## Abstract

Probabilistic risk assessment systems for tropical cyclone hazards rely on large ensembles of model simulations to characterize cyclones tracks, intensities, and the extent of the associated damaging winds. Given the computational costs, the wind field is often modeled using parametric formulations that make assumptions that are based on observations of tropical systems (e.g., satellite, or aircraft reconnaissance). In particular, for the Northern Hemisphere, most of the damaging contribution is assumed to be from the right of the moving cyclone, with the left-hand-side winds being much weaker because of the direction of storm motion. Recent studies have highlighted that this asymmetry assumption does not hold for cyclones undergoing extratropical transitions around Japan. Transitioning systems can exhibit damaging winds on both sides of the moving cyclone, with wind fields often characterized as resembling a horseshoe. This study develops a new parametric formulation of the extratropical transition phase for application in risk assessment systems. A compromise is sought between the need to characterize the horseshoe shape while keeping the formulation simple to allow for implementation within a risk assessment framework. For that purpose the tropical wind model developed by Willoughby et al. is selected as a starting point and parametric bias correction fields are applied to build the target shape. Model calibration is performed against a set of 37 extratropical transition cases simulated using the Weather Research and Forecasting Model. This newly developed parametric model of the extratropical transition phase shows an ability to reproduce wind field features observed in the western North Pacific Ocean while using only a restricted number of input parameters.

## 1. Introduction

Whether it serves disaster adaptation and mitigation organizations or the insurance industry, there is an increasing demand for probabilistic estimates of the potential impact from rare natural catastrophic events. For that purpose, risk assessment systems rely on very large ensembles of model simulations that extrapolate historical records. They are designed to represent extremely long periods of natural hazard activity [e.g., ~(10^{4}–10^{5}) yr] to ensure very high return period events are sampled. Within this context, there are trade-offs to be made between complexity and computational efficiency. The processes under study need to be reduced to a limited set of characteristic parameters, and in the case of tropical cyclone risk (Powell et al. 2005; Emanuel et al. 2006; Vickery et al. 2009a) these include information about the cyclone trajectory (e.g., cyclone position, heading direction, translational speed), its intensity (e.g., central pressure deficit, maximum wind speed), and the extent of the associated wind field using size indicators such as the radius of maximum wind and other model-specific shape parameters [e.g., parameter *B* in the Holland (1980) formulation, *X* in the Rankine vortex formulation of Phadke et al. (2003), or X1 and *n* in Willoughby et al. (2006)]. In most cases, probability distributions are fitted to the available data to characterize the climatology of each parameter while statistical models are used to constrain their interdependency and time evolution for a given track (e.g., Vickery et al. 2000, 2009b; James and Mason 2005; Hall and Jewson 2007). Since these modeled variables are only defined on a discrete set of track points, a spatial formulation is required to specify the wind hazard field around the cyclone center. When the wind field extends over land, roughness coefficients are applied to obtain estimates of damaging gusts.

A wide range of wind formulations has been used, which all involve specific input shape parameters (MacAfee and Pearson 2006). While the first category of tropical cyclone (TC) models have focused on methods to simulate winds from the specification of the pressure gradient field and gradient wind assumptions (e.g., Holland 1980; Georgiou 1985), more recent attempts have targeted the wind field directly (e.g., DeMaria et al. 1992; Phadke et al. 2003; MacAfee and Pearson 2006; Willoughby et al. 2006). As with all parametric modeling exercises, there is a clear trade-off between the complexity of the formulation and the number of parameters involved. Given the data requirements associated with the development of statistical models, the number of shape parameters has to be kept to a minimum while allowing for a formulation able to characterize wind field variability. Regardless of the model, the common assumption is that TC winds result from the superposition of an axisymmetric profile (i.e., when expressed within the framework of the moving TC) and the forward motion of the storm. Consistent with TC wind observations from satellite products and aircraft reconnaissance (Shea and Gray 1973; Powell et al. 1998; Willoughby and Rahn 2004), such a framework leads to wind fields that exhibit a strong left–right asymmetry in wind magnitude relative to the track path. For risk assessment purposes, this implies that the most damaging winds only extend to the right of the moving TC (in the Northern Hemisphere).

However, some recent studies focusing on the western North Pacific Ocean (WNP) have shown that wind fields from cyclones undergoing an extratropical transition can exhibit very distinct characteristics when compared with mature tropical systems (Fujibe and Kitabatake 2007; Kitabatake and Fujibe 2009; Loridan et al. 2014). Several factors may contribute to this change in wind field structure: as the TC moves into the midlatitude environment, it faces an increase in the environmental baroclinicity and vertical wind shear while it may also interact with existing mesoscale systems or a trough in the upper-level jet stream (Harr and Elsberry 2000; Jones et al. 2003). As a consequence, the complex interactions that occur between the TC vortex structure and the midlatitude circulation can lead to a significant increase in surface wind field asymmetries, with important implications for risk assessment. For instance, using 31 yr of reanalysis data, Loridan et al. (2014) find that the asymmetry assumption at the core of TC parametric models is violated for ⅔ of extratropical transition cases around Japan. Such cases are labeled left-hand-side contributions (LHSCs) and the associated wind fields exhibit strong winds on both sides of the moving cyclone. The maximum magnitude can be located either to the right or to the left in a pattern described as a horseshoe (see, e.g., Fig. 1a). Given that ~40% of WNP cyclones transition (Kitabatake 2011), these findings imply that risk assessment systems focused on the WNP are in need of a new methodology for the extratropical transition phase that can capture the full extent and variability of damaging winds. More specifically for Japan, which represents the main area of exposure for the WNP, analysis of the information reported by the International Best Track Archive for Climate Stewardship (IBTrACS; Knapp et al. 2010) reveals that ~85% of storms making landfall experience an extratropical transition at some point in their lifetime.

Although using fully numerical weather prediction (NWP) models might appear as a natural step to better represent complex wind field structures, a large number of challenges still exist before such types of models can reasonably be considered for the generation of large risk assessment ensembles (Vickery et al. 2009a). Setting aside the obvious computing and run-time limitations, the lack of control regarding the simulated track paths and wind intensities (Davis et al. 2010; Cavallo et al. 2013) represents a major issue. When developing TC track ensembles, there is a need to ensure that the proportion of simulated events making landfall in a given area and with a given intensity matches what has been observed or can be extrapolated from historical records (e.g., see Figs. 3, 5, and 6 in Vickery et al. 2009b). An even more challenging and often targeted aspect is to maintain a similar level of correlation in TC intensity between regions of high exposure and what can be inferred from historical data. Therefore, once the track parameters have been simulated to match the desired historical statistics, it is essential that the wind model be able to maintain a wind field consistent with these values (e.g., location and magnitude of central pressure deficit and maximum wind velocity); although recent efforts to apply state-of-the-art data assimilation techniques to NWP provide promising results (Wang 2011; Weng and Zhang 2012; Schwartz et al. 2013), the current state of NWP does not yet provide a tool flexible enough to satisfy these criteria over the range of simulations required by a risk assessment system. For these reasons a methodology to account for wind field specificities during the extratropical transition phase using parametric models is preferred. The simplicity of such formulations also facilitates their implementation as part of an ensemble data assimilation system [e.g., particle filters (van Leeuwen 2009)]; these are often involved when reconstructing past historical events based on surface wind observations.

In this study, we develop a new formulation designed specifically 1) to characterize wind fields from cyclones undergoing extratropical transition in the WNP and 2) that can be applied to generate large ensembles within a risk assessment system. The formulation aims to capture the horseshoe shape using a parametric approach based on a limited number of parameters. With most satellite wind products limited in their ability to represent wind fields during extratropical transition [e.g., see methods in Mueller et al. (2005)], and given the lack of aircraft reconnaissance of transitioning systems in the WNP (Kossin et al. 2007; Barcikowska et al. 2012), the development of such a model requires an alternative source of data. The WRF Model is here run for a set of selected historical transitioning WNP cyclones. The WRF Model simulations are presented in section 2. The extratropical transition wind field formulation is introduced in section 3, where it is calibrated against the WRF wind targets and tested for an example case. Note that although the current study focuses on the WNP, the methodology presented throughout the paper can be reproduced for other areas; in particular given the atypical wind fields observed from the recent case of Hurricane Sandy (2012), and given the concentration of risk exposed to transitioning system in the northeast of the United States, attention should be given to the situation in the North Atlantic Ocean.

## 2. Calibration targets: WRF simulations

The overarching goal of this study is to introduce a methodology for capturing the wind field shape from extratropical transition cases in the WNP. In this section, the steps taken to assemble a database of relevant target shapes are discussed.

First, a set of 37 historical cyclones experiencing extratropical transition around Japan between 2000 and 2011 are run using version 3.3.1 of the Advanced Research core of the WRF Model (ARW; Skamarock et al. 2008). These are selected to cover a range of transitioning cases capable of producing damaging winds over Japan. The physics options selected in WRF are the Yonsei University planetary boundary layer scheme (Hong et al. 2006), the Kain–Fritsch cumulus parameterization (Kain and Fritsch 1990), the Rapid Radiative Transfer Model longwave radiation model (Mlawer et al. 1997), the Dudhia shortwave radiation scheme (Dudhia 1989), and the WRF single-moment five-class microphysics scheme (Hong et al. 2004). The simulations use two nested domains (two-way nesting) with 36 vertical levels. The innermost, vortex-following domain covers 2316 km^{2} with a grid spacing of 12 km while the outermost fixed-parent domain has a resolution of 36 km. Initial and boundary conditions are taken from the Global Forecast System final analysis (1° resolution; 6 hourly), and analysis nudging is applied to the parent domain (nudging intensity = 0.003 s^{−1}). Note that further nests were initially considered in an attempt to better capture storm intensity. Yet, given the limited impact of increased resolution on the simulation of wind radii and asymmetries (Davis et al. 2010), the above settings were selected for reasons linked to computational costs and ease of implementation. As the main objective of the current study is to introduce a first formulation for a model of surface winds during extratropical transition, the priority is to capture the shape and extent of transitioning TC wind fields across a wide range of cases. The 12-km setting therefore provides a good compromise to ensure the whole extent of the storm is contained in the innermost domain (i.e., it contains at least 4 times the radius of maximum winds around the storm center; see section 3b) while still allowing simulation of an ensemble of 37 cases in a reasonable amount of time. Although there is a risk that the above setting might lead to a bias in simulated intensity for storms rated as category 3 or greater (Davis et al. 2010), these are rare for extratropical transition cases; note also that the model formulation presented in section 3 is designed to represent the wind field shape and decay independently of the maximum wind velocity and should, therefore, extend to storms of any category as long as the shape and decay rate of the wind fields simulated from the WRF runs can be assumed to represent that of more intense storms. All the methodology presented in this paper can be reproduced with higher-resolution WRF simulations in order to refine the proposed parametric formulation of section 3.

Similarly, while we aim to reproduce the historical tracks and wind field intensities as closely as possible with these model settings, it is important to note that there is no need for an exact match with observations. The objective here is to construct a database of realistic transitioning wind field targets. Deviations from the historical cases simulated are acceptable provided the physics of the extratropical transition phase is correctly represented (i.e., simulation of an alternative realization of the historical transition within the same midlatitude environment is suitable for the purposes of this study). The use of analysis nudging on the parent domain will ensure that the simulated track does not differ noticeably from the historical one, keeping the interactions of the TC with the midlatitude flow similar to the observed cases (e.g., see the discussion below on the position of the TC relative to the jet stream–baroclinic zone for both WRF and the CFSR product in Fig. 1). Past studies, both in the Atlantic (Davis et al. 2008a, 2010; Torn 2010; Cavallo et al. 2013) and the Pacific (Wu et al. 2010, 2012) basins, have shown the ability of WRF to simulate tropical cyclone dynamics with settings similar to those above. For the extratropical transition phase, Davis et al. (2008b) point out in their simulation of six Atlantic transitioning hurricanes that WRF realistically replicates the observed structural changes.

Second, as a way of providing some first-order confirmation of the ability of the simulations to capture the key dynamics of the extratropical transition phase and, more importantly, allow estimation of the start and end times of the transitioning window, the Cyclone Phase Space (CPS) analysis of Hart (2003) is applied to all 37 storms. The cyclone thermal symmetry parameter *B* [note that this is different than the *B* parameter of Holland (1980)], defined by Hart (2003) as the storm-motion-relative 900–600-hPa thickness asymmetry across the cyclone within 500-km radius, is used to characterize the frontal nature of the storm (symmetric/asymmetric structure; see vertical axis of Fig. 2); additionally, the vertical derivative of the horizontal height gradient for the same layer (−Vtl) characterizes the vertical structure of the cyclone at lower levels and allows distinction between cold-core and warm-core systems (horizontal axis in Fig. 2). The evolution of a cyclone through its life cycle is analyzed from the diagram of *B* versus −Vtl (see example from Fig. 2), where thresholds values of *B* = 10 and Vtl = 0 are used and a 4-h running-mean smoother is applied. The reader is referred to Hart (2003) for details on the topic and we here briefly describe the main case of interest: for a TC undergoing extratropical transition, the expected evolution is from the bottom-right quadrant defined as *B* < 10 and −Vtl > 0 (i.e., symmetric/warm core) through the top-right quadrant (*B* > 10, −Vtl > 0; i.e., asymmetric/warm core) and finishing in the top-left quadrant (*B* > 10, −Vtl < 0; asymmetric/cold core). The extratropical transition phase, which is the focus of the current study, corresponds to the top-right quadrant.

The overall analysis methodology is now illustrated with the case of Typhoon Rammasun from 2008 (Fig. 1), which was studied as an example of a typical LHSC storm by Loridan et al. (2014) using data from the Climate Forecast System Reanalysis (CFSR; Saha et al. 2010). The surface wind fields, synoptic situation, and upper-level dynamics from both the CFSR dataset and the WRF simulations at 0600 UTC 12 May 2008 are compared in the top and bottom panels of Fig. 1. According to the phase-space analysis (Fig. 2), at that time and date the TC is midway through its extratropical transition (i.e., in the top-right, asymmetric/warm-core quadrant; see red square in Fig. 2). Although the position of the TC simulated by WRF is not at the exact same location as for the CFSR, the general dynamics of the transitioning process are realistically simulated with the TC entering a strong baroclinic zone at the surface (Figs. 1b,e) while the top plot shows it is close to the right entrance of the jet stream (Figs. 1c,f). For the surface winds (Figs. 1a,d), the general shape and extent of the field simulated by WRF are consistent with CFSR. The maximum wind magnitude is higher than with the CFSR, which can be due to a higher horizontal resolution (12 km for WRF as compared with ~38 km for CFSR). Most important for our purpose, the area of strong winds on either sides of the track is well simulated with the maxima left of the track. The rest of the study focuses on the development of a parametric formulation able to reproduce similar types of patterns.

For each of the 37 cases selected the CPS is computed (i.e., as in Fig. 2) and WRF hourly snapshots of 10-m winds that fall into the top-right (asymmetric/warm core) quadrant are kept. This leaves a total of 1063 snapshots. Note further filtering of this database is applied before calibration (see section 3a). This setup is designed to specifically target the extratropical transition phase in the CPS, which is key to assessing wind damage risks for the main area of exposure in the WNP (Japan).

## 3. Parametric wind model

The proposed methodology is based on a simple parametric formulation following the work of Willoughby et al. (2006) along with a parametric bias correction designed to force the modeled wind field toward specific target shapes. After an overview of the methodology in sections 3a–c, a summary of the results is presented in section 3d with an example of application in section 3e.

### a. Categories of target transitioning wind fields

The WRF simulations from section 2 are used to calibrate the raw wind model presented in section 3b. For calibration purposes, all WRF wind snapshots are put on a standardized grid spanning 8 times the radius of maximum winds *R*_{max} in both directions (see, e.g., Fig. 3a). In the rest of the study all positions on this standard grid are referred to as an (*x*, *y*) pair in Cartesian coordinates, with *x* increasing from *x* = −4*R*_{max} to *x* = +4*R*_{max} (from left to right on the grid) and *y* increasing in the orthogonal direction from *y* = −4*R*_{max} to *y* = +4*R*_{max}. Each target is defined by positioning the storm center at the origin and rotating the wind field to relocate the maximum velocity at point (*x* = 0, *y* = *R*_{max}). Note that a Gaussian kernel smoothing is applied to the raw WRF wind outputs with a length scale of 10 km. A series of filtering criteria are also applied to the database of hourly WRF snapshots to keep only targets that 1) have maximum instantaneous 10-m wind > 22.35 m s^{−1}, 2) are over water, and 3) fit in the standard grid, after rotation (i.e., the WRF target domain has to cover at least a radius of 4(2)^{1/2} × *R*_{max} from the storm center). These criteria leave 557 WRF targets after filtering. Note that with this set of calibration targets the resulting model will be valid for overwater winds only and estimates over land require the addition of a roughness parameterization.

Given the variety of wind fields observed for the extratropical transition phase (Kitabatake and Fujibe 2009), several model formulations are here considered. This is done by splitting the target wind field shapes into subcategories. From 31 yr of reanalysis data in the WNP, Loridan et al. (2014) identified two groups of transitioning storms using the relative ratio between wind maxima on both sides of the track as a measure of asymmetry in the wind field. A similar ratio is used here for the WRF targets:

where *V*_{max} is the maximum wind located at (0, *R*_{max}) on the standard grid while *V*_{max.opp} represents the maximum in the *opposite* sector defined by *y* < −*R*_{max}. The Δ*V*_{max} values are used to split the WRF targets into the following two categories:

The right-hand side only (RHSO) category is composed of all WRF targets with Δ

*V*_{max}≥ 0.20 and a maximum wind magnitude located to the right of the moving storm (e.g., Fig. 3a; 204 snapshots).The left-hand side contribution (LHSC) category represents all other snapshots and is split into two subgroups: LHSC-RM have the maximum to the right of the moving storm (e.g., Fig. 8a; 175 snapshots) and for LHSC-LM it is located to the left (e.g., Fig. 13a; 178 snapshots).

### b. Wind profile formulation

To simulate the target shapes, the core model from Willoughby et al. (2006) is used as a starting point and two simple modifications are integrated (see below). Within the framework of the moving cyclone, the model can be expressed as

where *V*_{mf} is the tangential wind component (in the moving frame) at a radial distance *r* from the storm center and *V*_{max} and *R*_{max} represent the magnitude and radius of maximum wind while X1 and *n* are two shape parameters introduced by Willoughby et al. (2006). In physical terms, X1 conditions the exponential wind decay in the outer vortex, with smaller values leading to steeper gradients, and *n* drives the sharp increase in wind magnitude inside the eye (see Fig. 4). The Eq. (2) profile also introduces a new parameter (Y1) to provide additional control on the wind field decay in the far field. This modification acknowledges that the aircraft data used in the original study by Willoughby et al. (2006) were mostly limited in extent to less than 150 km while the extratropical transition cases we focus on are by nature larger systems. Figure 4 provides an example of the type of wind profiles generated from Eq. (2) with varying values of X1, *n*, and Y1. While an increase in X1 reduces the gradient in tangential wind magnitude throughout the entire area beyond *R*_{max} (shift from solid to dashed lines in Fig. 4), the introduction of Y1 (negative value) enables a more pronounced shift in the far field than in the vicinity of Rmax (cf. blue and red lines in Fig. 4). Additionally, and to simplify the formulation in view of its application within large ensembles of track simulations, the polynomial ramp function employed by Willoughby et al. (2006) has been omitted from Eq. (2). This introduces a sharper transition around the area of maximum winds, and to compensate for this, an extra exponential term is added inside the eye (*r* < *R*_{max}), ensuring the first derivatives are equal as *r* reaches *R*_{max}. As illustrated in Fig. 4 the transition around *r* = *R*_{max} remains very sharp; yet it becomes smoother as X1 increases (e.g., consider the green-shaded area in the inner core and the dashed lines in the outer field in Fig. 4). Note also that the bias-correction procedure detailed in section 3c will help reduce this type of systematic misrepresentation from the raw model formulation of Eq. (2).

Applying the formulation from Eq. (2) for all directions θ in polar coordinates on the standard grid, with X1, Y1, and *n* kept constant, leads to a perfectly axisymmetric two-dimensional flow pattern with tangential wind vectors . This structure characterizes the wind field within the frame of reference moving with the storm (or that of an idealized stationary vortex). To project this axisymmetric structure onto a fixed frame of reference, a correction term is introduced that accounts for the storm translational velocity ():

Within our framework β is set to a constant value (see Harper and Holland 1999) to be determined via model optimization (section 3c). Application of Eq. (3) will lead to a velocity field **V** that presents some level of asymmetries on the two-dimensional grid, with wind magnitudes for points located to the right of the storm motion vector enhanced relatively to the ones on the left (e.g., Fig. 3b). It is important to note that the model formed by Eqs. (2) and (3) is only representative of the mean wind flow pattern at a given height above the surface, without consideration of the impact of surface friction on the wind direction. As the WRF outputs used in our calibration are equivalent to 10-m-high, 1-min-averaged winds over water (see section 3c), this is what the model is considered valid for, and in the rest of the study we focus on the magnitude of the winds simulated by Eqs. (2) and (3) (**V**) without any concept of directionality.

### c. Calibration methodology

The raw wind model as defined by Eqs. (2) and (3) is set to approximate the WRF 10-m wind targets assuming the storm motion vector is along the *x* axis, to the left (e.g., Fig. 3b); that is, a 90° angle is assumed between the storm heading and the location of maximum wind (in other words, the black arrow in Fig. 3a, which represents the true heading direction, is assumed on the *y* = 0 axis, to the left, when first applying the raw model in Fig. 3b). Given that *R*_{max} and *V*_{max} are model inputs, this methodology ensures the core model perfectly captures the maximum winds while model biases can be stored on the common standard grid; these are then used to derive bias correction fields that help capture the rest of the target shape (see below). Because we consider that Y1 and β are part of the model formulation rather than target-dependent shape parameters, they are fixed to the pair of values that minimizes the mean root-mean-square error (RMSE) over the range of targets. The cost function (CF) used in the optimization phase follows the expected shape of the wind field with maximum weights around the location of maximum winds:

where *N* is the number of grid points, *V*_{i} is the wind speed modeled at grid point *i* according to Eqs. (2) and (3), *O*_{i} is the corresponding WRF wind estimate for the same grid point, and *d*_{i} is the distance to the location of maximum wind. In addition, *L* is a kernel length-scale parameter set to 3*R*_{max} to capture the extent of the transitioning wind fields. The term dampens the importance (weights) of the simulated winds as the distance to the *V*_{max} location increases.

For X1 and *n*, the best-fit values for each target are recorded. This forms a training dataset from which values can be sampled when applying the model as part of a risk assessment system. Similarly, the actual angle between the storm motion vector (e.g., black arrow in Fig. 3a) and the location of maximum winds is recorded for each target. The convention in this study is to express the rotation angle (*A*_{max}; Fig. 5) in terms of the departure from the 90° assumption: that is, *A*_{max} = 0° when the maximum winds are exactly at a right angle to the motion vector; to the right, *A*_{max} = 90° is the special case where the maximum winds are along the heading direction at the front of the storm; while for *A*_{max} = −90° they are at the back, opposite to the heading direction. When applying the model, the wind field can then be rotated around the storm center by sampling values from this dataset and relaxing the right angle assumption made during the first stage of calibration. Note that one might also consider using such training data to build statistical relationships and help constrain the evolution of the X1, *n*, and angle values (e.g., autoregression models) when implementing the model as part of a risk assessment system.

Using the overall optimum (Y1, β) pair, a relative bias field is computed for each target *t* on the standard 8*R*_{max} grid:

where *i* is the grid index, *V*_{i} is the wind field from the raw model of Eqs. (2) and (3) with target-dependent best-fit parameters X1(*t*) and *n*(*t*) (e.g., Fig. 3b), and *O*_{i}(*t*) is the WRF target wind estimate (e.g., Fig. 3a). In strict terms, to unbias the model, one would use the *mean relative bias* (MRB) field over the range of targets (e.g., Fig. 3c for the RHSO case). This leads to the following (unbiased) formulation for the wind field (e.g., Fig. 3d):

However, if storm characteristics appear relevant (e.g., if they show some signal that correlates to the model error), an alternative approach is to model the bias field. This will be discussed in more details in section 3d where storm-dependent bias corrections are introduced as a way to capture the horseshoe shape.

### d. Model calibration results

#### 1) RHSO wind model

We here only consider snapshots from the RHSO category (see section 3a). Iterative computation of the average RMSE from the raw model over the RHSO snapshots for a range of Y1 and β values shows Y1 = −0.70 and β = 0.25 as the optimum pair (Fig. 6). Note that although the values were set with a 0.05 increment resolution, refining the search to 0.01 increments only improves the RMSE by less than 0.1 m s^{−1}. The magnitude of the negative Y1 value selected ensures a wide extent for the wind field (see Fig. 4), which is enhanced to the right of the track as a result of the positive β [Eq. (3)]. Figure 3b shows how the application of the model formed by Eqs. (2) and (3) and the optimum (Y1, β) pair approximates the example RHSO WRF target in Fig. 3a. The MRB for the RHSO category (Fig. 3c) highlights the limitations of this type of axisymmetric formulations with large semicircular areas of positive biases. Note also the negative biases observed for *R*_{max} < *r* < 2.5*R*_{max} along the transect of maximum winds (i.e., *x* = 0). After subtraction of the MRB following Eq. (6), the characteristics of the unbiased modeled wind field (*V*_{ub}; Fig. 3d) become much closer to the RHSO target (Fig. 3a) than with the raw profile (Fig. 3b). In particular, the strong asymmetry in the magnitude of the wind on both sides of the track gets accentuated and the wind field is pushed to the right thanks to the negative MRB values beyond *V*_{max} (i.e., blue/green colors around *y* = 2*R*_{max}; Fig. 3c). The modeled wind structure is now consistent with the observed characteristics of RHSO winds.

The distributions of *A*_{max} (from the WRF simulations) and X1, *n* values (from model optimization) are reported in Fig. 7 along with those for the other two target categories, which will be discussed later. For *A*_{max} (Fig. 7a), the 0°–30° bin is the one with the most weight (i.e., maximum winds to the right, slightly at the front of the storm; see Fig. 5), followed by the bin from −60° to −30° and the 30°−60° bin. In terms of X1 (Fig. 7b), close to 90% of the weight is located in the 100–250-km range with values in the 100–150-km bin most frequently selected from the optimization procedure, while best-fit values of *n* (Fig. 7c) are mostly concentrated between 0.5 and 3 with a peak for the one to two value range. A comparison with the best-fit values from the other two categories is provided in section 3d(2).

Note that this RHSO case is not drastically different from the wind field shape observed during the tropical phase (strongest winds restricted to one side of the storm). Although this is not the point of this study, the methodology can also be applied to tropical targets to provide a tropical wind field model that uses the same core model formulation. Given the better data availability, such a model calibration could be performed with targets from satellite products such as the multiplatform data (Knaff et al. 2011) or H*WIND in the North Atlantic (Powell et al. 1998).

#### 2) LHSC wind model

When reproducing this analysis with the LHSC targets, some complications arise for all snapshots with a maximum to the left of the track as they present a very distinct shape (from those with a maximum to the right) when put on the standard grid (cf. LHSC-RM from Fig. 8a and LHSC-LM from Fig. 13a). The opposite orientation in the curvature of the wind field shape (i.e., to the right or left) does not allow the calibration of all LHSC targets together. They are split into two groups (see section 3a) and two distinct models are built.

##### (i) LHSC-RM

The optimum (Y1, β) pair for the LHSC-RM category is Y1 = −0.55 and β = 0.05. The larger Y1 compared to RHSO ensures the wind field is more compact and the very low β maintains winds of similar magnitude on both sides of the storm (Fig. 8b). With LHSC-RM snapshots characterized by a “horseshoe” shape to the rear of the moving storm (e.g., Fig. 8a), the MRB clearly presents larger biases in the bottom-left quadrant of the standard grid than are seen in the bottom-right example (Fig. 9). A closer analysis of the bias for selected grid points shows a dependency on the translational speed. Scatterplots of the model bias as a function of TS for three grid points (see location in Fig. 9) are presented in Fig. 10. With the storm moving left relative to the standard grid, faster TS implies larger wind magnitude around point 1 and translates into larger negative biases (Fig. 10a). This suggests the underestimation from the raw model is accentuated as the forward motion of the storm increases. On the other hand, for points 2 and 3 a systematic positive bias appears (Figs. 10b,c) as a direct consequence of the axisymmetric assumption at the core of the raw model (see Fig. 8b). The bias increases as TS increases and the trend is most pronounced for point 3.

To help capture the horseshoe shape, a storm-dependent bias correction model (BCM) is developed using the translational speed as the key storm characteristic: after computation of the linear regression for *all grid points* on the standard grid, the bias can be decomposed into an intercept field (Fig. 11a) and a slope field (Fig. 11b). The slope field highlights the areas for which the wind magnitude should increase (positive values) or decrease (negative) as the storm moves faster. This methodology also introduces some extra variability into the range of modeled wind fields, which is an advantage when generating large ensembles within the context of risk assessment. The BCM for a given target *t* can then be computed as

The BCM component is then substituted for the MRB in Eq. (6). Figure 12 presents an example of this parametric bias field for a given LHSC-RM snapshot (Fig. 12a) and shows how it impacts the raw model in Fig. 8b according to Eq. (7) to help shape the horseshoe pattern characteristic of LHSC snapshots (Fig. 12b). As for the RHSO case, a large area of negative biases appears along the transect of maximum winds (*x* = 0) for *R*_{max} < *r* < 2.5*R*_{max}. This highlights a systematic underestimation of the wind magnitude from the raw profile of Eq. (2) in this area. Note a similar bias decomposition was tested for the RHSO model but is not selected as it has very little impact on the results.

A shift in the location of the maximum winds to the right rear of the storm is noticeable from the analysis of the *A*_{max} values in Fig. 7a, with a larger contribution of all the negative bins. The largest difference occurs for the bin from −90° to −60°, which indicates that a number of the LHSC-RM cases have their maximum winds at the back of the moving storm. For X1 the best-fit values selected tend to be larger than in the RHSO case, with much of the weight moving to the >250-km range and with the contribution from the 100–150-km bin much less dominant. The analysis of the best-fit estimates for *n* reveals a clear shift toward larger values, therefore indicating a much steeper wind gradient inside the eye (see Fig. 4) than for RHSO targets.

##### (ii) LHSC-LM

The methodology is here repeated for the LHSC-LM category (see Fig. 13). The optimum (Y1, β) pair is Y1 = −0.55 and β = −0.15. The Y1 parameter is identical to the LHSC-RM case, showing some consistency for the extent of the LHSC wind fields. Now, β is negative, which suggests that although the maximum winds are to the left of the track the main area of strong winds is still to the right. As for LHSC-RM, it is via a parametric bias correction field [see Eq. (6)] that the target shape is best approximated (Fig. 13). Note the Fig. 13d wind field has to be rotated clockwise by >90° in the implementation phase to ensure the maximum winds end up to the left of the moving storm (*y* < 0 on the standard grid).

As could be expected from a left-hand-side wind model, the LHSC-LM case clearly stands out from Fig. 7a with most of the *A*_{max} weights in the bin from −120° to −90°. This corresponds to maximum winds located in the left-rear quadrant, as is often observed for horseshoe wind fields (Kitabatake and Fujibe 2009). The distributions of best-fit values for X1 and *n* are mostly in line with the LHSC-RM case.

### e. Applications

As an illustration of how the formulation captures the most important aspects of the wind field shape during extratropical transition, and to complement the examples from Figs. 3d, 12b, and 13d, the case of Rammasun (2008) presented in Fig. 1 is here reconstructed using Eqs. (2)–(7). According to the classification from section 3a, the wind field simulated by WRF is an LHSC-RM case at 0000 UTC 12 May 2008 (Fig. 14a) whereas it is an LHSC-LM at 0600 and 1200 UTC (Figs. 14d,g). Note that this behavior, in which the wind maxima flips from one side of the track to the other during the extratropical transition of a given storm, occurs for most of the LHSC cases in the subset of the CFSR database studied in Loridan et al. (2014).

As this exercise is designed to test the wind field formulation only, the location (*R*_{max}, *A*_{max}) and magnitude (*V*_{max}) of the maximum wind, as well as the best-fit values for X1 and *n*, are provided as input for each of these three targets. The bias fields and (Y1, β) values are taken from the LHSC-RM model for the first target (Fig. 14b) while the LHSC-LM model is applied to the two subsequent targets (Figs. 14e,h). Comparison of the WRF targets with the parametric profile outputs shows the ability of the method to approximate the shape and extent of the LHSC transitioning wind fields. Strong winds are maintained on both sides of the track and the magnitude of the wind decay is realistic when compared to the WRF simulation. To complete the discussion from section 3b, the wind profiles along the transect of maximum winds (white lines in Figs. 14a,d,g) from the storm center up to 3 times the radius of maximum winds are shown for both the raw model of Eqs. (2) and (3) and the bias-corrected model from Eqs. (6) and (7). As highlighted by Figs. 12a and 13c, the largest contribution from the BCM occurs inside the eye for *r* < 0.5*R*_{max}, where the wind magnitude is increased by up to 20% of the *V*_{max} value. For the area immediately after *r* = *R*_{max}, an increase in the wind speeds for the three profiles can also be observed; farther out, the BCM acts to reduce the wind magnitude for the two LHSC-LM cases (Figs. 14f,i) while the contribution is still toward a wind speed increase for the LHSC-RM case (Fig. 14c).

From the point of view of risk assessment studies this suggests that, when provided with realistic inputs, the parametric formulation is able to capture the key aspects of transitioning wind fields (i.e., location and extent of damaging winds). For the formulation to be applicable as part of a risk assessment ensemble, a method is required to sample physically reasonable input estimates. For that purpose the training dataset summarized by Fig. 7 can be used to develop simple statistical models. Additionally, a procedure to link the three distinct model formulations (RHSO, LHSC-RM, and LHSC-LM) consistently with observed climatology is required. For instance one can imagine merging the three components with statistics matching the results presented in Loridan et al. (2014) for their CFSR climatology of extratropical transition cases: for ⅓ of transitioning storms simulated within an ensemble of events the wind field would be modeled using the RHSO formulation (and associated input parameter estimates), while for the other ⅔ a combination of LHSC-RM and LHSC-LM would be used. Although a detailed analysis of the climatology of the LHSC-RM and LHSC-LM phases would be useful to characterize the drivers of the shift in maximum wind location for LHSC cases (e.g., as in Fig. 14), as a first approximation one could assume the wind maxima randomly flips from one side to the other within the transitioning lifetime of a given LHSC storm. To implement such a pattern of behavior as part of an ensemble system, a climatology of durations for the LHSC-RM and LHSC-LM phases within LHSC storm is needed; here again, one can use a dataset such as the CFSR product to assemble a training dataset and allow the development of statistical models for each phase duration.

Some limitations of the model also need to be highlighted when analyzing Fig. 14. Although the model is able to capture the main location of maximum winds, cases like Fig. 14a where two maxima of similar magnitude exist on both sides of the track are not well captured. The parametric formulation presented in this section is designed to show the best skill in modeling the location and area of maximum winds (which most matters for risk assessment). It results from a range of assumptions and simplifications that will never be fully fulfilled. A natural step to build up from this work is therefore to model the uncertainty associated with our approach. Khare et al. (2009) for instance provide a methodology to reconstruct the residual wind field model error using truncated empirical orthogonal functions (EOFs). A similar methodology should be considered in a follow-up work to add variability in the range of wind fields simulated by our approach when generating large ensembles. As a result, the risk associated with cases like Fig. 14a with multiple areas of similar wind magnitude would be sampled in these ensembles.

Note that the full methodology presented in this section was reproduced with only a subset of the entire WRF database (i.e., ~25% of snapshots were removed) to ensure that the three models are not overly reliant on a particular target selection. Although some small localized differences appeared in the bias fields, their general shapes and magnitudes were very similar to those presented. The values of Y1 and β were identical. This gives good confidence that the number of targets in our database is sufficient for the purpose of this study.

## 4. Conclusions

Recent studies have shown important differences in the shape and extent of the wind fields from extratropical transition cases in the WNP when compared with the tropical phase. One important implication within the context of risk assessment is that damaging winds are not restricted to the right of the moving cyclone during the transition period. Failure to account for this variability can result in an important misrepresentation of the risk profile. This is perhaps best pictured by considering the swath of damaging winds that an extratropical transition case like Rammasun in 2008 (Figs. 1 and 14) would leave if it was to make landfall in Japan; the sequence of events formed by Meari, Ma-on, and Tokage in late September–early October 2004 also provides important examples of cases where damaging gusts were recorded on both sides of the storm tracks. Additionally, consideration should also be given to the indirect impact on other perils such as storm surge/waves as well as TC-driven rainfall given that their modeling is heavily conditioned by the quality of the wind forcing [e.g., see modeling systems such as the ones presented in Grieser and Jewson (2012) for TC rain or Haigh et al. (2014) for TC-induced storm surges].

To account for this wind field variability, and constrained by our inability to rely on NWP models to develop risk assessment ensembles, a new parametric formulation is here introduced. It is based on a limited amount of inputs (shape parameters) and uses a parametric bias correction field to achieve the wind target shape. Calibration of the proposed model against 37 WRF simulations of extratropical transition cases in the WNP leads to a set of three formulations that cover the range of wind fields described in recent studies (Fujibe and Kitabatake 2007; Kitabatake and Fujibe 2009; Loridan et al. 2014). First results reveal the potential of the method to replicate the features specific to the transition phase, such as a wide extent of damaging winds on both sides of the cyclone track with a maximum wind magnitude that can be located on either side.

With the WRF cases selected to represent events capable of producing damaging winds in Japan, the model presented here is mostly designed to be applicable for risk assessment of extratropical transition cases in the WNP basin. Although we do not expect the main pattern to be drastically different, the modeling of transitioning risk in other basins would require a recalibration with relevant target wind fields. In particular, a closer look at the situation in the northeast United States and along the Australian coast is advised given the potential similarities of the midlatitude environment conditions with the case presented here.

The modeling of damaging winds from fully extratropical systems has not been discussed. As a first step, we believe the transitioning component can be applied throughout the lifetime of a storm. However, given the differences between the dynamics of transitioning and extratropical storms (e.g., cold core/frontal system), one might consider building a separate model to deal with the fully extratropical phase. Mixing the three (tropical, transitioning, and extratropical) model components in a risk assessment tool would then allow us to capture the full variability in damaging winds during the entire life cycle of a cyclone.

The extent to which the change in wind field pattern during extratropical transition impacts the modeling of other perils such as storm surge or rainfall should also be given particular attention in follow-up studies. For wave/surge risk in particular, the wind fetch over ocean will significantly be impacted by the reverse circulation that LHSC wind patterns can provoke.

## Acknowledgments

The authors thank Dr. John Knaff for his useful comments and advice, as well as three anonymous reviewers for their constructive suggestions. We also thank UCAR/NCAR/MMM for provision of the WRF Model, NCEP for provision of the CFSR reanalysis, and JMA/IBTrACS for the track data.

## REFERENCES

*23rd Conf. of Hurricane and Tropical Meteorology,*Dallas, TX,

*Wea. Forecasting,*

**21,**990–1005, doi:.

*J. Wind Eng. Indust. Aerodyn.,*

**77–78,**53–64.

*J. Struct. Eng.,*

**135,**301–320, doi:.