a. Predictive model

The predictive model for hurricane simulations is based on the Los Alamos National Laboratory High Gradient (HIGRAD) large-eddy smooth cloud model (Reisner and Jeffery 2009).

2) Bulk microphysical model

The mass conservation equation for a given particle type, ρ_part = ρ_c, ρ_r, ρ_i, ρ_s, ρ_g, within the bulk microphysical model can be written as follows:

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where are fluid velocities in each spatial direction and the last term represents the turbulent diffusion of a particle type using a diffusion coefficient diagnosed from a turbulent kinetic energy (TKE) equation, see Eq. (3) in Reisner and Jeffery (2009).

The conservation equation for either cloud droplet number N_c or ice particle number N_i, N_part = N_c, N_i, can be written as follows:

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where wfall_part, fdensity_part, and fnumber_part represent the fall speed, density, and number sources or sinks from the bulk microphysical model for a given particle type, a hybrid of the activation and condensation model found in Reisner and Jeffery (2009) together with all of the other relevant bulk parameterizations found in Thompson et al. (2008). Note, because of significant differences in the particle distributions between winter storms and hurricanes, the slope–intercept formulas were modified following McFarquhar and Black (2004).

3) Parameters of interest

The initialization of the environmental or background horizontal homogeneous potential temperature, water vapor, and total gas density fields for all Guillermo simulations was achieved by examining vertical profiles from ECMWF analyses obtained near the time period of the dual-Doppler radar data (1830 UTC 2 August to 0030 UTC 3 August 1997) and using a representative composite. Though some uncertainty exists within the thermodynamic fields with regard to the actual environment versus the perturbed environment obtained from the ECMWF soundings, the impact of this uncertainty was deemed to be smaller than that associated with the momentum fields, i.e., the simulated vortex is sensitive to small changes in wind shear. So to quantify this sensitivity, the horizontal velocity fields, and , were initialized as follows

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where φ_shear is a tuning coefficient that determines the shear impacting Hurricane Guillermo within a range of 0 and 1, is the height, and ecmwf_u and ecmwf_υ represent mean soundings calculated from the ECMWF analyses.

Given the delicate balance in nature that is needed for a sheared hurricane to intensify, it is not entirely obvious whether numerical models, which are necessarily limited in resolution, can accurately represent boundary layer processes that are responsible for supplying water vapor to eyewall convection. The accurate representation of boundary layer processes implies the model has been somewhat tuned to represent the impacts of waves, sea spray, and air bubbles within the water; likewise, the accurate treatment of energy release in eyewall convection implies that the upward movement of, for example, moisture is being reasonably simulated by the hurricane model.

To examine this uncertainty, the diffusion coefficient for surface momentum calculations was specified as follows:

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where κ_{surface friction} is a tuning coefficient that ranges from 0.1 to 10 m² s⁻¹ and V_h is the near-surface horizontal wind speed. A no-slip boundary condition was utilized in the horizontal momentum equations with the magnitude of κ_{surface friction} the determining factor with regard to the impact of this boundary condition on the intensity and structure of Guillermo. Note, unlike for the horizontal momentum equations, all scalar equations use a diffusion coefficient estimated from the TKE equation within calculations of surface fluxes.

Another uncertain boundary layer process that has a significant impact on intensification rate is surface moisture availability and the unresolved vertical transport of this water vapor with the first term being formulated as follows:

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where q_υs is the saturated vapor pressure over water and is a tuning coefficient that ranges in value from 0.0 to 0.2. This term enters into surface diffusional flux calculations of water vapor in discrete form as follows:

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where is the specific humidity of the first grid cell in the vertical direction. To address the uncertainty associated with the turbulent transport of water vapor (and all other fields) from the surface to the free atmosphere, the turbulent length scale was modified as follows:

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where the tuning coefficient φ_turb ranged from 0.1 to 10, and the eddy diffusivity now being .

b. Parameter estimation model

In this section the EnKF method is briefly described. In the current work, the EnKF is mainly utilized to estimate the model parameters.

1) Parameter estimation with EnKF

The EnKF is a Monte Carlo approach of the Kalman filter that estimates the covariances between observed variables and the model state variables through an ensemble of predictive model forecasts. The EnKF for the geosciences was first introduced by Evensen (1994) and is discussed in detail in Evensen and van Leeuwen (1996) and in Houtekamer and Mitchell (1998). For the current study, only the parameters will be estimated, not the state vector of the model. The EnKF procedure is directly applied to the parameters, that is, the state vector contains only the parameter values. Nevertheless, the model covariance matrix is still required for the innovation with observations. The following EnKF description will be concerned with model parameters.

Let be a vector holding the different model parameters and be the model state forecast. Let for i = 1 … N be an ensemble of model parameters and state forecasts, and a vector of m observations, and then the estimated parameter values given by the EnKF equations are

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where the matrix is a modified Kalman gain matrix (see appendix B), is the model forecast covariance matrix, is the cross-correlation matrix between the model forecast and parameters, is the observations covariance matrix, and is an observation operator matrix that maps state variables onto observations. In the EnKF, the vector is a perturbed observation vector, defined as

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where is a random vector sampled from a normal distribution with zero mean and a specified standard deviation σ. Usually σ is taken as the variance or error in the observations.

One of the main advantages of the EnKF is that the model forecast covariance matrix is approximated using the ensemble of model forecasts as shown:

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where is the model forecast ensemble average. The use of an ensemble of model forecast to approximate enables the evolution of this matrix for large nonlinear models at a reasonable computational cost. Additionally, the cross-correlation matrix is defined as

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where is the parameter ensemble average.

For our particular implementation, the system of Eqs. (9) and (10) is rewritten as

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for i = 1, … , N, where is the solution of the linear system (14) for ensemble i. For our implementation, is taken as a diagonal matrix, with σ in its main diagonal.

2) Considerations for parameter estimation with EnKF

Several studies have utilized the EnKF data assimilation to simultaneously estimate the model state and parameter. Among them are the studies by Aksoy et al. (2006a,b), Tong and Xue (2008), Hu et al. (2010), and Nielsen-Gammon et al. (2010). Typically, the parameters are included as part of the model state in the assimilation. This combination of evolving (dynamic) and nonevolving elements within the analysis introduces some difficulties for parameter estimation, such as parameter variance collapse and assimilation divergence. To mitigate these difficulties, the parameters are inflated to a prespecified variance, so as to avoid the collapse and keep a reasonable spread in the parameters. These techniques have proven to be effective to estimate parameters, but they require adjustments and tuning of the inflation to obtain good estimates.

The particular approach taken in this work is to use the EnKF as a tool only to estimate the model parameters using the available data. This approach is significantly different from the studies mentioned above, in the sense that only the parameters are estimated with the EnKF; that is, the model state is not being estimated. The motivation behind this approach is that model parameters are assumed to be constant; they do not evolve through the model, although they affect the dynamics of the solution. For this reason, determining parameters can be viewed as a stationary or static optimization. Our objective is to estimate a constant parameter value for the given dataset, over the given time window. To achieve this, the assimilation of the data is performed on each time instance, where observations are available, independently. The reasoning behind this technique is to treat the parameters as constants and nonevolving elements in the model; hence, for each time period we compute an estimated parameter value.

The procedure used to estimate the parameters is the following: Let t₁, … , t_k be the time instances where observations are available. For each time instance t_j, j = 1, … , k; the EnKF data assimilation [Eqs. (14) and (15)] provides parameter estimates for the ensemble, . A final parameter estimate is then computed by first taking the ensemble average and then the time average of the parameters, that is,

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One advantage is that this approach avoids the problem of parameter collapse and filter divergence, since the data assimilation is used to estimate the parameters at each time instance independently. Additionally, since the state is not being updated in the assimilation, and only the parameter is being estimated, localization is not required for the EnKF.

The number of available data points for Hurricane Guillermo is about 200 000 at any given time. This rich dataset can be used to investigate a number of aspects of hurricane intensification. To exploit this dataset for assimilation, we used an efficient matrix-free EnKF algorithm developed by Godinez and Moulton (2012). The algorithm works by efficiently solving the linear system in Eq. (14) using a solver based on the Sherman–Morrison formulas. In their paper, Godinez and Moulton (2012) show that this algorithm is more efficient than traditional implementations of the EnKF, by several orders of magnitude, and enables the assimilation of vast amounts of data. Additionally, the algorithm provides an analysis that is qualitatively and quantitatively the same as more traditional implementations. Readers are referred to the work of Godinez and Moulton (2012) for more details.

a. Derived Doppler data fields

The primary driver of a hurricane is the release of latent heat in clouds, which arises mainly from condensation. Latent heat cannot be observed directly and instruments, such as Doppler radars, only measure the reflectivity and radial velocity of precipitation particles averaged over the pulse volume. As a result, retrievals of dynamically relevant quantities (e.g., the Cartesian wind components and latent heat) are required. Guillermo’s 3D wind field was retrieved using a variational approach on a system of equations that includes the radar projection equations, the anelastic mass continuity equation, and a Laplacian filter (Gao et al. 1999; Reasor et al. 2009). This wind field and estimates of the precipitation water content (derived from the reflectivity measurements) are used to retrieve the latent heat of condensation/evaporation following Guimond et al. (2011). There are two main steps in the latent heat retrieval algorithm: 1) determine the saturation state at each grid point in the radar analysis using the precipitation continuity equation and 2) compute the magnitude of heat released using the first law of thermodynamics and the vertical velocity estimates described in Reasor et al. (2009). There are several potential sources of error in the latent heat retrievals, and a detailed treatment of these errors can be found in Guimond et al. (2011). Here, we summarize the most relevant information.

The uncertainty in the latent heat retrievals reduces to uncertainties in two main fields: reflectivity and vertical velocity. Guimond et al. (2011) were able to reduce and/or document the uncertainty in these fields to a level where the latent heat retrievals have a reasonably acceptable accuracy. For example, Guimond et al. (2011) focus on the inner portion of the eyewall and often use two aircraft to construct the radar analyses (Reasor et al. (2009), which reduce the effects of attenuation. In an attempt to correct the known calibration bias in the National Oceanic and Atmospheric Administration (NOAA) P-3 tail radar reflectivity, 7 dB was added to the fields (J. Gamache and P. Reasor 2010, personal communication). More importantly, however, the reflectivity is only used to determine the condition of saturation in the latent heat retrieval. Thus, the algorithm is not dependent on the precise value of the reflectivity, rendering the retrievals somewhat insensitive to errors (Guimond et al. 2011).

The uncertainties in the magnitude of the retrieved heating are dominated by errors in the vertical velocity. Using a combination of error propagation and Monte Carlo uncertainty techniques, biases are found to be small, and randomly distributed errors in the heating magnitudes are 16% for updrafts greater than 5 m s⁻¹ and 156% for updrafts of 1 m s ⁻¹ (Guimond et al. 2011). Even though errors in the vertical velocity can lead to large uncertainties in the latent heating field for small updrafts/downdrafts, in an integrated sense, the errors are not as drastic. Figure 2 shows example horizontal views (averaged over all heights) of the latent heating rate of condensation/evaporation for 4 of the 10 aircraft sampling periods of Guillermo.

Horizontal views (averaged over all heights) of the latent heating rate (K h⁻¹) of condensation/evaporation retrieved from airborne Doppler radar observations in Hurricane Guillermo (1997) at four select times of the dual-Doppler radar data. Pass 5 corresponds to 2117 UTC from Fig. 6 of Reasor et al. (2009). Note that grid points without latent heating were assigned zero values after the vertical averaging. Vertical profile of the azimuthal mean latent heating rate at the RMW (30 km) is shown above each contour plot. The first level of data is at 1 km due to ocean surface contamination.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Horizontal views (averaged over all heights) of the latent heating rate (K h⁻¹) of condensation/evaporation retrieved from airborne Doppler radar observations in Hurricane Guillermo (1997) at four select times of the dual-Doppler radar data. Pass 5 corresponds to 2117 UTC from Fig. 6 of Reasor et al. (2009). Note that grid points without latent heating were assigned zero values after the vertical averaging. Vertical profile of the azimuthal mean latent heating rate at the RMW (30 km) is shown above each contour plot. The first level of data is at 1 km due to ocean surface contamination.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Horizontal views (averaged over all heights) of the latent heating rate (K h⁻¹) of condensation/evaporation retrieved from airborne Doppler radar observations in Hurricane Guillermo (1997) at four select times of the dual-Doppler radar data. Pass 5 corresponds to 2117 UTC from Fig. 6 of Reasor et al. (2009). Note that grid points without latent heating were assigned zero values after the vertical averaging. Vertical profile of the azimuthal mean latent heating rate at the RMW (30 km) is shown above each contour plot. The first level of data is at 1 km due to ocean surface contamination.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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For assimilation, only latent heat data where reflectivity observations are available is included into the dataset. The errors for the retrieved wind fields are set to 5.0%–6.0%, and the errors in the retrieved latent heating, as a percentage, are specified following Guimond et al. (2011):

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where δw = 1.56 m s⁻¹ represents the overall uncertainty in the vertical wind velocity field w Reasor et al. (2009) and is the observation error for the latent heat fields. It is worth noticing that observational latent heat errors are sometimes overestimated or underestimated. Thus, in an integral sense, the errors are not so drastic; the bias is only of +0.16 m s⁻¹.

b. Guillermo ensemble setup

Since the primary goal is to examine the impact of various model parameters containing high uncertainty on the intensity and structure of Guillermo, but not the track, all simulations comprising the ensemble have been undertaken in which a mean wind of 1.5 m s⁻¹ was added or subtracted to the respective environmental wind components to prevent the movement of Guillermo from a region containing high spatial resolution found in the domain center. Specifically, this high-resolution patch in Cartesian space, and , is defined as follows:

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where determines how quickly the grid spacing changes from 7 km near the model edges to 1 km near the center with Ngp_i′ the number of grid points in either direction and x*, y* representing grid values for a normalized grid with a domain employing 0.5Ngp_i′ away from a center location in which x* = y* = 0. Like the horizontal direction, the vertical direction also employs stretching with the highest resolution near the ocean boundary, approximately 50 m, and the coarsest near the model top, 500 m, with 86 vertical grid points being utilized to resolve a domain extending upward to 21 km. Note, because of the relatively high vertical spatial resolution, time step size was limited to 1 s to avoid any instabilities associated with exceeding the advective Courant number limit.

The ensemble is generated by perturbing only the four parameters discussed in section 2a(3), which are φ_shear, κ_{surface friction}, , and φ_turb. The parameter values are generated by the Latin hypercube sampling technique with a uniform distribution, where each parameter is sampled over a specified interval. All ensembles have the same initial, where the background fields are initialized as described in section 2a(3), where the background winds that are independent of the wind field associated with Guillermo are initialized using ECMWF data. Whereas to initialize the wind field associated with Guillermo, a composite of the radar winds and a bogus vortex is employed within a nudging procedure over a 1-h period. Afterward, the hurricane model is simulated in free mode for 5 h to become balanced with the parameters, and develops a hurricane vortex. It is important to mention that at this stage, the behavior of the ensemble simulations is mainly dominated not by the initial conditions but by the parameter values. Hence, utilizing derived fields from the same hurricane for a vortex initialization does not bias the results of the parameter estimation experiments, as long as the subsequent spinup is sufficiently long to allow the parameters to dominate the long-term behavior of the simulation.

c. EnKF and observation selection

The EnKF data assimilation is used to estimate only the parameters of interest. The time distribution of the parameters is obtained by assimilating each time period independently, as stated in section 2b(2). A final parameter value is obtained by averaging in time the results of the assimilation. The particular algorithm used for the parameter EnKF is a matrix-free algorithm presented in Godinez and Moulton (2012).

The initial set of parameter values is generated using a Latin hypercube sampling strategy with a uniform distribution. The interval for each parameter, shown in Table 1, is chosen according to the values mentioned in section 2a, which are based on prior sensitivity simulations and from a physical intuition of their interaction with the model. Figure 3 shows the initial distribution for each parameter of interest.

Table 1.

Initial parameter intervals for sampling with Latin hypercube strategy using a uniform distribution.

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Parameter spread of the 120 ensemble members obtained by utilization of the Latin hypercube sampling strategy within the limits shown in Table 1.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Parameter spread of the 120 ensemble members obtained by utilization of the Latin hypercube sampling strategy within the limits shown in Table 1.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Parameter spread of the 120 ensemble members obtained by utilization of the Latin hypercube sampling strategy within the limits shown in Table 1.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Although the optimal ensemble size for estimating reliable model uncertainty is still under active research, an ensemble of 120 members was deemed appropriate to capture essential model statistics for parameter estimation. Upon completion of the 120-member ensemble and due to the small movement of simulated hurricanes within the ensemble, information from each ensemble member was independently interpolated to the center location of the grid so as to be coincident with the observations. Note, only a portion of the computational domain was utilized within the EnKF corresponding to the portion of the domain that contains the derived fields of winds and latent heat. Finally, after this interpolation step, all interpolated model and derived data fields were read into the EnKF to conduct the parameter estimation for a given time period.

The data selection procedure followed in this paper is similar to the rank parameter-observation correlation procedure presented by Tong and Xue (2008). The parameter cross-correlation matrix in Eq. (13) provides information about the correlation between the model state variables and the parameters. By applying the observation operator to the model state, we have

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which is a cross-correlation matrix that can be used to identify regions of high correlation between parameters and model state variables in observation space. To identify the location of the observations, we add the correlations in and sort them from largest to smallest. To clarify, let . Each of the columns c_i contains the correlation of parameter p_i to the state variable in observation space. We add all of these columns to obtain , which is then sorted from smallest to largest. This will provide the location of the largest correlation of the model in observation space to assimilate. It is important to note that the regions of high correlation can be identified from different model variables fields, such as latent heat, horizontal winds, or reflectivity. This enables the use of one of these fields as a proxy for observation localization in order to better capture the physical processes of interest.

a. Ensemble spread and structure

Since the EnKF method depends on proper model statistics to determine optimally the parameter values (Anderson 2001), this section will illustrate this necessary variability across the various members of the ensemble. Likewise, for reasonable parameter estimation, it is important that the ensemble produces statistics that are within the range of the observations and this is another aspect of the ensemble that will be presented. For example, Fig. 4 shows the simulated pressure traces for the 120 ensemble members (blue lines), the ensemble average (black line), and the 3-hourly observations from the National Hurricane Center (NHC) advisories (red dots), with a large spread in hurricane intensity being denoted within the ensemble. Further, even though this result may be somewhat fortuitous, the ensemble-averaged pressure trace is in remarkably good agreement with observations with a difference of less than 5 hPa. To highlight differences between the ensemble average and a given member, in addition to displaying ensemble-average statistics, results from a control simulation that also reasonably reproduced the observed pressure trace will be shown (parameter values shown in Table 4). The parameter values for the control run were obtained through brute-force trial-and-error experiments until a hurricane simulation that matches the observed minimum sea level pressure was obtained.

Minimum sea level pressure vs simulation time for each ensemble member (blue lines), ensemble average (black line), and observations (red dots), for (top left) ensemble 1–30, (top right) ensemble 31–60, (bottom left) ensemble 61–90, and (bottom right) ensemble 91–120.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Minimum sea level pressure vs simulation time for each ensemble member (blue lines), ensemble average (black line), and observations (red dots), for (top left) ensemble 1–30, (top right) ensemble 31–60, (bottom left) ensemble 61–90, and (bottom right) ensemble 91–120.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Minimum sea level pressure vs simulation time for each ensemble member (blue lines), ensemble average (black line), and observations (red dots), for (top left) ensemble 1–30, (top right) ensemble 31–60, (bottom left) ensemble 61–90, and (bottom right) ensemble 91–120.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Because Hurricane Guillermo was embedded in an environment characterized with vertical wind speed shear, its observed eyewall horizontal structure was asymmetric with a dominant wavenumber-1 mode, for example, Reasor et al. (2009) and Sitkowski and Barnes (2009). To illustrate the models’ ability to reproduce this asymmetry, Fig. 5 shows both the ensemble-averaged layer-averaged vertical velocities and the corresponding layer-averaged fields from the control at two different layers. As evident in this figure, the vortex in both the average sense and for the control is asymmetric with a dominant wavenumber-1 mode being readily apparent. Likewise, the impact of averaging across all members is clearly evident in Fig. 5, with a significant smoothing and reduction in vertical motions being noted with regard to the vertical motion fields produced by the control simulation. Furthermore, using a similar Fourier spectral decomposition procedure as in Reasor et al. (2009), Fig. 6 reveals significant amounts of the vertical motion fields being in wavenumber-0 and -1 components, with the magnitude of the wavenumber-1 vertical motion field from the control reasonably agreeing with the observations [see Fig. 15a of Reasor et al. (2009)].

Ensemble average vertical motion fields at 2300 UTC (11 h into the simulations) for ensemble averaged (a) between 1 and 3 km and (c) between 5 and 7 km. Corresponding layer-averaged vertical motions fields from a control simulation (b) between 1 and 3 km and (d) between 5 and 7 km.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Ensemble average vertical motion fields at 2300 UTC (11 h into the simulations) for ensemble averaged (a) between 1 and 3 km and (c) between 5 and 7 km. Corresponding layer-averaged vertical motions fields from a control simulation (b) between 1 and 3 km and (d) between 5 and 7 km.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Ensemble average vertical motion fields at 2300 UTC (11 h into the simulations) for ensemble averaged (a) between 1 and 3 km and (c) between 5 and 7 km. Corresponding layer-averaged vertical motions fields from a control simulation (b) between 1 and 3 km and (d) between 5 and 7 km.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Time-averaged axisymmetric (black solid line) and azimuthal wavenumber-14 amplitudes of vertical velocity in the 13-km layer within a 200-km radial distance from the storm center from the (a) ensemble average and (b) control simulation. Averaging times are between 2230 and 2300 UTC.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Time-averaged axisymmetric (black solid line) and azimuthal wavenumber-14 amplitudes of vertical velocity in the 13-km layer within a 200-km radial distance from the storm center from the (a) ensemble average and (b) control simulation. Averaging times are between 2230 and 2300 UTC.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Time-averaged axisymmetric (black solid line) and azimuthal wavenumber-14 amplitudes of vertical velocity in the 13-km layer within a 200-km radial distance from the storm center from the (a) ensemble average and (b) control simulation. Averaging times are between 2230 and 2300 UTC.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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To provide another view of the simulated storm structure, both the ensemble-averaged and control simulation azimuthal structures were compared to observations and are presented for two flight legs in Figs. 7 and 8. Common disagreement with observations can be seen in both figures: First the simulated radius of maximum wind (RMW), and hence eyewall size, is about 5–10 km smaller than in the observations. While the simulated ensemble-averaged azimuthal tangential component of the wind lies within about 10 m s⁻¹ of the observations, more notable differences are seen in the radial component of the wind with magnitudes rarely exceeding 10 m s⁻¹ in the observations and the simulation consistently producing magnitudes exceeding 15 m s⁻¹. Similar overestimation is produced in the latent heat fields, with values exceeding 25 000 or even 30 000 K h⁻¹ in the simulation and with the observations showing values marginally reaching 20 000 K h⁻¹.

Comparisons between azimuthally averaged profiles for the ensemble average (contours) and observations (shaded) for (a),(b) tangential winds, (c),(d) radial winds, and (e),(f) latent heat. Time periods for comparisons are at flight legs 5 (2117 UTC) and 9 (2333 UTC).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Comparisons between azimuthally averaged profiles for the ensemble average (contours) and observations (shaded) for (a),(b) tangential winds, (c),(d) radial winds, and (e),(f) latent heat. Time periods for comparisons are at flight legs 5 (2117 UTC) and 9 (2333 UTC).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Comparisons between azimuthally averaged profiles for the ensemble average (contours) and observations (shaded) for (a),(b) tangential winds, (c),(d) radial winds, and (e),(f) latent heat. Time periods for comparisons are at flight legs 5 (2117 UTC) and 9 (2333 UTC).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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As in Fig. 7, but for the control simulation (contours) and observations (shaded).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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As in Fig. 7, but for the control simulation (contours) and observations (shaded).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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As in Fig. 7, but for the control simulation (contours) and observations (shaded).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Despite these noteworthy differences, the HIGRAD model (section 2a) is able to reproduce the slope/tilt of radial, tangential, and latent heat fields with a reasonable degree of realism. Moreover, the heights above sea level of the contours encompassing the largest simulated values of those three fields are in overall good agreement with observations. Note that the larger values of latent heat are required to compensate for the impact of spurious evaporation (Reisner and Jeffery 2009) common to most cloud models, with the consequences of this evaporation being discussed later in this manuscript. Additional plots were made for the remaining eight flight legs (not shown) and displayed similar attributes.

b. Twin experiments for parameter estimation

To assess the reliability of parameter estimation within the current context, and the amount of observational data needed for the estimation, a series of twin experiments was performed. A synthetic observational dataset is produced from a reference model run with the specific parameter values given in Table 2, and initialized according to section 3b. These reference parameters were selected near the ensemble average with white noise added to them.

Table 2.

Parameter values used for the reference run, from where synthetic data are used for the twin experiments.

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In their paper Tong and Xue (2008) found that to estimate simultaneously the model state and five parameters with the ensemble square root filter, using their rank parameter-observation procedure, they needed only 30 observational data points. Given that our parameter estimation setting is significantly different from their approach, it is not entirely obvious whether only small amounts of data are required to conduct the parameter estimation or whether, in the other limiting situation, more data than are currently available are needed for undertaking the estimates. To address this issue, nine parameter estimation experiments, named TE1–TE9, were performed for different amounts of data, given in Table 3. The data being used for parameter estimation are the latent heat field from the reference run, where the data are selected according to the procedure described in section 3c. The first observation set was taken at t = 6 h of the simulation time, afterward 9 more observation sets were taken at 30-min intervals, which makes a total of 10 observational dataset over a 5-h window. For each experiment, the parameters are estimated according to section 2b(2); that is, parameter estimates for the ensemble are computed with the EnKF at each observational time, and then a final parameter estimate is computed by averaging over ensemble and then time, as in Eq. (16). Figure 9 shows the parameter estimates for each experiment, where the vertical lines indicate the time variance of the parameter estimate. The figure clearly shows the impact of the additional data on the parameter estimates with a noticeable reduction in the error for all for parameters. Furthermore, it is only when approximately 200 000 observations are used that all four parameters converge to the correct values. This is highly relevant since it indicates that a significant amount of data is required, in this context, to estimate correctly the values of the parameters. A similar result was obtained when the horizontal wind field or reflectivity was used to estimate the parameters. Additionally, this twin-experiment parameter estimation result is consistent with the results of Yussouf and Stensrud (2012), where the simultaneous estimation of all parameters leads to an improved analysis compared to single-parameter estimation.

Table 3.

Number of observations (obs) used in the twin experiments for parameter estimation.

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EnKF parameter estimation as a function of number of latent heat observations assimilated. Latent heat observations were added in locations where the ensemble is most sensitive to changes in the parameters (section 3c).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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EnKF parameter estimation as a function of number of latent heat observations assimilated. Latent heat observations were added in locations where the ensemble is most sensitive to changes in the parameters (section 3c).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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EnKF parameter estimation as a function of number of latent heat observations assimilated. Latent heat observations were added in locations where the ensemble is most sensitive to changes in the parameters (section 3c).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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c. Parameter estimation experiments with Hurricane Guillermo observations

Given the large ensemble spread in various model fields, such as derived intensity, and the ability of the model to reproduce observed data suggests that estimation of the four model parameters is not only possible with the EnKF but it should produce parameter estimates that hopefully reduce model forecast errors. The sensitivity of the hurricane simulation, with regard to structure and intensity, is shown in appendix A. In that appendix, it is observed that the intensity of the storm is highly sensitive to the turbulent length scale, surface friction, and surface moisture parameters, while the structure of the storm is highly sensitive to the turbulent length scale wind shear and surface friction parameters. The sensitivity results suggest that all four parameters must be estimated to obtain both structure and intensity correct. This is consistent with Yussouf and Stensrud (2012), who found that the simultaneous estimation of all parameters will provide a more accurate set of parameter values, in the presence of model error, compared to single-parameter estimation experiments with the EnKF.

The number of observations used for parameter estimation, using the EnKF in all subsequent experiments, were those identified with the highest ensemble sensitivity located at 200 000 model grid points (see selection of observations in section 3c). The assimilation is performed over latent heat (DA1), horizontal winds (DA2), or both fields (DA3) started 6 h into the ensemble simulation (corresponding to 1900 UTC). With the inclusion of DA3, an assessment regarding how the EnKF procedure weights two different observational datasets and model results can be made and analyzed. Hence, this section will not only highlight how the various parameter estimates change when using different observational fields but also how these estimates change over time.

Figure 10 shows the time distribution of the ensemble-average parameter estimates with EnKF using the 10 data time periods for DA1, DA2, and DA3. The largest temporal oscillations in the parameter estimates are associated with DA2 and suggest that the wind fields produced by the ensemble tend to oscillate more in time than the latent heat fields. Likewise, the surface moisture and the wind shear parameter estimates do not appear to change significantly in time, whereas the turbulent length scale and surface friction estimates either increase or decrease with time. Note that the temporal changes in these two parameters could be due to numerical errors and/or the impact of initial condition errors. For example, Fig. 7 shows that with time the areas of maximum latent heating and/or winds from the ensemble are expanding outward and hence this outward expansion, probably the result of numerical diffusion, could explain the temporal changes in these two parameter estimates.

Time distribution of the ensemble-average parameter estimates with EnKF from DA1 (blue line, latent heat), DA2 (red line, horizontal winds), and DA3 (green line, both latent heat and horizontal winds).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Time distribution of the ensemble-average parameter estimates with EnKF from DA1 (blue line, latent heat), DA2 (red line, horizontal winds), and DA3 (green line, both latent heat and horizontal winds).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Time distribution of the ensemble-average parameter estimates with EnKF from DA1 (blue line, latent heat), DA2 (red line, horizontal winds), and DA3 (green line, both latent heat and horizontal winds).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Figure 11 shows the control parameter values and the time-averaged parameter values for each of the assimilation experiments (SDA1–SDA3). The vertical lines indicate the 95% confidence interval for each parameter value. The parameter estimate for the surface moisture parameters shows the most difference between the assimilation experiments, mainly between the control, SDA1, and SDA2. These small differences in the parameters do lead to rather significant differences in both the structure and intensity of the simulated hurricanes (see appendix A for a brief sensitivity-to-parameters study).

Analysis parameters averaged over time for the control simulation, DA1, DA2, and DA3. Vertical lines from the dots indicate 95% confidence intervals of the parameter estimates for each experiment.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Analysis parameters averaged over time for the control simulation, DA1, DA2, and DA3. Vertical lines from the dots indicate 95% confidence intervals of the parameter estimates for each experiment.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Analysis parameters averaged over time for the control simulation, DA1, DA2, and DA3. Vertical lines from the dots indicate 95% confidence intervals of the parameter estimates for each experiment.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Table 4 show the time average parameter values from the assimilation experiments, as well as the parameter values used for control simulation. Figure 11 also reveals the various parameter estimates are different from the parameter values used in the control simulation, suggesting the importance of using a technique, such as the EnKF, to obtain the estimates, instead of simply using estimates obtained from a simulation that matches an observable, such as minimum sea level pressure. This ability of the EnKF to fit the parameter values reasonably to the chosen observational dataset is also illustrated by the time-averaged estimates from DA3 that, as expected, lie somewhere between the parameter estimates from DA1 and DA2, except for the turbulent length scale parameter.

Table 4.

Time-average parameter values for each of the experiments DA1–DA3, and the parameter values for the control simulation.

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d. Parameter error assessment

To illustrate the ability of the parameter estimates to reduce model forecast error, three simulations (SDA1, SDA2, and SDA3) were run using the time-average parameter estimates from DA1, DA2, and DA3 shown in Table 4. The setup for each simulation is the same as described in section 3b. Both qualitatively (see Figs. 12 and 13) and quantitatively (see Fig. 14) the parameter estimates produce model fields that are in better agreement with observed fields, especially the estimates associated with DA1 or those derived from using the latent heat fields. Although it is not surprising that SDA1 best matches the observed latent heat fields, what was surprising was the errors associated with the wind fields were lower in SDA1 than those associated with SDA2. This suggests the possible utility of using latent heat instead of more traditional observational fields, such as horizontal winds or radar reflectivity within data assimilation procedures to estimate model parameters and/or portions of the model state vector. In fact, since for sheared hurricanes the majority of convection can occur on one side of the hurricane, the utilization of latent heats fields may be especially important for parameter estimation as compared to the incorporation of the much smoother and more symmetric wind fields that are a consequence of the asymmetric convective activity. Thus, for Hurricane Guillermo, it is important to estimate properly the parameters responsible for the observed asymmetric convective activity, and by doing so the overall model error is reduced not only in the latent heat fields but also in the wind fields.

Comparisons of the azimuthally averaged profiles between a model simulation (contours) using estimated parameters from DA1 and observations (shaded). Plots for (a),(b) tangential winds, (c),(d) radial winds, and (e),(f) latent heat for flight legs 5 (2117 UTC) and 9 (2333 UTC).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Comparisons of the azimuthally averaged profiles between a model simulation (contours) using estimated parameters from DA1 and observations (shaded). Plots for (a),(b) tangential winds, (c),(d) radial winds, and (e),(f) latent heat for flight legs 5 (2117 UTC) and 9 (2333 UTC).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Comparisons of the azimuthally averaged profiles between a model simulation (contours) using estimated parameters from DA1 and observations (shaded). Plots for (a),(b) tangential winds, (c),(d) radial winds, and (e),(f) latent heat for flight legs 5 (2117 UTC) and 9 (2333 UTC).

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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As in Fig. 12, but from DA2 and observations.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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As in Fig. 12, but from DA2 and observations.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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As in Fig. 12, but from DA2 and observations.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Error estimates as a function of time computed using Eq. (20) of Reisner and Jeffery (2009) for the control simulation, SDA1, SDA2, and SDA3.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Error estimates as a function of time computed using Eq. (20) of Reisner and Jeffery (2009) for the control simulation, SDA1, SDA2, and SDA3.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Error estimates as a function of time computed using Eq. (20) of Reisner and Jeffery (2009) for the control simulation, SDA1, SDA2, and SDA3.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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When a combination of both observational fields are utilized for parameter estimation, error estimates from SDA3 reveal that the response is weighted toward producing results closer to SDA2, suggesting the dominance of the horizontal wind observational data in the parameter estimates. This behavior can also be appreciated in the parameter estimates themselves, as seen in Figs. 10 and 11. The dominance of the horizontal wind field in SDA3 may be explained by the relative observational error associated with each type of observational data [see section 3a and Eq. (17)]; that is, the errors associated with the horizontal wind fields are lower than the errors in the latent heat observational field, leading the EnKF algorithm to rely more on the horizontal wind fields than the latent heat fields.

Figure 14 shows the L₂ error norm of the SDA1–SDA3 simulations. As seen from the figure, SDA1 (the parameter estimates obtained using latent heat) produces lower errors than SDA2 and SDA3, which may be counterintuitive. Nevertheless, we should keep in mind that one of the main drivers in hurricane structure is latent heat release, which will also affect the horizontal wind fields. This is the main reason that, by estimating the parameters using latent heat fields, the simulation is able to obtain a very reliable structure of the hurricane, including the horizontal wind fields. Hence, the L₂ error norm of SDA1 is much smaller than that of SDA2 and SDA3. A key remark on the results shown in Fig. 14 is that of observation sensitivity, that is, the quality of information provided by the type of observations being used to estimate the parameters of interest. Not all observations provide the same quality of information. While some observations may be abundant and very reliable, their information may not be sufficient to estimate correctly important aspects of a hurricane. Such is the case with latent heat observations: they provide good information regarding the structure of the hurricane.

Though Fig. 14 demonstrates that SDA1 produces lower errors in the structure of the storm than SDA2 and SDA3, Fig. 15 illustrates that the intensity of SDA1 is significantly weaker than SDA3 and slightly weaker than SDA2. This finding may suggest that in order for SDA1 to reproduce accurately the intensity of Guillermo, additional observational data are needed below the range of the radar (approximately 1 km in height) or that other errors, such as numerical errors, are contributing to the intensity differences, that is, numerical diffusion and spurious evaporation associated with large numerical errors found near cloud boundaries. For example, the bottom panels of Figs. 12 and 13 show areas of evaporative cooling occurring immediately to the left of regions of strong positive latent heat release that do not have an analog in the observed fields. But, though SDA1 appears to suffer from relatively large numerical errors that are also common to all hurricane models, the end result of the parameter estimation procedure is still a reduction in overall model forecast error for structure.

Minimum sea level pressure for the control simulation, SDA1, SDA2, and SDA3 along with the observed pressure from Hurricane Guillermo.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Minimum sea level pressure for the control simulation, SDA1, SDA2, and SDA3 along with the observed pressure from Hurricane Guillermo.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Minimum sea level pressure for the control simulation, SDA1, SDA2, and SDA3 along with the observed pressure from Hurricane Guillermo.

Citation: Journal of the Atmospheric Sciences 69, 11; 10.1175/JAS-D-12-022.1

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Another remark is that, like the control simulation, SDA2 produces more latent heat than the observed amount of latent heat (see Fig. 13), suggesting the simulation is indeed having to compensate for large amounts of spurious evaporative cooling. Also note that both SDA2 and SDA3 in both plots in Fig. 14 are very close to each other, suggesting that the parameters used in both simulations provide very similar results.

Similar to previous studies (Bryan 2012; Emanuel and Rotunno 2011), the primary sensitivity in intensity and structure is with regard to changes in surface friction and the turbulent length scale. The surface friction coefficient in the model leads to rather large changes in the size of the eyewall (structure) as well as the impact of shear on the intensity; that is, a smaller more rounded eyewall is less susceptible to wind shear. Note, it is entirely possible to get the same intensity but completely different structures for two simulated hurricanes; that is, a large broken eyewall produced by a small surface friction parameter and a large turbulent length scale versus a small rounded eyewall produced by a large surface friction coefficient and a small turbulent length scale. Thus, the importance of using both sea level pressure and L₂ error estimates with regard to ensuring that the estimated parameters are getting the right result (intensity) for the right reason (structure).