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  • View in gallery
    Fig. 1.

    Illustration of the observation coverage in the vicinity of the TC basin from 1200 UTC 26 Jun 2004. The surface and sounding observations are much more dense over land and were not included in this figure for clarity. The best-track locations of Typhoon Mindulle (10) and Typhoon Tingting (11) are indicated by TC symbols.

  • View in gallery
    Fig. 2.

    Limited-area domain used for the limited-area analyses.

  • View in gallery
    Fig. 3.

    (a) Position and (b) intensity RMSE for the global LETKF analysis and the differences between the (c) position and (d) intensity errors in the truncated operational analysis and the LETKF analysis. Each bar shows the RMSE error or the difference between the RMSE for a particular storm (numbered), where the mean is computed over the life cycle of the TC or the RMSE error difference averaged over all times for all storms [AVG in (c) and (d)]. Shaded bars in (c) and (d) indicate storms for which the difference is statistically significant. For the numbering of the storms, see Table 1. A positive value indicates that the LETKF analysis is more accurate than the reanalysis or the truncated operational analysis.

  • View in gallery
    Fig. 4.

    Comparison between the errors in the coarse resolution limited-area LETKF analyses and the global LETKF analysis. The format of the figure is as in Fig. 3. A positive value indicates that the limited-area analysis is more accurate.

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    Fig. 5.

    Comparison between the errors for the limited-area analyses of different resolutions using the format of Fig. 3. A positive value indicates that the higher-resolution analysis is more accurate. The resolution increases from top to bottom.

  • View in gallery
    Fig. 6.

    Stratification of the distribution of the position analysis errors by storm intensity. The distributions are obtained by grouping errors for (top) all category 3 and 4 cyclones, (middle) category 1 and 2 cyclones, and (bottom) tropical storms and depressions. Each box plot represents the distribution of the analysis errors for a different analysis system.

  • View in gallery
    Fig. 7.

    As in Fig. 6, but for the intensity errors.

  • View in gallery
    Fig. 8.

    Background vs analysis root-mean-square position error. Each dot indicates a pair of background and analysis errors for a single analysis. Results are shown for (rows) different configurations of the LETKF and (columns) different storm intensities.

  • View in gallery
    Fig. 9.

    Difference between the root-mean-square steering flow errors for the global analysis and the limited-area analyses at horizontal resolutions of (a) 200, (b) 100, (c) 48, and (d) 36 km. The mean in the computation of the root-mean-square error is taken over all verification times. Positive values indicate a superior regional analysis performance. The typhoon tracks have been marked. Note the difference in the shading interval between (a) and the rest of the panels. Shading is only plotted for those values that indicate a significant difference between the mean of the time series at each grid point.

  • View in gallery
    Fig. 10.

    Difference between the root-mean-square steering flow background errors for the global and limited-area backgrounds at horizontal resolutions of (a) 200, (b) 100, (c) 48, and (d) 36 km. Computation is as in Fig. 9.

  • View in gallery
    Fig. 11.

    The difference between the magnitudes of the background error reduction by the assimilation of observations in the global and the limited-area analyses. Results are shown for the limited-area analyses of (a) 200-, (b) 100-, (c) 48-, and (d) 36-km resolution. Positive values indicate that the limited-area system reduced the background error more than the global system.

  • View in gallery
    Fig. 12.

    Comparison between the evolution of the root-mean-square error in the global (black) and limited-area (gray) forecast position. Results are shown for the initially (a) strong, (b) moderate, and (c) weak storms.

  • View in gallery
    Fig. 13.

    Domain-averaged difference between the RMS errors in the global and limited-area forecasts of the steering flow. Positive values indicate a superior regional analysis performance.

  • View in gallery
    Fig. 14.

    Difference between the root-mean-square errors in the global and the limited-area forecasts of the steering flow at forecast times (a) 0, (b) 24, (c) 48, (d) 72, (e) 96, and (f) 120 h. The mean in the computation of root-mean-square error is taken over all verification times. Positive values indicate a superior regional analysis performance. The typhoon tracks have been marked.

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    Fig. 15.

    Distributions of the errors in the forecast of the position of the initially weak storms.

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Can a Moderate-Resolution Limited-Area Data Assimilation System Add Value to the Global Analysis of Tropical Cyclones?

Christina R. HoltTexas A&M University, College Station, Texas

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Istvan SzunyoghTexas A&M University, College Station, Texas

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Gyorgyi GyarmatiTexas A&M University, College Station, Texas

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Abstract

This study investigates the benefits of employing a limited-area data assimilation (DA) system to enhance lower-resolution global analyses in the northwest Pacific tropical cyclone (TC) basin. Numerical experiments are carried out with a global analysis system at horizontal resolution T62 and a limited-area analysis system at resolutions from 200 to 36 km. The global and limited-area DA systems, which are both based on the local ensemble transform Kalman filter algorithm, are implemented using a unique configuration, in which the global DA system provides information about the large-scale analysis and background uncertainty to the limited-area DA system. The limited-area analyses of the storm locations are, on average, more accurate than those from the global analyses, but increasing the resolution of the limited-area system beyond 100 km has little benefit. Two factors contribute to the higher accuracy of the limited-area analyses. First, the limited-area system improves the accuracy of the location estimates for strong storms, which is introduced when the background is updated by the global assimilation. Second, it improves the accuracy of the background estimate of the storm locations for moderate and weak storms. Improvements in the steering flow analysis as a result of increased resolution are modest and short lived in the forecasts. Limited-area track forecasts are more accurate, on average, than global forecasts, independently of the strength of the storms up to five days. This forecast improvement is due to the more accurate analysis of the initial position of storms and the better representation of the interactions between the storms and their immediate environment.

Corresponding author address: Christina R. Holt, Texas A&M University 1204 Eller O&M, 3150 TAMU, College Station, TX 77843. E-mail: crh403@tamu.edu

Abstract

This study investigates the benefits of employing a limited-area data assimilation (DA) system to enhance lower-resolution global analyses in the northwest Pacific tropical cyclone (TC) basin. Numerical experiments are carried out with a global analysis system at horizontal resolution T62 and a limited-area analysis system at resolutions from 200 to 36 km. The global and limited-area DA systems, which are both based on the local ensemble transform Kalman filter algorithm, are implemented using a unique configuration, in which the global DA system provides information about the large-scale analysis and background uncertainty to the limited-area DA system. The limited-area analyses of the storm locations are, on average, more accurate than those from the global analyses, but increasing the resolution of the limited-area system beyond 100 km has little benefit. Two factors contribute to the higher accuracy of the limited-area analyses. First, the limited-area system improves the accuracy of the location estimates for strong storms, which is introduced when the background is updated by the global assimilation. Second, it improves the accuracy of the background estimate of the storm locations for moderate and weak storms. Improvements in the steering flow analysis as a result of increased resolution are modest and short lived in the forecasts. Limited-area track forecasts are more accurate, on average, than global forecasts, independently of the strength of the storms up to five days. This forecast improvement is due to the more accurate analysis of the initial position of storms and the better representation of the interactions between the storms and their immediate environment.

Corresponding author address: Christina R. Holt, Texas A&M University 1204 Eller O&M, 3150 TAMU, College Station, TX 77843. E-mail: crh403@tamu.edu

1. Introduction

The recent improvements in the forecasts of tropical cyclone (TC) tracks are usually attributed to increasing model resolution, improved data assimilation techniques, and the rapid increase in the number of routinely assimilated observations over oceans (Rappaport et al. 2009). Improvements in the intensity analyses and forecasts of TCs have been much more modest, which is not surprising considering that the eyewall radius of a TC is about 25–50 km (Kimball and Mulekar 2004), and from 4 to 10 grid points are needed to resolve a flow feature such as the eye of a TC (e.g., Fiorino and Elsberry 1989; Grasso 2000; Skamarock 2004). The 5-km nominal resolution that would be required, at minimum, to capture the structure of the eye of a TC is expected to remain unattainable in an operational real-time global analysis/forecast system until about 2020 (Källén 2012).

In this study we investigate the effect of model resolution on the accuracy of TC analyses. In particular, we study the situation in which a limited-area data assimilation system is employed to enhance a lower-resolution global analysis in the TC region. Since the resolutions we consider are coarser than 10 km, at which the flow feature in the model is a warm-core vortex rather than a TC, we can hope to see significant improvements in the analysis of the position, but not in the analysis of the intensity. To further explain our motivation for this study, it is important to note that the superiority of a fully cycled limited-area analysis to a lower-resolution global analysis is not self-evident. For instance, the National Centers for Environmental Prediction (NCEP) has struggled for many years to develop a limited-area data assimilation system that would provide a better analysis for their limited-area model than their global assimilation system. Their current limited-area data assimilation system is based on a partial cycling strategy: a cold start of the regional data assimilation system is started from a global analysis 12 h prior to the actual analysis time and cycled over four 3-h analysis time windows to produce the limited-area analysis (Rogers et al. 2009). The most likely explanation for the difficulties with fully cycling the limited-area analysis is that propagating information about the large-scale flow through the lateral boundaries for an extended time is a challenging task. In addition, the gainful assimilation of satellite radiance observations, which has shown promising results for the global setting, is still an open problem of limited-area data assimilation, because, among other issues, no effective algorithm currently exists to estimate the bias in the radiance observations within the framework of a limited-area data assimilation system (Schwartz et al. 2012).

The particular region selected for this study is the northwest Pacific TC basin. We use the analysis/forecast system described in Merkova et al. (2011) to assimilate observations collected in the summer of 2004. This system consists of a research data assimilation system based on the local ensemble transform Kalman filter (LETKF) algorithm, the NCEP Global Forecast System (GFS) model and Regional Spectral Model (RSM). The somewhat unusual choice of using the RSM as the regional model is motivated by the fact that it was designed to be as consistent with the GFS as possible, which makes it particularly suitable to isolate the effect of resolution.

We use a configuration of the data assimilation system, in which, first the global and then the regional analysis is prepared. In addition to this configuration, Merkova et al. (2011) also considered another one, in which the high-resolution regional background information was used in both the limited-area and the global data assimilation processes. While that configuration of the data assimilation system is not used in this study, one motivation for the research reported here has been to develop a better understanding of the potential benefits of the availability of the higher-resolution model information in a limited-area domain for both the global and the limited-area data assimilation processes. In the future, we hope to utilize the knowledge we gain to build a data assimilation system that would analyze the joint states of the global and limited area following an approach that was successfully tested by Yoon et al. (2012) on a toy model system.

We use an ensemble-based data assimilation approach, in part, because it provides a simple framework to introduce information about the large-scale uncertainty into the limited-area data assimilation process. Another motivation to use an ensemble-based system is the growing body of evidence that ensemble-based data assimilation techniques are particularly well suited for TC data assimilation (e.g., Torn and Hakim 2009; Torn 2010; Hamill et al. 2011; Zhang et al. 2011). It is important to point out, however, that because of the lack of assimilating satellite radiance observations, and because of the low resolution of our global data assimilation system, our results have limited direct value for operational numerical weather prediction. The main value of our results is that they provide an assessment of the value of the higher resolution of a limited-area system in a well-controlled experiment setting.

The paper is organized as follows. Section 2 gives an overview of the experimental design, while section 3 describes our analysis and forecast verification methods. Section 4 describes the analyses, while section 5 presents the forecast verification results. Section 6 is a summary and discussion of the main results.

2. Experiment design

a. Analysis/forecast system

The data assimilation system is an implementation of the LETKF algorithm (Ott et al. 2004; Hunt et al. 2007) on the NCEP GFS model (Szunyogh et al. 2005, 2008) and the NCEP RSM (Merkova et al. 2011). The GFS is a spectral-transform model, and we integrate it using a triangular truncation with a cutoff wavenumber of 62 and 28 vertical sigma levels (T62L28).

The RSM is a nested limited-area version of the GFS model (Juang and Kanamitsu 1994), which uses a one-way nesting; that is, the global solution affects the regional solution, but the regional solution has no effect on the global solution. The vertical levels in the RSM are identical to the 28 sigma levels of the GFS. The horizontal resolution of the RSM in our experiments, which is defined by the grid spacing, varies from 200 to 36 km. The particular implementation of the analysis/forecast system used here was first tested on winter storms over the United States and is described by Merkova et al. (2011). To obtain the LETKF analyses, all observations that were assimilated by NCEP in real time in the summer of 2004 are used, excluding satellite radiance observations and Tropical Cyclone Vitals Database (TCVitals) information. (An example of the locations of such observations for a typical analysis cycle is shown in Fig. 1.) In addition, the procedure to relocate the TCs in the background (first guess of the analysis), which has been used in the operational systems of NCEP (Liu et al. 2000), is not used in our system. Hence, adjustment of the position of the storm by the data assimilation is due entirely to the assimilation of observations that do not provide direct information about the TC position.

Fig. 1.
Fig. 1.

Illustration of the observation coverage in the vicinity of the TC basin from 1200 UTC 26 Jun 2004. The surface and sounding observations are much more dense over land and were not included in this figure for clarity. The best-track locations of Typhoon Mindulle (10) and Typhoon Tingting (11) are indicated by TC symbols.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

b. LETKF data assimilation

The LETKF, similar to other formulations of the ensemble Kalman filter, generates an ensemble of forecasts to estimate the background and covariance matrix of the background error. To be precise, the state update equation is
e1
where is the analysis; is the background, which is computed by taking the mean of the background ensemble; is the ensemble-based estimate of the Kalman gain matrix bT(bT + )−1; and is the observation innovation, where yo is the vector of observations and H is the observation function that maps the model gridpoint variables into observables at the observation locations. The matrix represents the linearization of H about and is the observation error covariance matrix. The ensemble provides the estimate of both the bT matrix of background covariance for the observation locations and the bT matrix of background covariance for the observation and grid point locations.

The unique features of the LETKF are that it assimilates all observations that may affect the estimate of a given state vector component simultaneously, and the analyses of the different state vector components are computed independently of each other (Hunt et al. 2007; Szunyogh et al. 2008; Merkova et al. 2011). In practice, the observations that may affect the analysis at the given grid point are selected by assimilating all observations from a prescribed local volume around the grid point. The definition of the local volume is a tunable parameter of the LETKF algorithm. In essence, the local volume has to be sufficiently small so that the ensemble can provide an efficient representation of the most important degrees of freedom in the space of the background errors, but sufficiently large, so the local volumes centered at neighboring grid points include similar subsets of the observations, which is necessary to ensure the smooth spatial variation of the analyzed fields.

c. Configuration of the data assimilation systems

In our configuration of the data assimilation system, each global ensemble member has a limited-area counterpart: the kth member of the global analysis ensemble at the previous analysis time, tn−1, provides the initial condition for the global forecast that (i) produces the kth member of the global background ensemble at the analysis time tn and (ii) provides the boundary conditions and the large-scale forcing for the limited-area forecast that produces the kth member of the limited-area background ensemble at tn. The initial condition for the computation of the kth member of the limited-area background ensemble at tn is the kth member of the limited-area analysis ensemble at tn−1. The global data assimilation process at tn produces the members of the global analysis ensemble at tn, while the limited-area data assimilation process provides the members of the limited-area analysis ensemble at tn. In what follows, we refer to the ensemble mean of the global analysis ensemble as the global analysis and the ensemble mean of the limited-area analysis ensemble as the limited-area analysis.

The tunable parameters of the LETKF are set to values that were found to provide a near-optimal performance of the analysis system given the available computational resources by the earlier studies with the same system (Szunyogh et al. 2008; Merkova et al. 2011). In particular, all experiments are carried out with K = 40 ensemble members and observations are assimilated within 800 km of each grid point in both the global and the limited-area system. In addition, covariance inflation is applied to the ensemble-based estimates of the background error covariance matrix to compensate for the effects of sampling errors (due to the finite size of the ensemble) and nonlinearities in the evolution of the state estimation errors, and to account for the effects of model errors. The covariance inflation in the global system is variable with height and with latitude. In the Southern Hemisphere and tropics, the inflation ranges from 1.25 at the surface to 1.2 at the top model level. There is a transition zone near 25°N where the inflation tapers from 1.35 at the surface to 1.3 at the top. At latitudes higher than 25° in the Northern Hemisphere, the inflation factor is 1.5 at the surface tapering to 1.3 at the top level. The covariance inflation factor in the regional system varies only with height and ranges from 1.5 at the surface to 1.3 in the top level.

Four daily analyses are obtained by assimilating observations from an observation time window of Δt = 6 h. This approach provides analyses at 0000, 0600, 1200, and 1800 UTC. Analyses are prepared for the period between 0000 UTC 22 June 2004 and 1800 UTC 15 August 2004. We generate the initial global ensemble for 0000 UTC 22 June 2004, which is necessary to start the cycling of the analysis, with the 0000 UTC operational GFS analyses truncated to T62 resolution from 40 different days during the summer of 2004. During the cycling period, there were eight TCs in the northwest Pacific basin. Of these eight, there were a variety of storm intensities ranging from tropical depression (<34 kt; 1 kt = 0.5144 m s−1) to category 4 (113–136 kt) on the Saffir–Simpson scale (Atangan et al. 2004). Basic information for each of these storms is included in Table 1.

Table 1.

2004 typhoons and tropical storms (TS) included in this study. Storm number indicates the order in which the storm was named in the 2004 season. Data are taken from Atangan et al. (2004).

Table 1.

The domain for the limited-area calculations is chosen such that it includes the tropical northwest Pacific, a large portion of eastern Asia, and the northeast Pacific Ocean (Fig. 2). The resolution of the limited-area system starts at roughly the same scale as the global analysis (200 km) and is increased in increments to 36 km at which point it becomes too computationally expensive to further increase the resolution for the same domain.

Fig. 2.
Fig. 2.

Limited-area domain used for the limited-area analyses.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

Five-day deterministic forecasts from both the global and limited-area 48-km analyses are prepared every 12 h over the course of the nearly two-month period we study.

3. Verification methods

a. Nominal resolution

Analyzing and predicting the storm locations on a finite-resolution grid introduces an inevitable error component into the position analyses and forecasts. Assuming that the four grid points surrounding the storm location are the vertices of a Δx × Δx square, the magnitude of this error component can take any value between zero and with equal probability. The discretization error formally enters the data assimilation process by limiting the accuracy of the observation function H and the accuracy of the linearization of H.

b. Effective resolution

Resolution also has a major effect on the accuracy of the estimate of the background error covariance matrices bT and bT. These effects are controlled, not by the grid spacing, but the much coarser effective resolution of the model, because the ensemble can estimate the background error covariances only at those scales that the model can resolve. The effective resolution of the analyses and forecasts is coarser than the Δx grid spacing (nominal resolution) of the model. The exact difference between the effective and the nominal resolution depends on a number of factors, which include the discretization strategy, the truncation strategy (when a spectral discretization is used), and the parameterization schemes that represent the effect of processes at the subgrid scales on the processes at the resolved scales.

The NCEP GFS is a spectral transform model, which we integrate using a triangular truncation with cutoff wavenumber T = 62. The model uses an aliasing-free approach for the computation of the nonlinear terms, which requires that the number of grid points in the zonal direction M satisfies the condition M ≥ 3T + 1, while the number of grid points in the meridional direction Y satisfies Y ≥ 3T/2. To satisfy these conditions, we set the number of grid points to M = 192 and N = 94. In the region where the TCs are typically located, these choices for M and N provide a nominal resolution of Δx ≈ 200 km.

To provide an estimate of the effective resolution of our global analyses, we note that the LETKF analysis is obtained on the 192 × 94 grid, but then transformed to spectral space to obtain the spectral coefficients that provide the initial conditions for the T62 forecasts. This step amounts to a spectral filtering of the initial conditions; for instance, the cutoff wavenumber for the discrete Fourier transform of 192 gridpoint variables in the zonal direction would be 96 (>62), thus the spatial resolution associated with a cutoff wavenumber 62 is about 300 km. The effective resolution, however, is expected to be even lower than that: in an ensemble-based Kalman filter scheme, the analysis is a linear combination of the background ensemble members, which implies that the analysis cannot have a higher resolution than that of the background forecasts, which usually have little energy at the tail end of the spectrum due to the diffusive effect of the parameterization schemes. The overall effect of the parameterization schemes must be diffusive, because the energy, which would otherwise accumulate at the tail end of the spectrum, has to be removed (e.g., Durran 2010).

An estimate for the effective resolution, which accounts for the effect of diffusion and is usually considered a conservative one, can be obtained by assuming that diffusion wipes out most kinetic energy for wavenumbers larger than 2T/3. Applying this estimation approach to our T62 resolution global analysis fields, leads to an estimate of 460 km for the effective resolution. Based on similar arguments, the effective resolution of our 100-, 48-, and 36-km limited-area simulations are about 230, 110, and 80 km, respectively.

Even more pessimistic estimates of the effective resolution are obtained when it is defined as the smallest scale where the model correctly captures the power spectrum of the kinetic energy distribution in the atmosphere. Following this approach, Källén (2012) concluded that the global model of the European Centre for Medium-Range Weather Forecasts (ECMWF) at nominal resolutions 16, 10, and 5 km, had an effective resolution of 110, 70, and 30 km. Notice that this approach provides the same estimate of 110 km for the effective resolution at nominal resolution 16 km as our estimate based on the 2T/3 rule at nominal resolution 48 km. This relationship between the two estimates indicate that while a 48-km resolution model can represent flow features with a characteristic spatial scale of 110 km, an efficient representation of the nonlinear interactions between those features would require a spatial resolution of 16 km.

c. Estimation of the track and intensity error

The Joint Typhoon Warning Center’s (JTWC) best-track data (Atangan et al. 2004) are used as the verification dataset to assess the errors in the track and intensity analyses and forecasts. The best track is issued after the JTWC reassesses all available data once a storm has dissipated. The dataset includes both 6-hourly track and intensity information, but the intensity information has larger errors. The use of the Dvorak model is the main source of the discrepancy in intensity from real-time observations. Lookup tables based on the Dvorak intensity index have been derived from empirical data to provide corresponding estimates of minimum central pressure and maximum wind speed in given basins (Velden et al. 2006). Another discrepancy from other agencies’ best-track datasets arises from using the 1-min mean sustained wind speed. This procedure can lead to estimates that are higher than those provided by other agencies, which base their estimates on a 10-min mean (Chu et al. 2002). Since neither the best-track data, nor the real-time version (TCVitals) (Keyser 2007), were assimilated in our experiments, the best-track dataset provides independent verification information.

The best-track data are all point estimates and are compared to our gridded analysis and forecast data. Storm locations in the analyses and forecasts are determined using a technique described by Suzuki-Parker (2012, chapter 2): first we identify the location of the storm as the grid point with the maximum positive vorticity at the 850-hPa level in the model, then determine the storm intensity as the lowest SLP found within 1° of the vorticity maximum. Once the cyclone tracks and intensities are extracted for all analyses or forecasts at times included in the best-track data, root-mean-square errors (RMSEs) are calculated over all the locations along the track to provide a measure of the track and intensity error for the entire lifetime of each TC.

d. Discretization error

Because we use a 2.5° × 2.5° grid to verify the global analyses and forecasts, for which Δx ≈ 260 km, the maximum and the mean of the discretization error are about 180 and 90 km, respectively. For the 100-km resolution limited-area grid, the maximum of the discretization error is 90 km, while the mean is 45 km. The same numbers for the 48-km grid are 34 and 17 km, while for the 36-km grid they are 26 and 13 km.

e. Statistical significance test for autoregressive process

To test the statistical significance of the differences between the RMSE for the different analyses, we employ a method described in Szunyogh et al. (2008) and Aravéquia et al. (2011). The test is based on a paired t test with a null hypothesis that the means of the two time series are the same. Assuming that the difference between two time series of analysis errors Δ(t) is described by a first-order autoregressive process, the test statistic is
e2
where is the time average over Δ(t) and VΔ is the variance of the time series [Δ(t), t = 1, 2, …, T]. In Eq. (2), the effective sample size T′ is
e3
where the autocorrelation coefficient r is computed by
e4
Here, ΔT1 is the average over the first T − 1 time steps of Δ(t) and ΔT2 is the average over the last T − 1 time steps of Δ(t).

Under the assumption that the time series of errors is described by a first-order autoregressive process, autocorrelation, r should fall within the range [0, 1]. While the intensity analysis and forecast error statistics generally satisfy this assumption, the position analysis errors do not, because the analysis position error has a “short memory.” Thus, in the computation of the statistical significance of the difference between the track errors of each experiment, we assume that the effective sample size T′ is equal to the sample size T.

Once z is computed, the difference between the time series is deemed significant if the likelihood of obtaining this z is less than the significance level being tested. Differences between analysis time series of pressure or track error are sufficiently significant for our purposes if the probability of achieving the observed value of z, using a t distribution with T′ − 1 degrees of freedom, is greater than 90% (i.e., p < 0.05 or p > 0.95). An example of some of these metrics is provided in Table 2 for the comparison of the global LETKF and the limited-area 100-km LETKF experiments.

Table 2.

Number of time steps T, autocorrelation coefficient r, effective sample size T′, and p value of the test statistic for each of the storms from the comparison of the global LETKF and RSM 100-km experiments for TC intensity.

Table 2.

f. Steering flow error

A necessary condition for an accurate tropical cyclone track forecast, including the forecast that provides the background, is an accurate analysis and prediction of the steering flow. Here, we define the steering flow by the mass-weighted vertical average of the wind vector between the 850- and the 250-hPa pressure levels. We define the error in the deep layer wind, STWE, as
e5
where u and υ are the deep layer mean-weighted wind components in our analyses and forecasts; ut and υt are the proxies for the true values of u and υ, respectively; and the overbar represents the time mean over all verification times. The two-dimensional field of STWE is calculated for the entire limited-area domain. We use u and υ fields computed based on the NCEP operational analysis at 1° horizontal resolution as the proxies ut and υt. The flow associated with the TCs is not removed for the calculation of the environment flow.

g. Stratification by storm intensity

Verification statistics stratified by cyclone intensity are also prepared. This verification approach is, in part, motivated by Torn (2010), who found that the largest errors in the intensity analyses tended to occur for strong TCs (categories 3–5), while the largest track errors tended to occur for weak TCs (tropical depressions and storms). The intensity and the position errors are separated based on best-track wind speed into the three groups used in Torn (2010): Tropical depression/tropical storm (<63 kt), categories 1–2 (63–96 kt), and categories 3–5 (>96 kt).

h. NCEP operational analyses

To assess the quality of our LETKF analyses, we employ the NCEP operational analyses from 2004 as both benchmarks and verification data. For the computation of the verification statistics, the operational analyses are considered on a 2.5° × 2.5° grid for the comparison with the global LETKF analyses, and on a 1° × 1° (about 100-km resolution) grid for the comparisons with the limited-area analyses. The operational NCEP analysis used the same version of the model as we did, but with the spectral statistical interpolation (SSI) data assimilation system, a 3D variational analysis method (Parrish and Derber 1992). The operational analyses also has a higher, T254L64, resolution, which is roughly equivalent to a 50-km nominal resolution at 20°N. It also assimilated a large number of satellite radiance observations in addition to the observations assimilated in our system, and most importantly, it employed a TC relocation technique (Liu et al. 2000) based on the TCVitals. Thus, the main sources of error in the operational analyses are the discretization error and the differences between the TCVitals and the best-track data. The latter source of error was examined by Trahan and Sparling (2012), who found that for the 2004 season, the frequency of the TCVitals position being different from the JTWC best-track position by more than 40 km was just over 10%. They also found, however, that in particular cases, the difference between the best-track data and the TCVitals were surprisingly large. For instance, the difference between the JTWC best-track data and the TCVitals can be larger than 100 km for weak storms.

4. Analysis verification results

a. Verification of the global LETKF analyses

We start the discussion of the analysis verification results with the validation of the global LETKF system. The purpose of this validation exercise is to show that the limited-area analyses are compared to a global analysis of reasonable quality.

The global position RMSE for the different storms vary between about 150 and 450 km (Fig. 3a), while the intensity RMSE varies between 5 and 30 hPa for the LETKF analysis for each storm (Fig. 3b). When the position RMSE is compared for the global LETKF and the operational analysis at the resolution of the LETKF analysis, the difference between the two position analyses is much smaller than the RMSE for the LETKF analysis. As expected, the differences between the intensity RMSE for the two systems are small, as neither of the analyses considered here has sufficient resolution to resolve the dynamics of the most intense phase of the life cycle of a TC.

Fig. 3.
Fig. 3.

(a) Position and (b) intensity RMSE for the global LETKF analysis and the differences between the (c) position and (d) intensity errors in the truncated operational analysis and the LETKF analysis. Each bar shows the RMSE error or the difference between the RMSE for a particular storm (numbered), where the mean is computed over the life cycle of the TC or the RMSE error difference averaged over all times for all storms [AVG in (c) and (d)]. Shaded bars in (c) and (d) indicate storms for which the difference is statistically significant. For the numbering of the storms, see Table 1. A positive value indicates that the LETKF analysis is more accurate than the reanalysis or the truncated operational analysis.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

b. Comparison of the global and the regional LETKF analyses

We first compare the performance of the limited-area data assimilation system to that of the global LETKF system (Fig. 4). Results are shown only for the 200- and the 100-km resolution regional analyses. The comparison shows, as expected, that when the regional and global analyses have about the same resolution (200 km), the global analysis is more accurate. While we expect this result to be rather general, we also expect the magnitude of the difference between the errors in the global and the limited-area analyses of equal resolution to be strongly dependent on such factors as the resolution, the size of the regional domain, the choice of data assimilation system, and the observation density.

Fig. 4.
Fig. 4.

Comparison between the errors in the coarse resolution limited-area LETKF analyses and the global LETKF analysis. The format of the figure is as in Fig. 3. A positive value indicates that the limited-area analysis is more accurate.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

When the resolution of the limited-area analysis is doubled to 100 km, the limited-area analysis has a statistically significant advantage over the global system for three of the eight storms and shows some significant systematic advantage. For the rest of the storms, the limited-area analyses are more accurate, but not at the 90% confidence level. For all but two storms, the 100-km limited-area intensity analysis is also more accurate, but from a practical forecasting point of view, the advantage of the limited-area analysis is small.

c. Comparison of the different resolution regional analyses

In Fig. 5, the accuracy of the analyses is compared for each resolution to the accuracy of the analyses at the one step higher resolution (e.g., Figs. 5a and 5b compare the 200- and the 100-km analyses, Figs. 5c and 5d compare the 100- and the 48-km analyses, etc.). There is a clear improvement in the position analyses only when the resolution is increased from 200 to 100 km, but further increases of the resolution show no further significant improvement. There is also a statistically significant improvement in the intensity analysis when the resolution is increased from 200 to 100 km. While further increases of the resolution lead to further improvements in the intensity analysis (right panels), the magnitude of the improvement is small from a practical point of view. (The average reduction in the intensity RMSE due to increasing the resolution from 100 to 48 km, or from 48 to 36 km, is a mere 0.2%.)

Fig. 5.
Fig. 5.

Comparison between the errors for the limited-area analyses of different resolutions using the format of Fig. 3. A positive value indicates that the higher-resolution analysis is more accurate. The resolution increases from top to bottom.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

d. Stratification of the errors by storm intensity

The effect of storm intensity on the errors in the global and the limited-area position analyses is shown in Fig. 6. The statistics shown in this figure are based on 15 data points for strong storms, 48 data points for moderate storms, and 155 data points for weak storms. The results suggest that storm intensity has a major effect on the distribution of the position error. First, the mean and the median of the errors for the strong storms are smaller than for the moderate and weak storms for all configurations of the analysis system. Second, large errors, which appear as statistical outliers in Fig. 6 in the case of weak storms, are not nearly as large and do not occur with as great a frequency in the case of moderate and strong storms, as in the case of the weak storms. We note that the outliers for the weak storms are typically due to the given analysis not resolving a circulation at all. In addition, weaker systems often have less well-defined steering flow. Most likely, these two factors lead to the larger number of outliers for the weak storms.

Fig. 6.
Fig. 6.

Stratification of the distribution of the position analysis errors by storm intensity. The distributions are obtained by grouping errors for (top) all category 3 and 4 cyclones, (middle) category 1 and 2 cyclones, and (bottom) tropical storms and depressions. Each box plot represents the distribution of the analysis errors for a different analysis system.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

The maximum error for the global LETKF analyses of strong storms is 246 km, while the mean error is 136 km. The maximum is larger by about 60 km, while mean by about 46 km, than what we would expect if the errors were solely due to discretization errors. The distribution of the errors, however, is very similar to that for the operational global analysis truncated to the resolution of the global LETKF analysis. That is, at that resolution, the LETKF can compensate for the advantage the operational system has as a result of the use of TC relocation.

The limited-area analyses of the positions of the strong storms are clearly more accurate than the global LETKF analyses of the storms. According to a Kolmogorov–Smirnov two-sample test for distribution (e.g., Massey 1951), the distributions of the errors for both the 100- and the 36-km resolution systems, but not for the 48-km system, are significantly different from the distribution of position errors for the global LETKF at a 90% confidence level. Neither the distributions, nor the means of the errors are different for the three different resolution limited-area analyses. Nevertheless, the maximum of the position error (excluding outliers) indicates a slight decrease of the position error with increasing resolution: the maximum of the position error for the 100-, 48-, and 36-km resolution limited-area analyses is, respectively, 201, 177, and 94 km.

The mean (85 km) and the maximum (201 km) of the error for the 100-km resolution system is consistent with our estimate of the effective resolution (230 km) for that grid spacing, but the mean and the maximum for the 48- and 36-km resolution systems are larger than what would be expected based on the estimates of the effective resolution. This suggests that at resolutions finer than 100 km, the analysis system cannot take advantage of the smaller discretization errors and the higher effective resolution. This result suggests that either there is no sufficient observed information to take advantage of the higher-resolution grid and/or the quality of the estimates of the background covariances do not improve sufficiently to lead to a better use of the available observations. The operational analysis, which has a nominal resolution about the same as that of the 48-km resolution limited-area analysis, can achieve a higher accuracy than the limited-area analyses (a mean error of 58 km and a maximum error of 99 km, excluding outliers). The higher accuracy of the operational analysis is most likely primarily due to the use of TC relocation: the TCVitals are based on a large amount of observed information, which is not assimilated in our system. In addition, the TC relocation procedure does not rely on the estimates of the background covariances for the correction of the storm location, thus its accuracy is limited by the grid spacing and the accuracy of the TCVitals, rather than the effective resolution. Finally, the assimilation of a large number of radiance observations in the operational system, which are not assimilated in our system, may lead to a more accurate analysis of the steering flow.

The superiority of the higher-resolution limited-area analyses to the global LETKF analysis is most apparent for the weak storms: the distributions and means of the errors are significantly different from the global distribution of errors for all limited-area resolutions at the 90% confidence level.

For completeness, we also show the stratification of the intensity errors by storm intensity (Fig. 7). The most striking feature of the intensity errors is that they are larger for the strong storms than for the moderate storms, and for the moderate storms than for the weak storms. The differences between the performance of the global systems, the performance of the global and limited-area systems, and the performance of different resolution limited-area systems are small and not statistically significant. This result suggests that, as expected, analyzing the intensity is equally challenging for all investigated configurations of the system.

Fig. 7.
Fig. 7.

As in Fig. 6, but for the intensity errors.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

e. Stratification of the analysis increments by storm intensity

Next, we investigate the effect of storm intensity on the relationship between the background and the analysis errors. The results of our investigation are summarized in Fig. 8. In this figure, a dot over a diagonal indicates a case in which the assimilation of the observation improved the state estimate provided by the background, while a dot below the diagonal indicates a degradation by the assimilation of observations. The one-sentence segue of the results is that the assimilation of observations tends to help when the background error is large. In particular, in the case of strong storms, data assimilation has a positive effect on the accuracy of the state estimate only for the 200-km resolution data assimilation system, which is the one configuration that tends to provide a low-accuracy background for strong storms. The assimilation of the observations leads to the largest improvement in the 100-, 48-, and 36-km resolution analyses of moderate storms. There is a similar, but somewhat less dramatic improvement in the state estimates for the weak storms.

Fig. 8.
Fig. 8.

Background vs analysis root-mean-square position error. Each dot indicates a pair of background and analysis errors for a single analysis. Results are shown for (rows) different configurations of the LETKF and (columns) different storm intensities.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

f. Errors in the analysis of the steering flow

The accuracy of the analysis of the steering flow can have a major effect on the accuracy of the analysis of the location of the storms at later times through the background estimate of the storm locations. The geographical distribution of the RMSE of the difference between the analyses of the steering flow in the different limited-area experiments is shown in Fig. 9. (We recall that the proxy for the true steering flow in the computation of the error is the operational NCEP analysis at 1° resolution.) In the extratropics, the limited-area analyses of the steering flow are generally more accurate than the global analysis in all but the 200-km resolution limited-area analysis. In the tropics the advantage of the limited-area system is less obvious. The only regions where the limited-area analyses are clearly less accurate than the global analysis are near the western and eastern lateral boundaries in the extratropics. The magnitude of the degradation near the boundaries decreases with increasing resolution. This result is most likely due to the fact that in the higher-resolution system more grid points are used for the relaxation of the limited-area model solution to the global model solutions near the lateral boundaries.

Fig. 9.
Fig. 9.

Difference between the root-mean-square steering flow errors for the global analysis and the limited-area analyses at horizontal resolutions of (a) 200, (b) 100, (c) 48, and (d) 36 km. The mean in the computation of the root-mean-square error is taken over all verification times. Positive values indicate a superior regional analysis performance. The typhoon tracks have been marked. Note the difference in the shading interval between (a) and the rest of the panels. Shading is only plotted for those values that indicate a significant difference between the mean of the time series at each grid point.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

Is the improved analysis of the steering flow in the 100 km (and higher) resolution limited-area analyses due to an improved background and/or to a larger improvement of the state estimate in the update step? First, we compare the background errors for the different resolution analyses (Fig. 10). For the interpretation of the results, we recall that the background is the ensemble mean of the ensemble of 6-h forecasts. The results show that except for the 200-km resolution configuration, the background error is smaller for the limited-area systems, than the global system. In addition, while increasing the resolution from 200 to 100 km helps significantly, the further increase of the resolution cannot further enhance the quality of the limited-area background. This result suggests that the 100-km and finer resolution analyses are more accurate than the global analysis, in part, because the higher-resolution limited-area model provides a more accurate background. To see whether the higher-resolution limited-area analysis can also more efficiently assimilate the observations, we compare the magnitude of the corrections made by the limited-area and the global analyses to the background (Fig. 11). The results show that the limited-area systems more efficiently reduce the background error.

Fig. 10.
Fig. 10.

Difference between the root-mean-square steering flow background errors for the global and limited-area backgrounds at horizontal resolutions of (a) 200, (b) 100, (c) 48, and (d) 36 km. Computation is as in Fig. 9.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

Fig. 11.
Fig. 11.

The difference between the magnitudes of the background error reduction by the assimilation of observations in the global and the limited-area analyses. Results are shown for the limited-area analyses of (a) 200-, (b) 100-, (c) 48-, and (d) 36-km resolution. Positive values indicate that the limited-area system reduced the background error more than the global system.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

There are two potential factors that can contribute to the more efficient use of the observations by the limited-area data assimilation systems: the more accurate estimation of the background covariance and the more accurate computation of the observation operator. In a system where the observation operator implements only spatial and temporal interpolations, as is the case in our study, a higher-resolution grid is a guarantee for a more accurate observation operator. A more accurate observation operator can result in a more accurate analysis because it leads to more accurate computation of the innovation, δy. It can also lead to a more accurate weighting of the background and the observed information in the analysis, by allowing for a more accurate estimation of bT. Finally, it spreads the information more efficiently from the observation locations to the model grid points with the help of a more accurate estimate of bT. The small improvement found in the analysis of the steering wind when the resolution is increased beyond 100 km suggests that once that resolution is reached, the interpolation errors become secondary compared to the errors in the background and the estimates of the background error statistics and/or the limited availability of observed information.

5. Forecast verification results

a. Mean forecast error

We compare the error in the global forecasts started from the global LETKF analyses with the error in the 48-km forecasts started from the 48-km limited-area analyses. The mean error of the position forecasts for the first five forecast days is reduced independent of the storm strength at analysis time (Fig. 12). Are these forecasts improved because of the better analysis of the position of the storms or to the improvements in the analysis of the steering flow? First, in the domain averaged sense, the advantage of the limited-area forecast of the steering flow is short lived; about 18 h (Fig. 13). A more careful investigation of the error in the steering flow shows, however, that in the immediate vicinity of some of the storms the improvements in the wind forecasts survive up to about 72-h forecast time (Fig. 14). Thus, we conjecture that the forecast improvements are due to the more accurate analysis of the storm locations and the better representation of the interactions between the storms and their immediate environment.

Fig. 12.
Fig. 12.

Comparison between the evolution of the root-mean-square error in the global (black) and limited-area (gray) forecast position. Results are shown for the initially (a) strong, (b) moderate, and (c) weak storms.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

Fig. 13.
Fig. 13.

Domain-averaged difference between the RMS errors in the global and limited-area forecasts of the steering flow. Positive values indicate a superior regional analysis performance.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

Fig. 14.
Fig. 14.

Difference between the root-mean-square errors in the global and the limited-area forecasts of the steering flow at forecast times (a) 0, (b) 24, (c) 48, (d) 72, (e) 96, and (f) 120 h. The mean in the computation of root-mean-square error is taken over all verification times. Positive values indicate a superior regional analysis performance. The typhoon tracks have been marked.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

b. Stratification of forecast errors by storm intensity

We find statistically significant improvements only in the forecasts of the weak storms (Fig. 15). (The results for the moderate and the strong storms are not shown.) Even though the errors can be quite large in the forecasts of these storms, the forecast improvement in the mean error by the limited-area system is statistically significant up to the 72-h forecast time. Since the errors for the weak storms are larger than for the moderate and strong storms, their weight in the overall average of the errors is also larger. The combination of these relatively large improvements in the forecast of weak storms and the small (statistically not significant) improvements in the forecasts of the moderate and strong storms lead to the clear improvement in the overall forecast errors shown in Fig. 12.

Fig. 15.
Fig. 15.

Distributions of the errors in the forecast of the position of the initially weak storms.

Citation: Monthly Weather Review 141, 6; 10.1175/MWR-D-12-00093.1

6. Conclusions

We investigated the benefits of employing a limited-area data assimilation system to enhance the lower-resolution global analyses in the northwest Pacific TC basin. While several authors used ensemble-based data assimilation systems in the past to downscale information about the background uncertainty in limited-area systems employing multiple nests in the TC regions (Torn and Hakim 2009; Zhang et al. 2011), to the best of our knowledge, ours is the first study to use the approach in a setting where the outermost nest is a global model. The resolutions considered here were much lower than what would be necessary to resolve the inner core of the TCs and observations of the inner core were not assimilated. Since studies by others have shown that such capabilities are necessary to achieve significant reductions in the analyses and the ensuing forecasts of the intensity (e.g., Zhang et al. 2011), we did not expect to find significant differences between the quality of the limited-area and the global intensity analyses.

We found that the limited-area data assimilation system enhanced the accuracy of the analysis of the position of the storms, but the benefits of increasing the resolution beyond 100 km were limited. (The particular value of the critical resolution would most likely change with the resolution of the global model and the observational datasets assimilated.) Two factors contributed to the higher accuracy of the limited-area analysis:

  • in the case of strong (categories 3–5) storms, the assimilation of observations in the global system often degrades the accuracy of the analysis, while the effect of the assimilation of observations in the limited-area system is closer to neutral (last column of Fig. 8);

  • in the case of the moderate (categories 1–2) and weak (tropical storms and depressions) storms, the use of the limited-area system greatly reduces the number of unusually large (statistical outlier) errors (Fig. 6), because the assimilation of the observations can efficiently correct the large background errors in the limited-area systems (bottom three rows in the first and middle columns of Fig. 8).

We also found that the limited-area system improved the prediction of the storm tracks for the first five forecast days. Our analysis shows that the forecast improvement is due to the more accurate analysis of the position of the storms and the better representation of the interactions between the storm and their immediate environment.

Based on the results of the present study, can we expect that the global analysis would benefit from feeding back information from the limited-area system using the joint states approach of Yoon et al. (2012)? First, the higher accuracy of the analysis of the steering flow in the limited-area analyses suggests that the higher resolution of the limited-area analysis leads to a better interpretation of the observations in the case of moderate and weak storms. This result suggests that the information provided by the limited-area system about the position for the global system is potentially of lower value for the strong storms, for which the analysis error is closer to the discretization error than for the weaker storms.

Acknowledgments

We thank our collaborators at AER, Inc., Ross Hoffman and Mark Leidner. Russ Schumacher and Robert Korty provided much guidance through the duration of the work and deserve many thanks. The comments of two anonymous reviewers, as well as Ross Hoffman, were very helpful and made thought-provoking suggestions that ultimately led to a better presentation of our findings. The work was supported by ONR Grant N000140910589.

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