spectrum of waves is subsequently obtained. We will see that these two effects will lead to stronger TCs and, because of the wavenumber dependence, narrower, more compact TCs when the stochastic parameterization is evaluated at the beginning of each time step and without the Itô correction.
b. Numerical methods
We will describe two common ways of applying this parameterization in an NWP setting. For simplicity, and in the interest of brevity, we begin by writing the numerical procedure only for the stochastic terms; the integration method of the nonstochastic (dynamics) terms does not affect the integration of the stochastic terms (Hodyss et al. 2013).
c. NWP model formulation
The NWP model we use is the Navy Global Environmental Model (NAVGEM), which is a primitive equation spectral model with a hydrostatic, three-time level, semi-Lagrangian/semi-implicit dynamical core (Ritchie 1991). NAVGEM employs an extensive set of physical parameterizations, including a planetary boundary layer scheme based on Louis (1979), Webster gravity wave drag (Webster et al. 2003), a four-layer land surface model (Hogan 2007), simplified Arakawa–Schubert deep convection (Arakawa and Schubert 1974), the shallow convection scheme of the National Centers for Environmental Prediction Global Forecast System (Han and Pan 2011), the Rapid Radiative Transfer Model for general circulation models (Clough et al. 2005), and a two-species prognostic total cloud water scheme based on Zhao and Carr (1997). All integrations are run at a horizontal resolution of T239 (approximately 50-km resolution) and with 50 vertical levels. A time step of 360 s is used for these integrations. The initial conditions for NAVGEM are obtained from the Naval Research Laboratory (NRL) Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR), which is a four-dimensional variational data assimilation system (4DVAR; Xu et al. 2005; Rosmond and Xu 2006; Chua et al. 2009).
The terms in (2.19) are added to each of the five prognostic equations of the dynamical core of NAVGEM: relative vorticity, divergence, virtual potential temperature, specific humidity, and surface pressure. The two random numbers in 
3. Experiments
The experiments of this section are constructed to illustrate the differences between the stochastic calculus of Itô and Stratonovich in a NWP model, as well provide evidence of the broad potential a stochastic parameterization of the form illustrated in the previous section has on controlling the shape (mean and variance) of a TC track ensemble distribution.
The following experiments will make use of simulations of Hurricane Isaac (2012), which was a slow-moving tropical cyclone that caused severe damage in the Caribbean and along the northern Gulf Coast of the United States. The simulations presented below begin at 0000 UTC 22 August 2012 and extend for 5 days. During this time Isaac tracked along the Leeward Islands through Hispaniola to near Cuba. Figure 3 shows the NAVGEM (no stochastic parameterization) representation of Hurricane Isaac at three forecast lead times and in the 850-mb geopotential height field. The strong subtropical high to the east and north of Isaac will be shown to be a strong influence on the resulting track forecasts.
The 850-mb geopotential height fields from the control NAVGEM simulation at a forecast lead time of (a) 0, (b) 72, and (c) 120 h. Contour interval is 20 m.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
We track the location of the TCs using the Marchok (2002) tracker. This TC tracker uses a variety of input variables [including absolute vorticity, winds, mean sea level pressure (MSLP), etc.,] to track the TC. Because of the prediction from the stochastic theory that the intensity of the TC will be modified by the Itô calculus, we also present values for minimum central pressure and maximum surface wind speed to evaluate the intensity of the TC. We have also calculated volume-averaged values of kinetic energy, squared divergence, and squared surface pressure tendency in a volume centered on the TC and extending for a 1000 km around the TC, but do not show these norms as they behaved identically to the minimum central pressure and maximum surface wind speed.
a. Bias
The goal of this section is to show that the stochastic parameterization of the previous section can control the “bias” in the position of an ensemble of simulations. To this end, we set α = 0 and vary the bias parameters 
Note that a shift in the position of the TC of 800 n mi at τ = 120 h implies a speed bias of 3.4 m s−1, which corresponds to a shift of 1.2 kilometers per time step. Because we are shifting the TC a very small fraction of the distance of a grid cell each time step we need not concern ourselves with issues related to instability from violation of the Courant–Friedrichs–Lewy (CFL) condition. Last, all simulations begin with the identical initial condition of the control simulation such that all differences are a result of the new bias terms.
Figure 4 shows the result of shifting the TC track using the bias terms in (2.19) only. The result of the application of these terms to the track of the TC does not result in the exact specified shift 
Track bias simulations and intensity changes. (left) The north–south bias simulations and (right) the east–west bias simulations. (a),(b) The tracks for simulations using different desired track biases. (c),(d) The minimum central pressure as a function of forecast lead time in hours. (e),(f) The maximum surface winds (kt, 1 kt = 0.5144 m s−1) as a function of forecast lead time in hours.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
Realized vs desired bias. The simulations for the (a) north–south bias and (b) east–west bias. Open circles show the calculated north–south bias and crosses show the calculated east–west bias at a forecast lead time of 120 h.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
Figure 4 also shows the impact on intensity from the TC traveling along these different tracks using two common metrics: maximum surface wind speed and minimum central pressure. The variations in these two metrics reveal the variability induced by moving the TC into different environments. One can see that the storms that moved to the west and south were generally less intense than those that tracked to the east and north. The storms that tracked to the east and north deepened their central pressure quite significantly but did not increase their surface winds as significantly. We believe that this is due to the storms that moved east and, more importantly, north undergoing extratropical transition with a commensurate drop in central pressure but also an increase in radial extent of the storm such that the pressure gradient across the storm was maintained and hence no significant surface wind increase. In addition, a number of the simulations that tracked to the west lead the TC to track across Hispaniola and Cuba, which leads to significant interactions with topography that also lead to a weakening of the storm for those cases. A specific example of this can be seen in the 
b. Variance
In this subsection we will set 
Because the noise w has units of s−1/2 the variance parameter α has units of m s−1/2. Hence, we will again define this parameter relative to the τ = 120-h forecast lead time as 
Figure 6 shows the 20 ensemble member TC tracks obtained from the d = 100 n mi simulation for both the Itô and the Stratonovich algorithm. One can see in Fig. 6 that the character of the tracks is not distinctly different between the Itô and the Stratonovich algorithms. This feature was discussed in section 2 as the predicted differences between the two algorithms were entirely in the amplitude of the resulting TCs. So, while the character of the tracks is not distinctly different for the two stochastic calculi the intensity of the storms is clearly different (Fig. 7). Figure 7 shows that the use of the calculus of Itô leads to distinctly more intense TCs than that of Stratonovich. Further proof of this is shown in Figs. 8 and 9 where we compare ensemble members 1 through 5 for both the Itô and the Stratonovich simulations to the control simulation in absolute vorticity at the 850-mb level. The Stratonovich simulations of Fig. 9 reveal an ensemble of TCs that appear similar to the control simulation but shifted to new locations. In addition, there appears in both algorithms and the control a positive–negative vorticity couplet on the leeward side of Hispaniola consistent with vortex shedding from an obstruction in the counterclockwise flow of the TC. There are two main points to be taken from Figs. 8 and 9: 1) the application of the stochastic parameterization has not caused significant distortion to the shape of the TC and 2) the major difference between the TCs is that the Itô TCs are more intense and have a generally tighter vorticity profile consistent with the arguments of section 2. Last, note that the position of, say, member 1 (cf. Fig. 8a to Fig. 9a) is different between the two algorithms. This is not a result of differences in the stochastic calculi but actually a result of different random numbers being drawn by the random number generator during the simulation resulting in a different Brownian motion.
Tracks for the d = 100 n mi simulation. Red tracks are for the Itô stochastic calculus and blue tracks are for the Stratonovich calculus. Thick black line is the control simulation.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
Intensity measures for the d = 100 n mi simulation. Red tracks are for the Itô stochastic calculus and blue tracks are for the Stratonovich calculus. Thick black line is the control simulation.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
A comparison of ensemble members (a) 1, (b) 2, (c) 3, (d) 4, and (e) 5 for the Itô calculus to the control simulation for the 850-mb absolute vorticity at τ = 120 h. (f) The control is shown.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
As in Fig. 8, but for the Stratonovich calculus.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
Figure 10 shows the ensemble mean intensity for each of the different experiments of this section except for d = 200 n mi. These ensemble mean intensity measures reveal that the differences in intensity between the two stochastic calculi are well predicted by the Itô correction, in particular the quadratic dependency on ρ (and therefore d) can be seen by the larger difference between the Itô and the Stratonovich algorithms in the 100 n mi simulations than the 50 n mi simulations.
Ensemble mean intensity measures for all stochastic simulations. Solid lines are for the Itô stochastic calculus and dashed lines are for the Stratonovich calculus. Red is for d = 25 n mi, blue is for d = 50 n mi, and green is for d = 100 n mi.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
For the d = 200 n mi simulation we found that the behavior of the TCs became quite erratic and the intensity of the storms would spontaneously increase and decrease too rapidly to be physically sensible. We attribute this behavior, not to a difference in the stochastic calculi (indeed both calculi had this effect at this level of forcing), but in fact to simply CFL violations from inducing the TC to jump too far in a time step. Hence, the algorithm of section 2 cannot be used to increase the standard deviation of the ensemble’s track distribution by an amount too much above 100 n mi, at least for the time step and grid resolution chosen here. Nevertheless, because the underdispersiveness of real ensemble forecasting systems is typically less than 100 n mi at τ = 120 h, and this level of forcing produced realistic TCs, the parameterization is still viable to ameliorate issues with TC track distributions.
A possible manner in which to eliminate CFL violations from the use of these new terms is to reformulate the Brownian motion terms of (2.19) into the semi-Lagrangian advection calculation. This would entail modifying the departure point calculation routine to include the modified wind speed implied by the terms in (2.19). This would result in the possibility of performing substantially larger shifts without concern for CFL violations.
Finally, Fig. 11 presents a comparison of the realized ensemble standard deviation plotted against the desired standard deviation d at τ = 120 h. This figure shows that the realized standard deviation of the track errors approximates reasonably well the desired standard deviation d, at least to within what is reasonable for a 20-member ensemble.
Realized standard deviation vs desired standard deviation. Simulations using (a) the Itô calculus and (b) the Stratonovich calculus. Open circles are the north–south standard deviation and crosses are the east–west standard deviation.
Citation: Monthly Weather Review 142, 5; 10.1175/MWR-D-13-00299.1
4. Summary and conclusions
This article has shown that the choice of stochastic calculus is an important factor in designing a stochastic parameterization. The two most common stochastic calculi (Itô and Stratonovich) were used in a stochastic parameterization within an operational numerical weather prediction model. This stochastic parameterization induces a TC to undergo Brownian motion such that an ensemble derived from it may be tuned to produce an accurate TC track distribution. The methodology allows for the tuning of both the bias and variance of the TC track distribution to user-specified values.
The Itô correction was shown to be a useful tool to design a stochastic parameterization that delivers the Stratonovich result. In the case of stochastic advection terms the Itô correction implies a diffusion term. Note, however, that other stochastic terms will require a different form for the Itô correction and must be tailored to the user’s particular application. The result of the application of the Itô correction on the simulations presented here was to properly center the intensity of the ensemble of shifted TCs on the control (nonstochastic) simulation. Note, however, that because TCs at the typical resolutions of global, operational NWP models are generally too weak one might consider reducing the amplitude of the Itô correction to produce an ensemble with stronger TCs whose intensity may be more in line with the true intensity. Similarly, we feel it is important to point out that in this application we chose to create a stochastic parameterization consistent with the Stratonovich calculus and hence the Itô calculus delivered the “wrong” result. However, some physical processes may in fact be more consistent with the Itô calculus and, therefore, it may be easier to design a stochastic parameterization in those cases starting from the perspective of Itô. This examination of typical physics routines and the determination of whether they behave more like Itô or Stratonovich will be the subject of future work.
We envisage two types of practical applications of a stochastic parameterization of this form. In the first we envisage that this parameterization could be implemented in an ensemble forecasting system. Given estimates of the bias and underdispersiveness of the ensemble forecasting system this algorithm could deliver distributions that produce more accurate track uncertainty forecasts. To be operationally useful the algorithm would need to be configured to automatically track one or more TCs given an input file of the initial TC locations within the ensemble at the analysis time.
Second, we imagine one could use this type of parameterization and others like it, as a science tool to answer questions about the sensitivity of a forecast to small changes in the track of a forecasted flow feature. Note that there is nothing specific to TCs in the technique, but rather this parameterization could be applied to any localized flow feature that one wished to include greater track variance in the ensemble or desired greater understanding of the sensitivity to small changes in position. The life cycles of convective cells and cutoff lows at midlatitudes are two separate phenomena that might be interesting applications of these techniques. Work in these directions is already under way.
Acknowledgments
We gratefully acknowledge support from the Chief of Naval Research PE-0601153N.
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