• Aksoy, A., 2013: Storm-relative observations in tropical cyclone data assimilation with an ensemble Kalman filter. Mon. Wea. Rev., 141, 506522, doi:10.1175/MWR-D-12-00094.1.

    • Search Google Scholar
    • Export Citation
  • Aksoy, A., , S. D. Aberson, , T. Vukicevic, , K. J. Sellwood, , S. Lorsolo, , and X. Zhang, 2013: Assimilation of high-resolution tropical cyclone observations with an ensemble Kalman filter using NOAA/AOML/HRD’s HEDAS: Evaluation of the 2008–11 vortex-scale analyses. Mon. Wea. Rev., 141, 18421865, doi:10.1175/MWR-D-12-00194.1.

    • Search Google Scholar
    • Export Citation
  • Brown, B., , and G. J. Hakim, 2013: Variability and predictability of a three-dimensional hurricane in statistical equilibrium. J. Atmos. Sci., 70, 18061820, doi:10.1175/JAS-D-12-0112.1.

    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., , and R. Rotunno, 2009: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., 137, 17701789, doi:10.1175/2008MWR2709.1.

    • Search Google Scholar
    • Export Citation
  • Buizza, R., , P. L. Houtekamer, , G. Pellerin, , Z. Toth, , Y. Zhu, , and M. Wei, 2005: A comparison of the ECMWF, MSC, and NCEP global ensemble prediction systems. Mon. Wea. Rev., 133, 10761096, doi:10.1175/MWR2905.1.

    • Search Google Scholar
    • Export Citation
  • Cangialosi, J. P., , and J. Franklin, 2012: 2012 National Hurricane Center forecast verification report. NOAA, 79 pp. [Available online at http://www.nhc.noaa.gov/verification/pdfs/Verification_2012.pdf.]

  • Chen, Y., , and C. Snyder, 2006: Assimilating vortex position with an ensemble Kalman filter. Mon. Wea. Rev., 135, 1828–1845, doi:10.1175/MWR3351.1.

    • Search Google Scholar
    • Export Citation
  • DeMaria, M., , and J. Kaplan, 1994: A Statistical Hurricane Intensity Prediction Scheme (SHIPS) for the Atlantic basin. Wea. Forecasting, 9, 209220, doi:10.1175/1520-0434(1994)009<0209:ASHIPS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • DeMaria, M., , M. Mainelli, , L. K. Shay, , J. A. Knaff, , and J. Kaplan, 2005: Further improvements to the Statistical Hurricane Intensity Prediction Scheme (SHIPS). Wea. Forecasting, 20, 531543, doi:10.1175/WAF862.1.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., 2003: The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dyn., 53, 343367, doi:10.1007/s10236-003-0036-9.

    • Search Google Scholar
    • Export Citation
  • Gaspari, G., , and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723757, doi:10.1002/qj.49712555417.

    • Search Google Scholar
    • Export Citation
  • Hendricks, E. A., , M. S. Peng, , X. Ge, , and T. Li, 2011: Performance of a dynamical initialization scheme in the Coupled Ocean-Atmosphere Mesoscale Prediction System for Tropical Cyclones (COAMPS-TC). Wea. Forecasting, 26, 650663, doi:10.1175/WAF-D-10-05051.1.

    • Search Google Scholar
    • Export Citation
  • Hodyss, D., 2011: Ensemble state estimation for nonlinear systems using polynomial expansions in the innovation. Mon. Wea. Rev., 139, 35713588, doi:10.1175/2011MWR3558.1.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., , M. A. Bender, , R. E. Tuleya, , and R. J. Ross, 1995: Improvements in the GFDL hurricane prediction system. Mon. Wea. Rev., 123, 2791–2801, doi:10.1175/1520-0493(1995)123<2791:IITGHP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., , R. E. Tuleya, , and M. A. Bender, 1998: The GFDL hurricane prediction system and its performance in the 1995 hurricane season. Mon. Wea. Rev., 126, 13061322, doi:10.1175/1520-0493(1998)126<1306:TGHPSA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lawson, W. G., , and J. A. Hansen, 2005: Alignment error models and ensemble-based data assimilation. Mon. Wea. Rev., 133, 16871709, doi:10.1175/MWR2945.1.

    • Search Google Scholar
    • Export Citation
  • Liu, Q., , T. Marchok, , H.-L. Pan, , M. Bender, , and S. Lord, 2000: Improvements in hurricane initialization and forecasting at NCEP with global and regional (GFDL) models. NOAA Tech. Procedures Bull. 472, 7 pp.

  • Mainelli, M., , M. DeMaria, , L. K. Shay, , and G. Goni, 2008: Application of oceanic heat content estimation to operational forecasting of recent Atlantic category 5 hurricanes. Wea. Forecasting, 23, 316, doi:10.1175/2007WAF2006111.1.

    • Search Google Scholar
    • Export Citation
  • Rogers, R., and Coauthors, 2006: The intensity forecasting experiment: A NOAA multiyear field program for improving tropical cyclone intensity forecasts. Bull. Amer. Meteor. Soc., 87, 15231537, doi:10.1175/BAMS-87-11-1523.

    • Search Google Scholar
    • Export Citation
  • Torn, R. D., 2010: Performance of a mesoscale ensemble Kalman filter (EnKF) during the NOAA high-resolution hurricane test. Mon. Wea. Rev., 138, 43754392, doi:10.1175/2010MWR3361.1.

    • Search Google Scholar
    • Export Citation
  • Torn, R. D., , and G. J. Hakim, 2009: Ensemble data assimilation applied to RAINEX observations of Hurricane Katrina (2005). Mon. Wea. Rev., 137, 28172829, doi:10.1175/2009MWR2656.1.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924, doi:10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Wu, C.-C., , G.-Y. Lien, , J.-H. Chen, , and F. Zhang, 2010: Assimilation of tropical cyclone track and structure based on the ensemble Kalman filter. J. Atmos. Sci., 67, 38063822, doi:10.1175/2010JAS3444.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Y. Weng, , J. A. Sippel, , Z. Meng, , and C. H. Bishop, 2009: Cloud-resolving hurricane initialization and prediction through assimilation of Doppler radar observations with an ensemble Kalman filter. Mon. Wea. Rev., 137, 21052125, doi:10.1175/2009MWR2645.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Y. Weng, , J. F. Gamache, , and F. D. Marks, 2011: Performance of a convection-permitting hurricane initialization and prediction during 2008–2010 with ensemble data assimilation of inner-core airborne Doppler radar observations. Geophys. Res. Lett., 38, L15810, doi:10.1029/2011GL048469.

    • Search Google Scholar
    • Export Citation
  • Zou, X., , and Q. Xiao, 2000: Studies on the initialization and simulation of a mature hurricane using a variational bogus data assimilation scheme. J. Atmos. Sci., 57, 836860, doi:10.1175/1520-0469(2000)057<0836:SOTIAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Squared error in pressure (hPa2) of the ensemble-mean posterior as a function of localization radius. Spread in the prior is fixed at 36 km, and 100 observations are assimilated. Results are the average of 20 random simulations.

  • View in gallery

    Squared error in pressure (hPa2) of the storm-centered ensemble-mean posterior as a function of observed position error; 100 observations are assimilated.

  • View in gallery

    Normalized squared error in pressure of the ensemble-mean posterior as a function of position spread in the prior for the conventional and LH algorithms; 100 observations are assimilated.

  • View in gallery

    Squared error in pressure (hPa2) of the ensemble-mean posterior as a function of the number of observations. Spread in the prior is fixed at 36 km. Results for the conventional EnKF with and without the assimilation of a position observation are given by the thick and thin solid lines, respectively, and the LH and storm-centered approaches by the thick and thin dashed lines, respectively.

  • View in gallery

    Leading EOFs of ensemble-mean analysis increments for (left) conventional and (right) storm-centered assimilation. Gray lines indicate negative values and black lines positive values. The fraction of the total variance accounted for by each pattern is given in the bottom-left corner of each frame.

  • View in gallery

    Normalized squared error in pressure of the (top) azimuthal mean and (bottom) asymmetries in the ensemble-mean posterior as a function of prior position spread, computed at a radius of 50 km from storm center. Results for the conventional and LH ensembles are normalized by the storm-centered results.

  • View in gallery

    Power (hPa2) in the ensemble-mean posterior as a function of azimuthal wavenumber, computed at a radius of 50 km. Results are shown for the conventional, LH, and storm-centered algorithms and compared with the reference storm. Conventional and LH curves apply to prior spreads of 36 km.

  • View in gallery

    Comparison of power (hPa2) in the prior and posterior for azimuthal wavenumbers 1 and 2 as a function of position spread in the prior. Wavenumber 1 is given by solid lines and wavenumber 2 by dashed lines. Thick lines denote the prior and thin lines the posterior. Results apply at a radius of 50 km from storm center.

  • View in gallery

    Squared error in the ensemble-mean free-surface height (m2) for a vortex drifting on a beta plane as a function of time. Solid line denotes the conventional results; dashed line denotes the storm-centered results. Vertical lines denote the change in the posterior to the prior due from assimilation of measurements. Assimilation is performed every 6 h.

  • View in gallery

    Second shallow-water experiment, showing a vortex interacting with four surrounding, stationary anticyclones. Units of free-surface height are given in meters.

  • View in gallery

    As in Fig. 9, but for a cyclonic vortex interacting with four surrounding anticyclonic vortices in the environment. Bold solid line denotes the result calculated using the storm-centered algorithm. Assimilation is performed every 6 h.

  • View in gallery

    As in Fig. 11, except errors are computed relative to the storm center.

  • View in gallery

    Variance in free-surface height tendency (m2 s−2) averaged over the domain during the initial part of the forecast for the storm centered (dashed line), conventional assimilation (black solid line), and reference state (gray solid line). Storm-centered and conventional curves represent averages of the 30 ensemble members from one trial of the experiment with a cyclonic vortex interacting with four surrounding anticyclonic vortices.

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Storm-Centered Ensemble Data Assimilation for Tropical Cyclones

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Abstract

A significant challenge for tropical cyclone ensemble data assimilation is that storm-scale observations tend to make analyses that are more asymmetric than the prior forecasts. Compromised structure and intensity, such as an increase of amplitude across the azimuthal Fourier spectrum, are a routine property of ensemble-based analyses, even with accurate position observations and frequent assimilation. Storm dynamics in subsequent forecasts evolve these states toward axisymmetry, creating difficulty in distinguishing between model-induced and actual storm asymmetries for predictability studies and forecasting. To address this issue, a novel algorithm using a storm-centered approach is proposed. The method is designed for use with existing ensemble filters with little or no modification, facilitating its adoption and maintenance. The algorithm consists of 1) an analysis of the environment using conventional coordinates, 2) a storm-centered analysis using storm-relative coordinates, and 3) a merged analysis that combines the large-scale and storm-scale fields together at an updated storm location. This algorithm is evaluated in two sets of observing system simulation experiments (OSSEs): first, no-cycling tests of the update step for idealized three-dimensional storms in radiative–convective equilibrium; second, full cycling tests of data assimilation applied to a shallow-water model for a field of interacting vortices. Results are compared against a control experiment based on a conventional ensemble Kalman filter (EnKF) scheme as well as an alternative EnKF scheme proposed by Lawson and Hansen. The storm-relative method yields vortices that are more symmetric and exhibit finer inner-core structure than either approach, with errors reduced by an order of magnitude over a control case with prior spread consistent with the National Hurricane Center (NHC)’s mean 5-yr forecast track error at 12 h. Azimuthal Fourier error spectra exhibit much-reduced noise associated with data assimilation as compared to both the control and the Lawson and Hansen approach. An assessment of free-surface height tendency of model forecasts after the merge step reveals a balanced trend between the storm-centered and conventional approaches, with storm-centered values more closely resembling the reference state.

Corresponding author address: Erika L. Navarro, University of Washington, 408 Atmospheric Sciences/Geophysics (ATG) Building, Box 351640, Seattle, WA 98195-1640. E-mail: enavarr4@atmos.uw.edu

Abstract

A significant challenge for tropical cyclone ensemble data assimilation is that storm-scale observations tend to make analyses that are more asymmetric than the prior forecasts. Compromised structure and intensity, such as an increase of amplitude across the azimuthal Fourier spectrum, are a routine property of ensemble-based analyses, even with accurate position observations and frequent assimilation. Storm dynamics in subsequent forecasts evolve these states toward axisymmetry, creating difficulty in distinguishing between model-induced and actual storm asymmetries for predictability studies and forecasting. To address this issue, a novel algorithm using a storm-centered approach is proposed. The method is designed for use with existing ensemble filters with little or no modification, facilitating its adoption and maintenance. The algorithm consists of 1) an analysis of the environment using conventional coordinates, 2) a storm-centered analysis using storm-relative coordinates, and 3) a merged analysis that combines the large-scale and storm-scale fields together at an updated storm location. This algorithm is evaluated in two sets of observing system simulation experiments (OSSEs): first, no-cycling tests of the update step for idealized three-dimensional storms in radiative–convective equilibrium; second, full cycling tests of data assimilation applied to a shallow-water model for a field of interacting vortices. Results are compared against a control experiment based on a conventional ensemble Kalman filter (EnKF) scheme as well as an alternative EnKF scheme proposed by Lawson and Hansen. The storm-relative method yields vortices that are more symmetric and exhibit finer inner-core structure than either approach, with errors reduced by an order of magnitude over a control case with prior spread consistent with the National Hurricane Center (NHC)’s mean 5-yr forecast track error at 12 h. Azimuthal Fourier error spectra exhibit much-reduced noise associated with data assimilation as compared to both the control and the Lawson and Hansen approach. An assessment of free-surface height tendency of model forecasts after the merge step reveals a balanced trend between the storm-centered and conventional approaches, with storm-centered values more closely resembling the reference state.

Corresponding author address: Erika L. Navarro, University of Washington, 408 Atmospheric Sciences/Geophysics (ATG) Building, Box 351640, Seattle, WA 98195-1640. E-mail: enavarr4@atmos.uw.edu

1. Introduction

In the last two decades, operational forecasts of tropical cyclone (TC) track have improved significantly, while advances in intensity forecasts have been more modest. Dynamical models showed some improvement at short lead times (i.e., less than 72 h) during 2012 (Cangialosi and Franklin 2012), but historically they have been comparable or inferior to statistical models (DeMaria and Kaplan 1994; DeMaria et al. 2005; Mainelli et al. 2008). The impact of initial condition errors at the storm scale may contribute to this lack of improvement in intensity forecasts (Torn and Hakim 2009; Rogers et al. 2006). An attractive solution is the use of data assimilation (DA) (Torn and Hakim 2009); however, most operational data assimilation systems use quasi-fixed error statistics, which are often not appropriate for the TC environment. Given this challenge several other techniques have emerged, including dynamic initialization to observed surface pressure (Hendricks et al. 2011), “vortex bogusing” (Kurihara et al. 1995, 1998), “bogus data assimilation” (Zou and Xiao 2000), and vortex relocation (Liu et al. 2000); another method is to take no additional action, as for the European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble (Buizza et al. 2005). Although these techniques have demonstrated some success, they do not capture mesoscale storm features, especially at short lead times. Resolving such structures requires a DA scheme that can effectively spread storm-scale information contained in the observations. Here, we seek to improve upon current ensemble DA techniques by adopting a storm-centered approach to address this issue.

One method that appears well suited to this problem is the ensemble Kalman filter (EnKF) (Evensen 2003), which has the potential to produce both high-quality mesoscale analyses and the initial conditions for ensemble forecasting. This method combines state estimates from a numerical model with observations through a Kalman filter analysis update, with statistics drawn from the ensemble. These flow-dependent statistics are especially critical for TCs, since the flow near these features departs substantially from time-averaged conditions. Torn and Hakim (2009) cycled an EnKF over the lifetime of Hurricane Katrina (2005) and found reductions in forecast errors over previous forecasts of over 50% for both track and intensity. Moreover, the authors demonstrate that errors in the ensemble-mean 6-h forecasts of minimum pressure are 70% larger when dropsonde observations taken near the core of the storm are not assimilated. Torn (2010) continuously cycled an EnKF combined with the Advanced Research Weather Research and Forecasting Model (ARW) for 10 TCs occurring during the National Oceanic and Atmospheric Administration (NOAA)’s high-resolution hurricane (HRH) test and found reduced track errors over some operational models; errors in TC minimum sea level pressure and maximum wind speed, however, were reduced only for the nonmajor TCs (lower than category 3 on the Saffir–Simpson scale). Zhang et al. (2009) show that assimilating coastal radar data with an EnKF improved forecasts of Hurricane Humberto (2007), while Wu et al. (2010) show that by continually assimilating storm position, EnKF analyses are able to resolve secondary eyewall formation for Typhoon Sinlaku (2008). Further, Wu et al. (2010) also show that other experiments assimilating less data could not resolve this feature. Studies conducted with large samples of TCs using an EnKF to perform high-resolution vortex initialization at the vortex scale show improvements in storm intensity of 20%–30% using airborne Doppler radar observations in the ARW (Zhang et al. 2011) and the Hurricane Weather Research and Forecasting Model (HWRF) (Aksoy et al. 2013). While the results of these studies are very encouraging, there is limited evidence to verify the impact of frequent and accurate assimilation on azimuthal storm structure and, in particular, storm asymmetries.

A significant problem with the conventional EnKF approach for TC assimilation is that the ensemble statistics are strongly dominated by uncertainty in storm position. This effect is manifest in both the ensemble mean as well as the covariance estimates. For the mean, averaging yields an estimate with smooth, ill-defined mesoscale structure near the storm core. However, for ensembles exhibiting large position spreads, averaging can result in much worse artifacts, such as double vortices (Chen and Snyder 2006). In any event, the ensemble mean may not resemble any of the individual members. For the covariance estimates, small shifts in the large gradients near the eyewall of mature storms create perturbations about the ensemble mean that are much larger than those that arise naturally in the storm’s lifetime, such that these statistics are dominated by information on position rather than structure or intensity (Chen and Snyder 2006; Torn and Hakim 2009). If the finescale structure of the storm is not well constrained by measurements, then spurious asymmetrical structures can result. For example, Torn and Hakim (2009) demonstrate a whiter azimuthal Fourier spectrum in the posterior compared to the prior forecast, and attribute this effect to storm position. Forecasts made from analyses with these induced structural artifacts must first recover from artificial asymmetries through an axisymmetrization process that may be incomplete by the next assimilation cycle. Storm structure is thus compromised, and false asymmetries can be mistaken for truth. In the absence of a practical solution to this problem, the use of EnKF analyses in TC predictability studies requires careful consideration, since genuine features in subsequent forecasts and analyses are indistinguishable from artifacts due to data assimilation.

Several approaches have been proposed to reduce problems related to storm position. For example, Lawson and Hansen (2005, hereafter LH) propose a two-step method, where the first step reduces error in alignment, and the second step applies a standard EnKF update. This method shows improvement over traditional methods under a variety of scenarios, but the technique is problematic when the first step fails to remove all non-Gaussian effects. Hodyss (2011) makes use of a polynomial expansion in the innovation, using exponential powers to create a linear update with the second, third, and fourth moments, and relates this structure back to the prior distribution. Better state estimates of curvature of nonnormal priors are produced with a quadratic filter (i.e., one that uses square products of the innovation) as opposed to standard methods. Hodyss also concludes that forecasts for state estimates in situations with significant skewness are improved as powers of the innovation expansion are increased. These techniques are useful, but they are more expensive compared to conventional schemes, and require additional consideration for problems where localized features are embedded in complex environments. A recent study by Aksoy (2013) redefines observations in a storm-relative framework and, using the assumption of simultaneity of measurements over a given flight window, to produce a more spatially homogeneous distribution of observations. Doppler radar radial wind observations are then assimilated in the HWRF Ensemble Data Assimilation System (HEDAS) in an observing system simulation experiment (OSSE), demonstrating improvement to the observed kinematic representation of the TC analysis. This study does not address the issue of vortex position uncertainty, however, and it does not consider the impact of storm-relative assimilation on the surrounding synoptic environment.

Here, we propose an algorithm that builds upon existing ensemble data assimilation schemes to fundamentally address problems associated with storm position. Storm-scale analyses are performed in a storm-centered framework that removes position information in the update. Conventional EnKF results approach the storm-centered results in the limit of vanishing storm position spread, so it is useful to estimate the spread that necessitates the storm-centered approach. A perturbation df to an axisymmetric field f due to position error dr can be approximated by the leading-order Taylor approximation,
e1
In order for dfdA, where dA is the intrinsic (storm centered) perturbation amplitude, it requires
e2
Taking f to be the azimuthal wind speed, a storm with a maximum wind speed of 50 m s−1 at a radius of maximum wind (RMW) speed of 25 km and dA = 5 m s−1 implies that dr ≪ 2.5 km. Since this value is less than the uncertainty in current position measurements, it is unlikely to be satisfied for most mature storms.

The outline of the remainder of the paper is as follows. The algorithm and OSSEs conducted to test its performance are described in section 2. Results from idealized experiments using a long simulation of a three-dimensional (3D) storm in statistical equilibrium just testing the update step are described in section 3. Results from cycling data assimilation experiments involving a field of vortices in a rotating shallow-water model are described in section 4. Section 5 gives a concluding summary.

2. Method

Here, we describe the storm-centered assimilation method, followed by two experiments designed to test its efficacy. The general outline of the algorithm follows these steps:

  1. conventional data assimilation,
  2. storm-relative data assimilation, and
  3. merge analysis fields.
In step 1, conventional data assimilation is performed on the entire computational domain, which provides the analysis for the storm environment. The region near the storm is replaced by the storm-scale analysis, which is determined in step 2 using a storm-centered approach.

In step 2, the location of the storm center may be determined using a variety of methods, such as minimum surface pressure or maximum circulation; here, we choose minimum surface pressure.1 Numerical grids are coaligned in step 2 by shifting each ensemble member to a common location; this is achieved either on the entire computational domain or over a subdomain, such as a smaller, nested grid. In subsequent experiments, the full domain is shifted. Observation estimates for each ensemble member in step 2 are then determined from their storm-relative locations, as defined by observations of storm position. We note that errors in storm position affect these storm-relative observations, and this is considered in Fig. 2. Such errors will be largest for observations near large gradients, such as the eyewall. Spread in storm position prior to coalignment is accounted for in a separate position update, where the coordinates of each ensemble member’s storm location are updated as parameters. These coordinates are used to relocate the members in step 3.

To complete the algorithm, each ensemble member analysis is shifted to the new location and merged with the corresponding environmental field obtained in step 1. This step is critical for maintaining an appropriate synoptic flow, since the storm-relative framework is suboptimal away from the storm. The merge process may take many forms, but here we elect for a simple solution that consists of a linear combination of fields over a finite area, defined by a concentric ring around the vortex. The weight on the conventional analysis is given by
e3
where
e4
Here, r denotes the radial distance from storm center, while the inner and outer radii of the ring are defined by Rin and Rout, respectively. The term Rout is the distance from the storm center to the location that completely contains the vortex, determined here by the last closed isobar of mean sea level pressure. The inner boundary is taken to be . At this radius, the weight on the conventional analysis is zero.2 An increasing amount of weight is placed on the conventional solution with increasing radius from Rin, reaching unity at r = Rout; similarly, the weight on the storm-centered solution decreases monotonically toward zero at radius Rout. For r < Rin, w = 0, and for r > Rout, w = 1. Forecasts are performed from the merged ensemble analysis, and the algorithm is repeated at the next available assimilation time.

Although the storm-centered algorithm has clear benefits, it also has drawbacks. One anticipated weaknesses concerns variability in the environment, such as near land surfaces; in these cases repositioning the storm in step 2 leads to artifacts where geographically fixed surface conditions have disturbed the atmosphere. Limiting the size of the storm-centered domain near land, and not using the algorithm when storms are over land, may address this issue. A second weakness is that the storm-centered algorithm is more costly than conventional data assimilation, since the update is performed twice, but this cost is typically small in comparison to the expense of the ensemble forecasts. A third issue concerns the choice of level for calculating storm center, which may present a problem for tilted storms; we will consider this issue in section 3a. Error in observed storm position is also a critical factor, as the algorithm relies heavily on the ability to measure the center. This issue is considered in section 2a.

To assess the functionality and stability of the storm-centered algorithm, two experiments are conducted. The first experiment tests only the analysis step of the Kalman filter, and uses data sampled randomly from a long, statistically steady, simulation of an idealized TC. The second experiment concerns cycling an EnKF in a shallow-water model. These two experiments are designed to test the storm-centered algorithm in a simple framework where the exact solution is known, which facilitates evaluation of the storm-centered algorithm and its impact on assimilated storm structure. Results for both experiments are compared to a conventional EnKF solution employing the square root version with serial observations assimilated as described by Whitaker and Hamill (2002); however, this method should work in most existing ensemble data assimilation systems with modest modification. Where appropriate, results are also compared to the method of LH.3 In the LH method, one obtains the full analysis by first using a single position measurement to estimate the position error, then shifting the prior members by this amount and performing standard data assimilation. Recall that in the storm-centered algorithm, all prior ensemble members are shifted to the same location before assimilation. In this way, our method is the limiting case of the LH method in the case of zero position error.

a. Experiments with 3D tropical cyclones in statistical equilibrium

In this OSSE, the update step of the Kalman filter algorithm is applied to data from an idealized TC simulation. The goal is to systematically investigate the impact of storm-relative data assimilation on the ensemble analyses, as well as to qualitatively assess the influence of uncertainty in storm position on storm statistics. Data are extracted from a 100-day TC simulation generated using the cloud-permitting numerical model of Bryan and Rotunno (2009). The domain consists of a single 1024 km × 1024 km grid with a uniform horizontal grid spacing of 4 km; it is bordered by a region in which the horizontal resolution gradually increases to 36 km at the lateral boundaries, yielding a full domain width of 2944 km. The vertical grid spacing is 250 m in the lowest 1.25 km and gradually increases to 1 km from 10 to 25 km. An initially weak vortex develops into a mature storm in radiative–convective equilibrium with its surrounding environment after approximately 20 days, with an average intensity of 52 m s−1. The reader is referred to Brown and Hakim (2013) for more details regarding properties of the solution.

An initial ensemble of 50 members is drawn by randomly selecting states from this long solution, designating one additional member as the reference (“truth”) state. From this reference storm, 100 gridded, evenly spaced observations of surface pressure are given by the sum of the true value plus Gaussian random error having an error variance of 10 hPa2. The distance between gridded observations is 10 km. Spread in storm location, a parameter that we will vary in the experiments, is controlled by adding Gaussian random perturbations to vortex x and y location for each ensemble member consistent with the specified variance. The domain used in the assimilation experiments consists of a subset of approximately two-thirds of the uniform portion of the full model domain described above.

Assimilation is performed in both conventional and storm-relative frameworks. The ensemble-derived covariances are localized based on Eq. (4.10) of Gaspari and Cohn (1999), using a value of 600 km for the experiments described in section 3a; errors are a weak function of localization radius, provided that the radius is larger than the vortex and smaller than the domain size (Fig. 1). Since, by design, the storm-centered method uses observations of storm position, these observations are also used in the conventional EnKF scheme for consistency. The storm-centered analysis is more or less insensitive to position errors below 20 km, and increases linearly with increasing position error up to 100 km, and rapidly increases for position error greater than 100 km (Fig. 2). Given that the 2008–12 5-yr average 0-h forecast track error in all Atlantic TCs is approximately 18 km (see the official 5-yr average forecast error chart for all tropical cyclones from nhc.noaa.gov), the storm-centered method performs with errors on the same order as that with no position error.

Fig. 1.
Fig. 1.

Squared error in pressure (hPa2) of the ensemble-mean posterior as a function of localization radius. Spread in the prior is fixed at 36 km, and 100 observations are assimilated. Results are the average of 20 random simulations.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

Fig. 2.
Fig. 2.

Squared error in pressure (hPa2) of the storm-centered ensemble-mean posterior as a function of observed position error; 100 observations are assimilated.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

The measure used to evaluate performance is the squared error in the ensemble mean, averaged over the domain:
e5
Here, Nx and Ny are the number of grid points in the x and y directions, respectively; is the true value of x at position (i, j); and is the ensemble-mean value at the same location. To average over sampling error, 20 random trials are performed for each method; all results in this section show the mean of these trials. In certain figures, results are normalized by the storm-centered errors in order to facilitate comparison; this is labeled “normalized squared error” where applicable.

b. Experiments with a field of vortices in the rotating shallow-water equations

While the experiments described in the previous section test the properties of the update step at individual times, they do not test the impact of the storm-centered algorithm on subsequent forecasts, or the cumulative effect of many assimilation cycles. Two such experiments are conducted here for shallow-water vortices that propagate and interact with features in the external environment. The goal of these experiments is to assess the cumulative effect of storm-centered assimilation, as well as to test the robustness of the grid-merging procedure described in step 3 of the algorithm. Any imbalance introduced by this procedure will be realized as increased gravity wave activity, which is manifest as large height tendency variance in the subsequent forecasts.

The first shallow-water experiment consists of an axisymmetric vortex on a beta plane. Intensity perturbations are created by multiplying the amplitude of the vortex free-surface height and the zonal and meridional wind fields by a random number drawn from a Gaussian distribution of mean zero and standard deviation equal to 5 m4. Other than the β effect, which causes the vortex to drift northwestward on the domain, there are no environmental influences on the cyclone. The second experiment involves a vortex interacting dynamically with four surrounding, weaker anticyclones. The location and intensity of the vortex is perturbed similarly to the first experiment to populate the initial ensemble, while the position and intensity of the four anticyclones are fixed. The Coriolis parameter here is constant across the domain. Interaction between the cyclone and the anticyclones causes the vortex to translate throughout the domain, exhibiting structural and intensity changes and thus providing a challenge to the position update and the grid-merging procedures of the storm-centered method.

In both experiments the domain is a single 2000 km × 2000 km grid, with approximately 20-km grid spacing; experiments with 10-km grid spacing give quantitatively similar results (not shown). The first assimilation cycle is performed after the model is integrated forward 6 h, and then continually performed every 6 h for 144 h (25 assimilation cycles). In each of the two experiments an ensemble of 30 members is used, with one additional member designated as the reference storm (truth). The mean depth of the fluid layer of the reference storm is perturbed by 50 m, so that both experiments are carried out in a slightly imperfect model scenario. Gridded, evenly spaced observations of free-surface height are drawn from this reference storm, with added error having a variance of 10 m2. The standard deviation of error for position observations is 20 km. In both experiments there is no covariance inflation, and the covariance-localization length scale is 625 km. The radius Rout of the merge window is set to 375 km, which encompasses the last closed height contour of the vortex in each case.

3. Results

Results for the experiment testing only the update step of the EnKF are presented first, followed by shallow-water experiments that test the cycling properties of the storm-centered algorithm.

a. Idealized 3D tropical cyclones in statistical equilibrium

Following the experimental design outlined in the previous section, the storm-centered update outperforms both the conventional (the control) and LH schemes (Fig. 3). Relative to the storm-centered case, errors for the control increase rapidly for prior position spread larger than 12 km, reaching a factor of 3 for position spread greater than 24 km. The LH method reduces errors over the control by approximately 40%, but these remain larger than the storm-centered algorithm by at least 50%. LH errors increase rapidly beyond spreads of 36 km, which is on the order of the radius of maximum wind. Operational forecast position errors can substantially exceed this value, even for short lead times; for example, the mean track error in 12-h forecasts given by the National Hurricane Center (NHC) is approximately 53 km, compared to 18 km at 0 h (see the official 5-yr average forecast error chart for all tropical cyclones from nhc.noaa.gov). These results suggest two conclusions. First, for ensemble position spread similar to NHC analysis uncertainty, the storm-centered approach cuts errors in half compared to the control. Furthermore, for ensemble position spread similar to NHC average 12-h forecast error, the storm-centered approach improves upon the control error by more than an order of magnitude. Second, these results are qualitatively consistent with the estimate based on (2): in order for the conventional EnKF assimilation to approach the performance of the storm-centered approach, position spread in the prior must be on the order of 2.5 km, or about 75% less than the error in position measurements for real storms. Although the LH method improves upon errors in the control by 30% for position spread similar to NHC average 12-h forecast error, these errors are still larger than the storm-centered method by an order of magnitude.

Fig. 3.
Fig. 3.

Normalized squared error in pressure of the ensemble-mean posterior as a function of position spread in the prior for the conventional and LH algorithms; 100 observations are assimilated.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

The dependence of the results in Fig. 3 on the observation network is shown in Fig. 4. Both the control and LH schemes have a fixed position spread of 36 km in the prior in this case. The storm-centered approach again outperforms the control and LH methods by an average of 60% over a wide range of observation densities. Moreover, the error of the storm-centered case with 100 observations is not reached by the conventional approach until over 2000 observations are assimilated.

Fig. 4.
Fig. 4.

Squared error in pressure (hPa2) of the ensemble-mean posterior as a function of the number of observations. Spread in the prior is fixed at 36 km. Results for the conventional EnKF with and without the assimilation of a position observation are given by the thick and thin solid lines, respectively, and the LH and storm-centered approaches by the thick and thin dashed lines, respectively.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

A summary example of the spatial influence of observations on the ensemble mean is shown in Fig. 5. Contoured fields represent the leading patterns of variability in the ensemble-mean analysis increments (i.e., posterior minus prior), defined by empirical orthogonal functions (EOFs). For analysis increments in the control, a dipole pattern is evident in both EOFs, indicating that a shift in storm position accounts for 75% of analysis-increment variance, similar to Fig. 4 of Torn and Hakim (2009). In a storm-centered framework, 53% of the variance is attributed to a nearly axisymmetric structure apparent in the leading EOF. The second EOF is predominantly an axisymmetric modification of inner-core structure, along with a wavenumber 1 asymmetry just outside the radius of maximum wind. By moving into a storm-centered framework, the dipole structure associated with shifts in storm position is eliminated, and observations affect mainly storm intensity and axisymmetric structure.

Fig. 5.
Fig. 5.

Leading EOFs of ensemble-mean analysis increments for (left) conventional and (right) storm-centered assimilation. Gray lines indicate negative values and black lines positive values. The fraction of the total variance accounted for by each pattern is given in the bottom-left corner of each frame.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

The impact of prior position spread on assimilated storm structure is diagnosed by examining the axisymmetric and asymmetric components of the ensemble-mean analysis separately. A Fourier decomposition of the ensemble mean is performed in the ground-relative framework at a radius of 50 km, which is just outside the radius of maximum wind. Only the first 20 wavenumbers are included for the axisymmetric component; after wavenumber 20, the amplitude of the signal is effectively zero. For prior position spreads less than 12 km, errors in the conventional scheme are within 50% of the storm-centered approach, but increase rapidly for larger displacements (Fig. 6). In particular, normalized errors in the asymmetries become much more pronounced once the position spread in the prior increases beyond 24 km, doubling by 48 km and increasing to more than an order of magnitude for larger spreads. Errors for the LH method in both the axisymmetric and asymmetric components are similar to the control.

Fig. 6.
Fig. 6.

Normalized squared error in pressure of the (top) azimuthal mean and (bottom) asymmetries in the ensemble-mean posterior as a function of prior position spread, computed at a radius of 50 km from storm center. Results for the conventional and LH ensembles are normalized by the storm-centered results.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

When the analyses are separated into individual azimuthal wavenumbers, low wavenumbers appear most sensitive to position spread (Fig. 7), which in this case is again fixed at 36 km. Given the statistical equilibrium property of these experiments, the analysis power spectrum should match the reference (truth) spectrum. Both the conventional and LH results violate this property, with approximately an order of magnitude increase in the power of wavenumbers 1–5. The storm-centered algorithm, in contrast, has a spectrum that closely matches truth. Focusing on the dependence of azimuthal wavenumbers 1 and 2 as a function of position spread in the prior reveals that, for spreads near or larger than the radius of maximum wind, the conventional approach exhibits an increase in amplitude by a factor of 10, especially in wavenumber 1 (Fig. 8); that is, conventional data assimilation has artificially increased storm asymmetry (cf. Torn and Hakim 2009). The storm-centered method provides the only analysis that does not exhibit an increase in power over the prior.

Fig. 7.
Fig. 7.

Power (hPa2) in the ensemble-mean posterior as a function of azimuthal wavenumber, computed at a radius of 50 km. Results are shown for the conventional, LH, and storm-centered algorithms and compared with the reference storm. Conventional and LH curves apply to prior spreads of 36 km.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

Fig. 8.
Fig. 8.

Comparison of power (hPa2) in the prior and posterior for azimuthal wavenumbers 1 and 2 as a function of position spread in the prior. Wavenumber 1 is given by solid lines and wavenumber 2 by dashed lines. Thick lines denote the prior and thin lines the posterior. Results apply at a radius of 50 km from storm center.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

Figures 7 and 8 demonstrate that the storm-centered algorithm improves axisymmetric vortex structure by at least 50% as compared to the control and LH methods. Real storms are three dimensional, though, and the impact of the storm-centering method on the vertical structure of the storm must also be considered. Here, we measure the effect of the surface-based centering procedure on the analysis of 500-hPa height as a function of storm tilt. Storm tilt is measured as the distance between the point of lowest pressure on the lowest model level and the point of lowest geopotential height at 500 hPa. Thus, tilt is zero for perfectly stacked storms, and increases with increasing displacement between the centers in any azimuthal direction. Storms exhibiting no tilt and moderate tilt are tested, with 20 random trials for each scenario. An ensemble of 50 members exhibiting variable amounts of tilt is drawn for each of the 20 trials.

The mean result of the 20 trials for each assimilation method is summarized in Table 1. The storm-centered method exhibits the smallest error of the three algorithms, outperforming the conventional scheme by more than 50% for both tilted and vertically stacked storms. The LH method reduces error over the conventional method by a factor of 2 for storms with no tilt, but it exhibits a larger error than the storm-centered method by a factor of 2–3 for storms with tilted centers. This result indicates that the storm-centered method applies in a three-dimensional framework for modestly tilted storms, and motivates future testing on real storms.

Table 1.

Conventional, LH, and storm-centered schemes as a function of tilt in the prior. Tilt is defined as the difference between the surface center and the projection of the 500-hPa center onto the surface level; a tilt of 0 km represents a vertically stacked storm, while a tilt of, e.g., 4 km describes a storm whose upper-level center is displaced one grid point in any direction from the surface center. Three scenarios are presented: a reference storm with no tilt (second column), a reference storm with 4-km tilt (third column), and a reference storm with 8-km tilt (fourth column).

Table 1.

b. Vortices in the rotating shallow-water equations

The previous section tested the update step of the EnKF for a realistic TC in a framework where uncertainty in storm position spread could be easily controlled. Here, the cycling performance of the storm-centered algorithm is tested for two cases involving the interaction of a vortex with a large-scale environment: 1) a cyclonic vortex drifting on the beta plane and 2) a cyclonic vortex interacting with a field of anticyclonic vortices. These experiments address the cumulative impact of storm-centered assimilation, as well its effect on the balance dynamics of subsequent model forecasts; that is, it tests whether the merge step introduces shocks that amplify gravity wave activity.

In experiment 1, the vortex drifts on the beta plane due to a wavenumber 1 asymmetry associated with meridional advection of the planetary vorticity gradient. Considering that all position variance is eliminated in a storm-relative framework, this experiment is designed to test the ability of the algorithm to correctly update storm position and to resolve the wavenumber 1 asymmetry. These results show that the storm-centered ensemble exhibits roughly 75% less error than the control in the first 6 h, with improvement on the domain averaging 50% for the next 60 h. After 48 h, the reference vortex drifts toward the northern boundary and the spread of the storm-centered forecasts begins to increase; as a result, conventional and storm-centered forecasts converge at longer lead times.5 (Fig. 9).

Fig. 9.
Fig. 9.

Squared error in the ensemble-mean free-surface height (m2) for a vortex drifting on a beta plane as a function of time. Solid line denotes the conventional results; dashed line denotes the storm-centered results. Vertical lines denote the change in the posterior to the prior due from assimilation of measurements. Assimilation is performed every 6 h.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

The second shallow-water experiment is a harder test, since it involves merging and dynamically changing fields. Figure 10 shows the initial setup of the vortex environment, which includes a strong vortex placed among four surrounding, weak anticyclones. Since interaction with these features will generate structural changes in the vortex, comparison with the LH method is particularly important because it does not present the same threat to balance dynamics as the storm-centered model forecasts. Results over the entire domain illustrate that both the LH as well as the storm-centered algorithm make considerable improvements over the control, improving errors by roughly a factor of 2 in the first 36 h (Fig. 11). The storm-centered algorithm, however, reduces errors over the LH algorithm by an average of 30% over the full 144 h of the experiment. Comparing these results to those of Fig. 9 suggests that the merge step accurately preserves the large-scale flow of the environment over a domain where the environment is dynamically changing (i.e., the field of four anticyclones). Subsequent forecasts confirm this interpretation, since they also maintain smaller errors relative to the control. To isolate the effect of each method on storm structure, errors are also evaluated in a storm-relative framework (Fig. 12). Storm-centered errors are significantly lower than both the conventional and LH methods, exhibiting improvements on the order of 80%. Surprisingly, the storm-scale structure of the LH algorithm is not much better than the conventional algorithm, suggesting that some artifacts of the initial uncertainty of storm position persist through the second step of the algorithm.

Fig. 10.
Fig. 10.

Second shallow-water experiment, showing a vortex interacting with four surrounding, stationary anticyclones. Units of free-surface height are given in meters.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

Fig. 11.
Fig. 11.

As in Fig. 9, but for a cyclonic vortex interacting with four surrounding anticyclonic vortices in the environment. Bold solid line denotes the result calculated using the storm-centered algorithm. Assimilation is performed every 6 h.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

Fig. 12.
Fig. 12.

As in Fig. 11, except errors are computed relative to the storm center.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

Finally, to test whether the merge step generates imbalance in the surrounding environment, the spatial variance of the free-surface height tendency in the storm-centered forecasts is evaluated, since this quantity is sensitive on short time scales to gravity wave activity. The calculation is performed for the initial 2 h of the forecast for both conventional and storm-centered techniques, and then averaged over all 30 ensemble members. Results show a similar trend between the storm-centered and conventional forecasts (Fig. 13), with a balance adjustment at the start of the forecast for both methods; this lasts 30 min before asymptotically approaching a slower, balanced, solution. This transient stage is more pronounced in the storm-centered method, but it follows the same trend as the conventional scheme by quickly converging to a steadier solution. The factor of 2 difference between the methods is due to the improved vortex structure in the storm-centered solution; note that the storm-centered solution more closely simulates truth than the control. It appears that the storm-centered merge process does not introduce additional imbalance.

Fig. 13.
Fig. 13.

Variance in free-surface height tendency (m2 s−2) averaged over the domain during the initial part of the forecast for the storm centered (dashed line), conventional assimilation (black solid line), and reference state (gray solid line). Storm-centered and conventional curves represent averages of the 30 ensemble members from one trial of the experiment with a cyclonic vortex interacting with four surrounding anticyclonic vortices.

Citation: Monthly Weather Review 142, 6; 10.1175/MWR-D-13-00099.1

4. Summary and conclusions

Research on TC structure and the role of asymmetries in transient intensity change is limited by the ability of data assimilation to effectively utilize storm-scale observations. This issue applies to ensemble filters, which are an attractive tool for this problem because flow-dependent statistics are particularly important on the storm scale. Here, we have shown that these statistics are, however, contaminated by position information, which leads to high-wavenumber noise that adversely affects analyses. This is true even if the forecast of storm position spread is on the order of observation errors for position. In this study we proposed a novel storm-centered ensemble data assimilation scheme to address this problem. The approach aims to provide reliable analyses of storm structure, which are needed for future investigations of the dynamics and predictability of storm asymmetries.

The storm-centered algorithm was tested in two ways: first, looking only at the update step of the EnKF using a steady-state idealized three-dimensional tropical cyclone simulation; and second, by cycling data assimilation for vortices in a shallow-water model. Storm-centered results were compared against a conventional EnKF scheme as well as the two-step method of LH. Domain-averaged improvements to the conventional assimilation scheme were large, with errors on average about half the value of those in the control. By comparison, the LH scheme reduced errors over the conventional scheme by about 50% or more for most experiments, but it did not resolve storm structure as accurately as the storm-centered scheme. Improvements to the LH scheme with storm-relative assimilation averaged between 30% and 50% across a wide range of position spreads and observation densities. Statistics of the analysis increments in the storm-relative framework revealed that the leading patterns were related to structure and intensity, rather than a shift of storm location. Analyses for the storm-centered update consistently displayed lower errors in simulated axisymmetric structure relative to the control and LH method, and displayed reduced assimilation-induced amplitude in asymmetries. We confirm the estimate given in (2), concluding that the spread in the prior must be much less than the width of the eyewall in order for conventional ensemble techniques to approach the performance of the storm-centered algorithm.

The cumulative impacts of storm-centered assimilation, and the sensitivity of the subsequent forecasts to the merge process of the storm-centered algorithm were tested for 2D shallow-water vortices interacting dynamically with an external environment. The first experiment involved a cyclonic vortex drifting on the beta plane, and the second involved a cyclonic vortex interacting with four surrounding anticyclones. The storm-centered algorithm consistently showed lower errors across the domain, with a reduction in error of 50%–70% compared to the conventional case. Errors on the scale of the cyclonic vortex in the storm-centered simulations reduced error by a factor of 6 over the control and a factor of 4 over the LH method. Moreover, the storm-centered analyses produced forecasts that were not only dynamically balanced, but also consistently maintained 50% lower error relative to the forecasts of these two methods.

These results motivate further research to implement the algorithm for fully three-dimensional, real tropical storms. While computing the update in a storm-centered framework significantly reduces the noise introduced by data assimilation as compared to the conventional method, the cumulative effect of this method on observed 3D storm structure must be analyzed in detail before the method can be operational. We have anticipated some challenges and described them here, but impacts such as the presence of multiple wind maxima and the effect of storm size remain to be considered. In particular, the effect of error in observations of storm position that is translated to the observation estimates in the storm-centered update must be carefully diagnosed. We believe that the impact of this aspect of the algorithm is small, and the larger issue is the correcting of the covariance estimates, which the storm-centered algorithm does address. Such experiments are beyond the scope of this work but are fruitful directions for future study.

Acknowledgments

The authors thank Dr. Chris Snyder for conversations related to this research. This research was supported by the Office of Naval Research through Award N000140910436 and by the NSF through Award AGS-0842384 made to the University of Washington. This work was part of the first author’s master’s thesis at the University of Washington.

REFERENCES

  • Aksoy, A., 2013: Storm-relative observations in tropical cyclone data assimilation with an ensemble Kalman filter. Mon. Wea. Rev., 141, 506522, doi:10.1175/MWR-D-12-00094.1.

    • Search Google Scholar
    • Export Citation
  • Aksoy, A., , S. D. Aberson, , T. Vukicevic, , K. J. Sellwood, , S. Lorsolo, , and X. Zhang, 2013: Assimilation of high-resolution tropical cyclone observations with an ensemble Kalman filter using NOAA/AOML/HRD’s HEDAS: Evaluation of the 2008–11 vortex-scale analyses. Mon. Wea. Rev., 141, 18421865, doi:10.1175/MWR-D-12-00194.1.

    • Search Google Scholar
    • Export Citation
  • Brown, B., , and G. J. Hakim, 2013: Variability and predictability of a three-dimensional hurricane in statistical equilibrium. J. Atmos. Sci., 70, 18061820, doi:10.1175/JAS-D-12-0112.1.

    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., , and R. Rotunno, 2009: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., 137, 17701789, doi:10.1175/2008MWR2709.1.

    • Search Google Scholar
    • Export Citation
  • Buizza, R., , P. L. Houtekamer, , G. Pellerin, , Z. Toth, , Y. Zhu, , and M. Wei, 2005: A comparison of the ECMWF, MSC, and NCEP global ensemble prediction systems. Mon. Wea. Rev., 133, 10761096, doi:10.1175/MWR2905.1.

    • Search Google Scholar
    • Export Citation
  • Cangialosi, J. P., , and J. Franklin, 2012: 2012 National Hurricane Center forecast verification report. NOAA, 79 pp. [Available online at http://www.nhc.noaa.gov/verification/pdfs/Verification_2012.pdf.]

  • Chen, Y., , and C. Snyder, 2006: Assimilating vortex position with an ensemble Kalman filter. Mon. Wea. Rev., 135, 1828–1845, doi:10.1175/MWR3351.1.

    • Search Google Scholar
    • Export Citation
  • DeMaria, M., , and J. Kaplan, 1994: A Statistical Hurricane Intensity Prediction Scheme (SHIPS) for the Atlantic basin. Wea. Forecasting, 9, 209220, doi:10.1175/1520-0434(1994)009<0209:ASHIPS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • DeMaria, M., , M. Mainelli, , L. K. Shay, , J. A. Knaff, , and J. Kaplan, 2005: Further improvements to the Statistical Hurricane Intensity Prediction Scheme (SHIPS). Wea. Forecasting, 20, 531543, doi:10.1175/WAF862.1.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., 2003: The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dyn., 53, 343367, doi:10.1007/s10236-003-0036-9.

    • Search Google Scholar
    • Export Citation
  • Gaspari, G., , and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723757, doi:10.1002/qj.49712555417.

    • Search Google Scholar
    • Export Citation
  • Hendricks, E. A., , M. S. Peng, , X. Ge, , and T. Li, 2011: Performance of a dynamical initialization scheme in the Coupled Ocean-Atmosphere Mesoscale Prediction System for Tropical Cyclones (COAMPS-TC). Wea. Forecasting, 26, 650663, doi:10.1175/WAF-D-10-05051.1.

    • Search Google Scholar
    • Export Citation
  • Hodyss, D., 2011: Ensemble state estimation for nonlinear systems using polynomial expansions in the innovation. Mon. Wea. Rev., 139, 35713588, doi:10.1175/2011MWR3558.1.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., , M. A. Bender, , R. E. Tuleya, , and R. J. Ross, 1995: Improvements in the GFDL hurricane prediction system. Mon. Wea. Rev., 123, 2791–2801, doi:10.1175/1520-0493(1995)123<2791:IITGHP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., , R. E. Tuleya, , and M. A. Bender, 1998: The GFDL hurricane prediction system and its performance in the 1995 hurricane season. Mon. Wea. Rev., 126, 13061322, doi:10.1175/1520-0493(1998)126<1306:TGHPSA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lawson, W. G., , and J. A. Hansen, 2005: Alignment error models and ensemble-based data assimilation. Mon. Wea. Rev., 133, 16871709, doi:10.1175/MWR2945.1.

    • Search Google Scholar
    • Export Citation
  • Liu, Q., , T. Marchok, , H.-L. Pan, , M. Bender, , and S. Lord, 2000: Improvements in hurricane initialization and forecasting at NCEP with global and regional (GFDL) models. NOAA Tech. Procedures Bull. 472, 7 pp.

  • Mainelli, M., , M. DeMaria, , L. K. Shay, , and G. Goni, 2008: Application of oceanic heat content estimation to operational forecasting of recent Atlantic category 5 hurricanes. Wea. Forecasting, 23, 316, doi:10.1175/2007WAF2006111.1.

    • Search Google Scholar
    • Export Citation
  • Rogers, R., and Coauthors, 2006: The intensity forecasting experiment: A NOAA multiyear field program for improving tropical cyclone intensity forecasts. Bull. Amer. Meteor. Soc., 87, 15231537, doi:10.1175/BAMS-87-11-1523.

    • Search Google Scholar
    • Export Citation
  • Torn, R. D., 2010: Performance of a mesoscale ensemble Kalman filter (EnKF) during the NOAA high-resolution hurricane test. Mon. Wea. Rev., 138, 43754392, doi:10.1175/2010MWR3361.1.

    • Search Google Scholar
    • Export Citation
  • Torn, R. D., , and G. J. Hakim, 2009: Ensemble data assimilation applied to RAINEX observations of Hurricane Katrina (2005). Mon. Wea. Rev., 137, 28172829, doi:10.1175/2009MWR2656.1.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924, doi:10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Wu, C.-C., , G.-Y. Lien, , J.-H. Chen, , and F. Zhang, 2010: Assimilation of tropical cyclone track and structure based on the ensemble Kalman filter. J. Atmos. Sci., 67, 38063822, doi:10.1175/2010JAS3444.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Y. Weng, , J. A. Sippel, , Z. Meng, , and C. H. Bishop, 2009: Cloud-resolving hurricane initialization and prediction through assimilation of Doppler radar observations with an ensemble Kalman filter. Mon. Wea. Rev., 137, 21052125, doi:10.1175/2009MWR2645.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Y. Weng, , J. F. Gamache, , and F. D. Marks, 2011: Performance of a convection-permitting hurricane initialization and prediction during 2008–2010 with ensemble data assimilation of inner-core airborne Doppler radar observations. Geophys. Res. Lett., 38, L15810, doi:10.1029/2011GL048469.

    • Search Google Scholar
    • Export Citation
  • Zou, X., , and Q. Xiao, 2000: Studies on the initialization and simulation of a mature hurricane using a variational bogus data assimilation scheme. J. Atmos. Sci., 57, 836860, doi:10.1175/1520-0469(2000)057<0836:SOTIAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
1

The impact of this centering approach on tilted storms will be considered in section 4a.

2

The weight on the storm centered analysis is given by 1 − w.

3

At this time it appears that our test of this method is the first for a vortical flow; previous tests applied to one-dimensional solitons.

4

Since the domain size is much larger than the depth, horizontal units are expressed in kilometers, and vertical units are given in meters.

5

In the second shallow-water experiment, features in the surrounding environment prevent the vortex from interacting with the domain boundaries.

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