• Anderson, J. L., 2012: Localization and sampling error correction in ensemble Kalman filter data assimilation. Mon. Wea. Rev., 140, 23592371, doi:10.1175/MWR-D-11-00013.1.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., , and Collins N. , 2007: Scalable implementations of ensemble filter algorithms for data assimilation. J. Atmos. Oceanic Technol., 24, 14521463, doi:10.1175/JTECH2049.1.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 14310 162, doi:10.1029/94JC00572.

    • Search Google Scholar
    • Export Citation
  • Greybush, S. J., , Kalnay E. , , Miyoshi T. , , Ide K. , , and Hunt B. R. , 2011: Balance and ensemble Kalman filter localization techniques. Mon. Wea. Rev., 139, 511522, doi:10.1175/2010MWR3328.1.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , Whitaker J. S. , , and Snyder C. , 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 27762790, doi:10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hodur, R. M., 1997: The Naval Research Laboratory’s Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS). Mon. Wea. Rev., 125, 14141430, doi:10.1175/1520-0493(1997)125<1414:TNRLSC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hogan, T. F., , and Rosmond T. E. , 1991: The description of the Navy Operational Global Atmospheric Prediction System’s spectral forecast model. Mon. Wea. Rev., 119, 17861815, doi:10.1175/1520-0493(1991)119<1786:TDOTNO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., , and Mitchell H. L. , 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796811, doi:10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lu, H. J., , Xu Q. , , Yao M. M. , , and Gao S. T. , 2011: Time-expanded sampling for ensemble-based filters: Assimilation experiments with real radar observations. Adv. Atmos. Sci., 28, 743757, doi:10.1007/s00376-010-0021-4.

    • Search Google Scholar
    • Export Citation
  • McLay, J. G., , Bishop C. H. , , and Reynolds C. A. , 2008: Evaluation of the ensemble transform analysis perturbation scheme at NRL. Mon. Wea. Rev., 136, 10931108, doi:10.1175/2007MWR2010.1.

    • Search Google Scholar
    • Export Citation
  • McLay, J. G., , Bishop C. H. , , and Reynolds C. A. , 2010: A local formulation of the ensemble transform (ET) analysis perturbation scheme. Wea. Forecasting, 25, 985993, doi:10.1175/2010WAF2222359.1.

    • Search Google Scholar
    • Export Citation
  • Meng, Z., , and Zhang F. , 2007: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part II: Imperfect model experiments. Mon. Wea. Rev., 135, 14031423, doi:10.1175/MWR3352.1.

    • Search Google Scholar
    • Export Citation
  • Meng, Z., , and Zhang F. , 2008a: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part III: Comparison with 3DVAR in a real-data case study. Mon. Wea. Rev., 136, 552540.

    • Search Google Scholar
    • Export Citation
  • Meng, Z., , and Zhang F. , 2008b: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part IV: Comparison with 3DVAR in a month-long experiment. Mon. Wea. Rev., 136, 36713682, doi:10.1175/2008MWR2270.1.

    • Search Google Scholar
    • Export Citation
  • Mitchell, H. L., , Houtekamer P. L. , , and Pellerin G. , 2002: Ensemble size, balance, and model-error representation in an ensemble Kalman filter. Mon. Wea. Rev., 130, 27912808, doi:10.1175/1520-0493(2002)130<2791:ESBAME>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Peng, M. S., , Ridout J. A. , , and Hogan T. F. , 2004: Recent modifications of the Emanuel convective scheme in the Navy Operational Global Atmospheric Prediction System. Mon. Wea. Rev., 132, 12541268, doi:10.1175/1520-0493(2004)132<1254:RMOTEC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and Hamill T. M. , 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
  • Xu, Q., , Lu H. , , Gao S. , , Xue M. , , and Tong M. , 2008a: Time-expanded sampling for ensemble Kalman filter: Assimilation experiments with simulated Radar observations. Mon. Wea. Rev., 136, 26512667, doi:10.1175/2007MWR2185.1.

    • Search Google Scholar
    • Export Citation
  • Xu, Q., , Wei L. , , Lu H. , , Qiu C. , , and Zhao Q. , 2008b: Time-expanded sampling for ensemble-based filters: Assimilation experiments with a shallow-water equation model. J. Geophys. Res., 113, D02114, doi:10.1029/2007JD008624.

    • Search Google Scholar
    • Export Citation
  • Xue, M., and et al. , 2001: The Advanced Regional Prediction System (ARPS)—A multiscale nonhydrostatic atmospheric simulation and prediction tool. Part II: Model physics and applications. Meteor. Atmos. Phys., 76, 143165, doi:10.1007/s007030170027.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Meng Z. , , and Aksoy A. , 2006: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part I: Perfect model experiments. Mon. Wea. Rev., 134, 722736, doi:10.1175/MWR3101.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Weng Y. , , Sippel J. , , Meng Z. , , and Bishop C. , 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
  • Zhao, Q., , Zhang F. , , Holt T. , , Bishop C. , , and Xu Q. , 2013: Development of a mesoscale ensemble data assimilation system at the Naval Research Laboratory. Wea. Forecasting, 28, 13221336, doi:10.1175/WAF-D-13-00015.1.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Three COAMPS domains for the 45-, 15-, and 5-km nested grids used in this study.

  • View in gallery

    Ensemble spreads [defined by (3.1)] of potential temperature (°C; colored areas) and horizontal wind speed (m s−1; contours with an interval of 1 m s−1) at the model vertical level of σ = 7800 m from the experiments of (a) Ens10, (b) Ens10-tes, and (c) Ens30 of the EnKF analyses at 1200 UTC 23 Jun 2005. (d) The difference between (b) and (a). Colored areas in (d) are the changes of potential temperature ensemble spread (°C), where positive values indicate increases in ensemble spread by TES. Similarly, solid (dashed) contours show increases (decreases) in ensemble spread of wind speed (with an interval of 0.5 m s−1).

  • View in gallery

    Geopotential height analysis (m; contours) at 1200 UTC 23 Jun 2005 and the changes in geopotential height (m; colored areas) in the last 12 h.

  • View in gallery

    Noncorrelation [%; defined by (3.4)] of (a) θ, (b) q, (c) u, and (d) υ between forecast hours 9 and 12 of the previous ensemble forecasts that were used as background fields for the EnKF data assimilation with TES at 1200 UTC 23 Jun 2005. The white contour outlines the areas with noncorrelations larger than 50%. The model vertical level is the same as in Fig. 2.

  • View in gallery

    As in Fig. 4, but for forecast hours 12–15.

  • View in gallery

    Vertical profiles of mean biases of the EnKF analyses for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves), Ens10-tes (blue curves), and Ens30 (green curves) verified at 1200 UTC 23 Jun 2005 against rawinsondes. The vertical coordinate is pressure (hPa).

  • View in gallery

    As in Fig. 6, but for RMS differences.

  • View in gallery

    RMS errors of the ensemble means of the COAMPS forecasts for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves), Ens10-tes (blue curves), and Ens30 (green curves) verified every 12 h starting at 1200 UTC 23 Jun 2005 against rawinsondes.

  • View in gallery

    Vertical profiles of biases from the EnKF analyses for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves) and Ens10-tes (blue curves) averaged over the 6-day period and verified every 12 h starting at 1200 UTC 23 Jun 2005 against rawinsondes. Ensemble means were used in the verification. The vertical coordinate is pressure (hPa).

  • View in gallery

    As in Fig. 9, but for RMS differences.

  • View in gallery

    RMS errors of the COAMPS forecasts for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves) and Ens10-tes (blue curves) averaged over the 6-day period and verified every 12 h starting at 1200 UTC 23 Jun 2005 against rawinsondes. Ensemble means were used in the verification.

  • View in gallery

    Average reductions of RMS errors of the ensemble means of the COAMPS forecasts over the 72-h forecast period by TES for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1), as a function of the sampling time interval τ. The average RMS error reductions are calculated from (3.5). A positive value means improvement in the ensemble forecast by TES. The average RMS error reductions are displayed for the COAMPS nested grids of 45 (blue curves), 15 (red curves), and 5 km (green curves). The sampling time interval is the time interval (h) between two adjacent sampling times.

  • View in gallery

    As in Fig. 11, but for the results from COAMPS’s three nested grids: (top) 45-, (middle) 15-, and (bottom) 5-km grid for (from left to right) T (°C), u (m s−1), υ (m s−1), and q (g kg−1).

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 9 9 2
PDF Downloads 3 3 0

Time-Expanded Sampling for Ensemble-Based Data Assimilation Applied to Conventional and Satellite Observations

View More View Less
  • 1 Marine Meteorology Division, Naval Research Laboratory, Monterey, California
  • | 2 National Storms Laboratory, Norman, Oklahoma
  • | 3 Marine Meteorology Division, Naval Research Laboratory, Monterey, California
© Get Permissions
Full access

Abstract

The time-expanded sampling (TES) method, designed to improve the effectiveness and efficiency of ensemble-based data assimilation and subsequent forecast with reduced ensemble size, is tested with conventional and satellite data for operational applications constrained by computational resources. The test uses the recently developed ensemble Kalman filter (EnKF) at the Naval Research Laboratory (NRL) for mesoscale data assimilation with the U.S. Navy’s mesoscale numerical weather prediction model. Experiments are performed for a period of 6 days with a continuous update cycle of 12 h. Results from the experiments show remarkable improvements in both the ensemble analyses and forecasts with TES compared to those without. The improvements in the EnKF analyses by TES are very similar across the model’s three nested grids of 45-, 15-, and 5-km grid spacing, respectively. This study demonstrates the usefulness of the TES method for ensemble-based data assimilation when the ensemble size cannot be sufficiently large because of operational constraints in situations where a time-critical environment assessment is needed or the computational resources are limited.

Corresponding author address: Dr. Qingyun Zhao, Naval Research Laboratory, 7 Grace Hopper Ave., Mail Stop II, Monterey, CA 93943. E-mail: allen.zhao@nrlmry.navy.mil

Abstract

The time-expanded sampling (TES) method, designed to improve the effectiveness and efficiency of ensemble-based data assimilation and subsequent forecast with reduced ensemble size, is tested with conventional and satellite data for operational applications constrained by computational resources. The test uses the recently developed ensemble Kalman filter (EnKF) at the Naval Research Laboratory (NRL) for mesoscale data assimilation with the U.S. Navy’s mesoscale numerical weather prediction model. Experiments are performed for a period of 6 days with a continuous update cycle of 12 h. Results from the experiments show remarkable improvements in both the ensemble analyses and forecasts with TES compared to those without. The improvements in the EnKF analyses by TES are very similar across the model’s three nested grids of 45-, 15-, and 5-km grid spacing, respectively. This study demonstrates the usefulness of the TES method for ensemble-based data assimilation when the ensemble size cannot be sufficiently large because of operational constraints in situations where a time-critical environment assessment is needed or the computational resources are limited.

Corresponding author address: Dr. Qingyun Zhao, Naval Research Laboratory, 7 Grace Hopper Ave., Mail Stop II, Monterey, CA 93943. E-mail: allen.zhao@nrlmry.navy.mil

1. Introduction

It is well known that ensemble size plays a critical role in the performance of ensemble-based data assimilation systems and forecasts (Whitaker and Hamill 2002; Mitchell et al. 2002). This is because of the requirement of adequately representing the probability density function (pdf) of the model state by an ensemble of state vectors in an ensemble-based filter. Theoretically, the ensemble size should be sufficiently large in order to appropriately estimate the forecast error covariance. With the help of widely used localization techniques (Houtekamer and Mitchell 1998; Hamill et al. 2001; Anderson and Collins 2007; Anderson 2012; Greybush et al. 2011), however, this requirement has been significantly relaxed. In an early study by Houtekamer and Mitchell (1998) with an ensemble Kalman filter (EnKF; Evensen 1994) for a low-resolution global model, an ensemble size of 32 gave a reasonable level of performance of the EnKF with a properly selected localization radius. As the ensemble size increased to above 64, the EnKF performed much better and the performance appeared to be less dependent on the localization radius. Similar results were also reported by Zhao et al. (2013) using a high-resolution EnKF for the U.S. Navy’s Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS1; Hodur 1997). In both studies, however, the ensemble size of 16 produced unacceptable results no matter what localization length scales were used. These studies suggest that, even with the help of an appropriately selected localization length scale, the ensemble size should be adequately large in order to obtain reliable error covariances for the EnKF analyses.

For most major global and regional numerical weather prediction (NWP) centers, the computational cost of running an ensemble data assimilation and forecast system with an ensemble size of 60 or above is no longer a problem because of the rapid increase in computational resources in recent years. For some short- or near-term applications such as rapid environment assessment (REA) or nowcasting, however, the use of ensemble-based analyses and forecasts can still present a major challenge. In these situations, there is a desire to use large amounts of data from conventional, in situ, and remotely sensed observations and computationally expensive ensemble-based methods, but quick turnaround time is desired, which limits the amount of data available and requires a fast analysis and forecast solution. This is particularly true when the analyses and forecasts must be performed at forward-deployed sites where communications bandwidth and computational resources are limited compared to major NWP centers.

A time-expanded sampling (TES) method was developed by Xu et al. (2008b) for ensemble-based filters. This approach samples a series of (preferably three) perturbed state vectors from each prediction run in an ensemble of forecasts at properly selected time levels in the vicinity of the analysis time. As all the sampled state vectors are used to construct the ensemble and compute the covariance (with localization), the number of required prediction runs and the associated computation costs can be greatly reduced. If the sampling time interval is properly selected, the proposed approach can improve the ensemble spread and enrich the spread structures so that the filter can perform well even though the number of forecasts is greatly reduced. The TES method was developed first with a shallow-water equation model (Xu et al. 2008b). It was then tested in the Advanced Regional Prediction System (ARPS; Xue et al. 2001) and a storm-scale EnKF for radar data assimilation with simulated and real radar observations (Xu et al. 2008a; Lu et al. 2011). All the experimental studies showed improved EnKF performance even though the ensemble size was significantly reduced.

The TES technique has been adopted and integrated into the Naval Research Laboratory (NRL) EnKF for COAMPS aimed at improving ensemble analyses and forecasts for rapid environmental assessment for the U.S. Navy. The system has been extensively tested with real observations from conventional meteorological networks and meteorological satellites. In particular, assimilation experiments have been designed and performed to investigate how the TES technique performs with conventional meteorological observations and satellite data on the synoptic scale, mesoscale, and storm-allowing scale. This paper reports the results from our recent experimental studies and demonstrates the potential usefulness of the TES technique for time-constrained ensemble analyses and forecasts for rapid and time-sensitive environment assessment. In section 2, we will describe briefly the TES technique. The observational data and the EnKF and COAMPS configurations are described in section 2 along with the experimental setup. Section 3 shows the results from the experiments, followed by discussions on how and to what extent the TES method is able to improve the ensemble analyses and forecasts with a limited number of ensemble members. Conclusions are given in section 4.

2. The TES technique and the experimental design for testing

As explained in Xu et al. (2008b), the idea behind the TES technique was simply inspired by the often observed fact that the model-predicted weather system usually develops or propagates either faster or slower than the observed system in the real atmosphere. As such, the predicted field at a time level before or after the analysis time may better represent the true field than the one at the analysis time. The difference between the predicted field sampled before or after the analysis time and the one at the analysis time may represent, to a certain extent, the model’s forecast errors in both intensity and location of the predicted weather system. The extra information so provided about the model forecast errors may help improve the estimation of the localized covariance especially when the ensemble size is not large enough for such a statistical calculation. Mathematically, if vector represents the spatially discretized state variables (potential temperature θ, horizontal winds u and υ, and water vapor mixing ratio q) valid at the analysis time t from the ensemble member i, the ensemble of state vectors from an ensemble forecast of size N can be expressed by the following matrix:
e2.1
where square brackets denote a matrix. Similarly, the ensemble of state vectors sampled from the same ensemble forecast but valid at three time levels, tτ, t, and t + τ, respectively, can be written as
e2.2
where τ is the time interval between two adjacent time samplings. Suppose we also have an ensemble forecast of size 3N, and then the ensemble of state vectors sampled at time t is
e2.3
Based on the above explanation, the added state vectors in contain extra information about storms that develop and/or propagate too fast while have the information of storms that develop and/or propagate too slowly. These additional state vectors in improve the coverage of the uncertainties in the model forecast errors, especially the phase errors, and make better in representing the model forecast errors at time t than . The extra information contained in is, however, neither the same as nor fully covered by the information contained in , because the 3N forecasts in are all integrated by the same imperfect model with the unknown model errors neglected and therefore they are subject to the same types of phase errors (caused by the unknown model errors) similarly to the N forecasts in .

An indirect but effective way to validate the above-explained idea of TES is to evaluate the ensemble analyses from the EnKF and the consequential ensemble forecasts from different ensemble experiments. For this purpose, three ensemble experiments are designed: one for ensemble of size N without TES, one for the same ensemble but with TES, and one for the ensemble with a size of 3N. The NRL EnKF is employed as the ensemble data assimilation system. The NRL EnKF is an ensemble square root filter (EnSRF; Whitaker and Hamill 2002) that has its origins in a research version of the EnKF originally developed for the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) and the Weather Research and Forecasting (WRF) Model (Zhang et al. 2006, 2009; Meng and Zhang 2007, 2008a,b). It was implemented at NRL first as a research tool for COAMPS ensemble data assimilation and forecasting. Numerous major changes and improvements have been made to the NRL EnKF since then, to make it more suitable for not only research (Zhao et al. 2013) but also operational applications such as rapid environment assessment and nowcasting. COAMPS is selected as both the target model for data assimilation and the forecast model in the ensemble experiments. Three COAMPS nested grids with 45-, 15-, and 5-km spacing are used in this study and the grid domains are given in Fig. 1.

Fig. 1.
Fig. 1.

Three COAMPS domains for the 45-, 15-, and 5-km nested grids used in this study.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

The Navy Operational Global Atmospheric Prediction System (NOGAPS; Hogan and Rosmond 1991; Peng et al. 2004) ensemble forecasts are used as boundary conditions for COAMPS ensemble forecasts and first guesses for cold starts of the EnKF analyses. The NOGAPS ensembles are produced using the latitude-banded ensemble transform technique as described by McLay et al. (2008, 2010). This setup avoids the need to randomly generate initial and lateral boundary perturbations for COAMPS ensemble members and, more importantly, enables the global ensemble to transmit flow-dependent boundary condition uncertainty information to the regional ensemble. After the cold start, all experiments are warm started with a 12-h update cycle and the ensemble forecasts from the previous forecast cycle are then used as the first guesses for the designed experiments. It should be mentioned that there is no need for a spinup period in this study since the NOGAPS ensemble has been run for a long period of time.

Conventional observations (from rawinsondes, pilot balloons, land surface stations, ships and buoys, and commercial aircrafts) and satellite products (mostly satellite winds and derived temperature profiles) within the analysis domains are selected to be assimilated into the EnKF for the study. The main reason for selecting these types of observations is that these data are well quality controlled with better error statistics than data from new types of sensors. Table 1 lists the three ensemble experiments mentioned earlier where the ensemble size N = 10 for the control experiment Ens10. We select this ensemble size mainly because in most REA applications, the number of ensemble runs usually cannot exceed 15. This is especially true when high-resolution analyses and forecasts are needed to resolve the storm-scale features and capture their rapid changes while the analyses and forecasts of the large-scale environment are also required to support the storm development. For simplicity, we use τ = 3 h as the value for the time interval between the two adjacent time samplings in the Ens10-tes experiment. More discussions about the selection of τ will be given in section 3c.

Table 1.

List of experiments.

Table 1.

In this study, localization is used in the EnKF for the estimation of background error covariance. Based on the study with COAMPS by Zhao et al. (2013), the horizontal localization cutoff radius is set to values of 675, 225, and 75 km for the COAMPS’s three nested grids of 45-, 15-, and 5-km grid resolution, respectively. In the vertical direction, however, all three nested grids use the same localization cutoff depth that covers up to 10 vertical grid points in one direction within the 30 COAMPS sigma levels as configured in Zhao et al. (2013). The use of this vertical-grid-dependent cutoff depth allows the vertical extent of the background error correlation to be localized according to the vertical resolution of the model-predicted background field. COAMPS sigma levels are unevenly distributed. With the cutoff depth set to 10 vertical levels, an observation at the surface can impact the model fields from the surface up to the top of the boundary layer, and an observation at 500 hPa can affect the model fields from the top part of the boundary layer to the top of the troposphere.

3. Experimental results

a. Improved representation of forecast error uncertainty by TES

The three ensemble experiments listed in Table 1 were cold started at 1200 UTC 22 June and then warm started at 0000 UTC 23 June 2005. After that, the 12-h ensemble forecasts from each of the ensemble experiments were used as first guesses for the EnKF data assimilation. For the EnKF data assimilation with TES, three time levels of ensemble forecasts at forecast hours 9, 12, and 15, respectively, were sampled and used as first guesses for the estimation of background error covariance in each assimilation cycle (every 12 h). At the end of each data assimilation cycle, 72-h ensemble forecasts of the same sizes were launched from each of the three experiments. The 12-h analysis–forecast cycle continued for 6 days ending at 1200 UTC 28 June 2005 for Ens10 and Ens10-tes, whereas Ens30 was ended at 1200 UTC 23 June because of the larger computational and data storage requirements.

Figures 2a–c display the ensemble spreads of potential temperature (colored areas) and horizontal wind speed (contours) of the EnKF analyses for the 45-km grid at the model vertical level of σ = 7800 m from the three experiments valid at 1200 UTC 23 June. Here, the ensemble spread is measured by the ensemble standard deviation SD that is computed by
e3.1
where N is the ensemble size, xi is a given state variable from the ith ensemble member, and
e3.2
Figures 2a–c show that for all three experiments, the major areas of large ensemble spread are located over the eastern United States and its coastal areas. Over the central part of the United States, the ensemble spreads from the three experiments are relatively small. Surface analyses (not shown) indicate that during the 24-h period ending at 1200 UTC 23 June, there was a surface cold front moving southward over the eastern part of the United States. Figure 3 gives the 500-hPa geopotential height analysis (the contours) at 1200 UTC 23 June and the past 12-h geopotential height changes (colored areas). As seen in Fig. 3, a long-wave trough in the 500-hPa geopotential height was located along the East Coast right off the coastline while the central part of the United States was dominated by a strong 500-hPa ridge. At the analysis time, the eastern part of the United States and the coastal waters were dynamically active areas where the atmosphere was undergoing a major weather pattern transition as the 500-hPa height was decreasing in front of the long-wave trough and increasing behind the trough. The Northwest was another dynamically active area. The 500-hPa geopotential height over the northern Rocky Mountains was decreasing as a weak 500-hPa trough was approaching. The large ensemble spreads in the eastern part of the United States and the nearby ocean in Figs. 2a–c represent the large uncertainties in the model’s forecasts of intensities and locations of the weather systems in these areas by the individual ensemble members for the three experiments used as background fields for the EnKF data assimilation. Compared to Ens30, however, the ensemble spreads from Ens10 are smaller. This could be caused by the smaller ensemble size that may have underrepresented the uncertainties in model forecast errors in the EnKF data assimilation. Ens10-tes shows increased ensemble spreads by TES (Fig. 2b) compared to Ens10 (Fig. 2a) and that the increased spreads are close to those from Ens30 (Fig. 2c). At some locations, the ensemble spreads in Fig. 2b are even larger than those in Fig. 2c. This difference may be caused by two factors. First, the 30 ensemble members may not have large enough spreads at some locations to represent the uncertainties in model forecast errors; second, the TES method could have induced spuriously large spreads at some locations as a result of the time-expanded samplings of the model states. Nevertheless, small ensemble spread caused by the small ensemble size in Ens10 has been substantially increased by TES. To better show the ensemble spread increases by TES, Fig. 2d gives the difference between Figs. 2a and 2b. The increases in ensemble spread by TES are mainly located in two areas: the above-mentioned eastern part of the United States and the northern Rocky Mountains (Fig. 2d) associated with an approaching weak upper-level trough at the analysis time (Fig. 3). The improved ensemble spread by TES is useful in improving the representation of the background error variance used in the data assimilation in storm-active areas especially when the actual ensemble size is small.
Fig. 2.
Fig. 2.

Ensemble spreads [defined by (3.1)] of potential temperature (°C; colored areas) and horizontal wind speed (m s−1; contours with an interval of 1 m s−1) at the model vertical level of σ = 7800 m from the experiments of (a) Ens10, (b) Ens10-tes, and (c) Ens30 of the EnKF analyses at 1200 UTC 23 Jun 2005. (d) The difference between (b) and (a). Colored areas in (d) are the changes of potential temperature ensemble spread (°C), where positive values indicate increases in ensemble spread by TES. Similarly, solid (dashed) contours show increases (decreases) in ensemble spread of wind speed (with an interval of 0.5 m s−1).

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

Fig. 3.
Fig. 3.

Geopotential height analysis (m; contours) at 1200 UTC 23 Jun 2005 and the changes in geopotential height (m; colored areas) in the last 12 h.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

In addition to the improved ensemble spread in representing the forecast error variance, the effectiveness of the TES is also reflected in its representation of another important aspect of the forecast error uncertainty: the independence of the forecast error. The ensemble representation of forecast error independence can be impacted positively (or negatively) if the two subsets of ensemble members sampled at the analysis time and the time selected before or after the analysis time (separated by τ = 3 h) are largely uncorrelated (or correlated). The impact may depend again very much on the activeness and time variations of the local weather system on the spatial scale. To verify this, the ensemble correlation between the two subsets of members sampled at the aforementioned two time levels (separated by τ = 3 h) is defined and calculated for each model variable at each grid point by the following formulation:
e3.3
where x1ik and x2ik denote the values at the kth grid point for the ith member (in each subset) of the particular variable at the analysis time t1 and the time of t2 = t1τ or t1 + τ, N is the subset ensemble size, and the overbar represents averaging of each subset over the ensemble members.
The noncorrelation is then defined and computed by
e3.4
where denotes the absolute value. The calculated noncorrelation fields at the model vertical level of σ = 7800 m between forecast hours 9 (3 h before the analysis time) and 12 (at the analysis time) are given in Fig. 4 for the state variables of θ, q, u, and υ. Figure 5 shows the same fields but between forecast hours 12 and 15 (3 h after the analysis time). It is very interesting to notice that in the dynamically active areas in the eastern part of the United States and over the northern Rocky Mountains, the noncorrelation fields are higher (>50%) than those in other areas. The noncorrelation in the central part of United States, where a strong high pressure system was located, is relatively low ( < 20%). This verifies the aforementioned hypothesis and, in particular, it reveals that in the dynamically active areas, the ensemble forecasts at the selected two time levels are less likely to be correlated to each other than those in a dynamically stable area. The larger values of reveal the rapid changes of the weather systems in the dynamically active areas and the large uncertainties in the ensemble forecasts associated with the rapidly evolving weather systems. In those dynamically active areas, with a properly selected τ value, the fields sampled at a different time level before or after the analysis time from the same ensemble runs can be basically treated as additional ensemble members that contain additional information about the model forecast errors. In the dynamically stable areas, however, the additional information gained by the TES is very limited. This can be seen by noting that major increases in the ensemble spread by TES shown in Fig. 2d are mainly restricted within the areas with > 50% in Fig. 3. Thus, the matrix in (2.2) provides a better representation of the model forecast error uncertainty than the matrix in (2.1) and can be nearly as good as the matrix in (2.3).
Fig. 4.
Fig. 4.

Noncorrelation [%; defined by (3.4)] of (a) θ, (b) q, (c) u, and (d) υ between forecast hours 9 and 12 of the previous ensemble forecasts that were used as background fields for the EnKF data assimilation with TES at 1200 UTC 23 Jun 2005. The white contour outlines the areas with noncorrelations larger than 50%. The model vertical level is the same as in Fig. 2.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

Fig. 5.
Fig. 5.

As in Fig. 4, but for forecast hours 12–15.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

b. Improved ensemble analyses and forecasts by TES

The improved representation of model forecast error uncertainty by the TES leads to improved EnKF analyses. Figure 6 shows the vertical profiles of bias for the state variables of temperature T, u, υ, and q of the ensemble mean from the EnKF analyses of the three experiments in Table 1 at 1200 UTC 23 June for the 45-km grid. Figure 7 gives the vertical profiles of the root-mean-square (RMS) differences between the ensemble mean and the observations. Rawinsondes at 1200 UTC 23 June were used as observations for the calculations. Here, we use the RMS difference instead of RMS error because the rawinsondes used in the calculations were also assimilated into the EnKF. Although the verification is not completely independent (because of the use of rawinsondes in the verification), we are still able to use the biases and RMS differences to measure the effectiveness of the EnKF in assimilating observations into the ensemble analyses. This can also be supported by the results, as shown later, that the reduced biases and RMS differences in the ensemble analyses lead to improved ensemble forecasts.

Fig. 6.
Fig. 6.

Vertical profiles of mean biases of the EnKF analyses for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves), Ens10-tes (blue curves), and Ens30 (green curves) verified at 1200 UTC 23 Jun 2005 against rawinsondes. The vertical coordinate is pressure (hPa).

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for RMS differences.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

As seen in Fig. 6, the impact of TES on reducing the bias of the EnKF analyses is notable, with the Ens10-tes (blue curves) giving overall smaller biases than Ens10 (the red curves) although the bias patterns from the two experiments look similar. The biases from Ens30 (green curves) are still the smallest among these three experiments. The impact of TES on reducing the RMS differences in Fig. 7, however, is more obvious. Overall, Ens10 has the largest RMS differences while the Ens30 has the smallest ones. Ens10-tes is somewhere between the two but is closer to the Ens30 than to the Ens10 at most vertical levels. The RMS reductions by TES for wind analyses in Figs. 7c and 7d are especially apparent. Figures 6 and 7 show that the extra information about the model forecast errors from the time-expanded samplings at three time levels in Ens10-tes can improve the ensemble analyses at least on the scale resolved by the assimilated observations (i.e., the radiosondes in this case).

The 72-h ensemble forecasts were launched from the ensemble analyses at 1200 UTC 23 June for all three experiments. Verifications of the ensemble forecasts were conducted against rawinsondes every 12 h during the 72-h forecast period. RMS errors were calculated for the state variables of T, u, υ, and q of the ensemble mean. The results from the three experiments are given in Fig. 8. As expected, the improved ensemble analyses also lead to improved ensemble forecasts. The Ens10 again has the largest RMS errors while Ens30 overall has the best forecasts. For wind and moisture forecasts, the Ens10-tes forecasts are almost as good as those from Ens30. For temperature, Ens10-tes lies between Ens10 and Ens30 and closer to Ens30. The results from this case again demonstrate that Ens10-tes has the potential to provide ensemble analyses and forecasts with an accuracy close to what the Ens30 can provide while the computational cost is just about one-third of that needed by Ens30 (the increase in computational cost by TES in EnKF data assimilation is basically negligible compared to the time needed for the ensemble forecasts).

Fig. 8.
Fig. 8.

RMS errors of the ensemble means of the COAMPS forecasts for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves), Ens10-tes (blue curves), and Ens30 (green curves) verified every 12 h starting at 1200 UTC 23 Jun 2005 against rawinsondes.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

As mentioned earlier, the two experiments, Ens10 and Ens10-tes, lasted for 6 days with continuous 12-h update cycles that produced 12 analyses during the test period. Figures 9 and 10 give the 6-day averages of vertical profiles of biases and RMS differences, respectively, of the 12 EnKF mean analyses validated against rawinsondes for the 45-km grid from the two experiments. The TES impacts on both the averaged biases and the RMS differences of the ensemble analyses become more apparent than those for the single analysis shown in Figs. 6 and 7. This could be attributed to the continuous data assimilation and forecast cycles where the improved ensemble forecasts from the previous cycle were used as background fields for the current data assimilation. This is also an indication that the TES technique could be more beneficial to applications where ensemble forecasts are continuously updated by the EnKF data assimilation. Again, the impacts on the wind analyses are larger than those on temperature and water vapor. This is probably because wind errors tend to be more sensitive to spatial and temporal offsets of weather features, so the impact of TES is stronger in the wind fields.

Fig. 9.
Fig. 9.

Vertical profiles of biases from the EnKF analyses for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves) and Ens10-tes (blue curves) averaged over the 6-day period and verified every 12 h starting at 1200 UTC 23 Jun 2005 against rawinsondes. Ensemble means were used in the verification. The vertical coordinate is pressure (hPa).

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for RMS differences.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

Figure 11 displays the 6-day averages of RMS errors of the 12 ensemble forecasts (produced from the aforementioned 12 analyses), again verified against rawinsondes, of T, u, υ, and q of the ensemble mean from Ens10 and Ens10-tes, respectively, for the 45-km grid. The TES-enhanced ensemble analyses lead to improved ensemble forecasts. It is even more interesting to note that the impact lasts over the whole 72-forecast-hour duration. Note that the differences between the two curves in Figs. 11 decrease gradually as the forecast time increases (especially after the 60th hour in Figs. 11b,d). This is probably mainly a result of the fact that as the forecast time increases, the air masses in the TES-improved areas of the forecasted fields are advected downstream and move eastward gradually over the Atlantic where the improvements cannot be fully and continuously evaluated by our method because of the lack of rawinsondes there. The gradual decrease in the differences between the two curves in Fig. 11 could also be partially explained by the gradually reduced predictabilities in both Ens10 and Ens10-tes (as the predictabilities will be eventually diminished as the forecast time further increases far beyond 72 h) that gradually eat away the impact of the improved ensemble analyses from TES. The impact of the slow sweeping of the lateral boundary conditions through the COAMPS domain could also be a cause.

Fig. 11.
Fig. 11.

RMS errors of the COAMPS forecasts for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1) from the experiments of Ens10 (red curves) and Ens10-tes (blue curves) averaged over the 6-day period and verified every 12 h starting at 1200 UTC 23 Jun 2005 against rawinsondes. Ensemble means were used in the verification.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

c. Selection of the sampling time interval τ and its impacts

Needless to say, the time interval τ between two adjacent time samplings is an important variable in the TES technique. Ideally, τ should be selected optimally and adaptively according to the spatial and temporal scales of the main weather system covered by each of the three nested grids. Interesting results on optimizing τ were obtained from observing system simulation experiments with a shallow-water equation model in section 4 of Xu et al. (2008b) and with a full-plume atmospheric model on a single grid in section 4 of Xu et al. (2008a). The results from those studies show that properly selecting τ can further improve the ensemble analyses and subsequent forecasts (measured by RMS errors), but the improvements are not very sensitive to the variation of τ as long as τ stays in a properly confined range.

In a similar way to the above studies, six experiments were conducted in this study to investigate the impact of τ on the TES performance with the EnKF and the COAMPS model as well as real conventional and satellite observational data. In these experiments, we set τ to 1, 2, 3, 4, 5, and 6 h, respectively. The storm case shown in Figs. 68 was selected for the study. In each experiment, EnKF data assimilation was carried out with a selected τ value and 10 ensemble members, followed by a 72-h COAMPS ensemble forecast of the same ensemble size. RMS errors of the ensemble mean over the 3D model domain were calculated every 12 h during the 72-h forecast period verified against rawinsondes. The RMS error from the experiment is then subtracted from the control run without TES (Ens10 in Fig. 8) at each of the seven verification times. Then, an average root-mean-square error reduction DRMS over the whole forecast period is calculated by simply averaging the differences between the two RMS errors over the 72-h forecast period; that is,
e3.5
where RMSC is the RMS error from the control run without TES while RMST is the error with TES. From (3.5), a positive DRMS means a reduction in model forecast errors by TES.

The blue curves in Fig. 12 give the calculated DRMS values for model state variables T, q, u, and υ from the six experiments for the 45-km grid. It is seen from Fig. 12 that τ has a notable impact on DRMS for T. The best value of τ for T is 5 h. This could be a result of the fact that the temperature field at 45-km resolution is relatively smooth and has mainly large-scale features, at least for this particular case, as indicated by the geopotential height in Fig. 5. The value of τ = 5 h can probably best represent the phase errors of T at about this scale. Compared to T, the DRMS for q, u, and υ are less sensitive to the variation of τ (from 1 to 6 h). The time interval τ has the best values of 4, 2, and 3 h for q, u, and υ, respectively, on the 45-km grids. A possible reason for these reduced best values of τ (relative to the best value of τ = 5 h for T) is that the wind and moisture errors contain, in addition to large-scale patterns, some mesoscale features. As mentioned earlier, they are usually more sensitive to spatial and temporal offsets of the weather systems. For a selected τ value, TES may represent the phase errors of one field slightly better than those from the other two. Inside a field, TES for a given τ may better represent the errors at one scale than those at other scales. The overall impact is the relatively flat curves of DRMS with τ and the different best values of τ for different fields. As we can see from Fig. 12, it is not straightforward to pick up a τ value that would best benefit all the model fields. The selection of τ = 3 h used in most experiments in this study is a compromise choice. The nice thing from Fig. 12, however, is that, similar to the results from Xu et al. (2008a,b), the DRMS changes with τ are smaller compared to the values of DRMS for any given τ in the 1–6-h range, implying that for a reasonable value of τ, TES always has an overall positive impact on the ensemble analyses and forecasts.

Fig. 12.
Fig. 12.

Average reductions of RMS errors of the ensemble means of the COAMPS forecasts over the 72-h forecast period by TES for (a) T (°C), (b) q (g kg−1), (c) u (m s−1), and (d) υ (m s−1), as a function of the sampling time interval τ. The average RMS error reductions are calculated from (3.5). A positive value means improvement in the ensemble forecast by TES. The average RMS error reductions are displayed for the COAMPS nested grids of 45 (blue curves), 15 (red curves), and 5 km (green curves). The sampling time interval is the time interval (h) between two adjacent sampling times.

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

To further investigate the impact of τ on different model grid resolutions, DRMS values are also calculated for the COAMPS 15- and 5-km grids and displayed in Fig. 12 (the red and green curves, respectively) for the same state variables. Clearly, the DRMS curves from the 15-km grid basically look similar to those from the 45-km grid. This is understandable because the 15-km grid covers the major part of the dynamically active regions of the atmosphere inside the 45-km domain. Furthermore, like the 45-km grid, the 15-km grid can also represent the weather systems from synoptic scale to mesoscale. It is interesting, however, to note that the green curves for the 5-km grid look quite different from those from the other two grids. First, the green curve in Fig. 12a has the lowest values among the three grids while those in Figs. 12b–d have the highest numbers. Second, the green curves have two or more peaks with τ for all four state variables. The reasons for the notable differences in both patterns and values between the green curves for the 5-km grid and the blue and red curves from the other two grids are not completely clear. A possible explanation is that the 5-km grid domain (see Fig. 1) is small and can basically represent the small-scale features of the weather systems only with the high grid resolution. As mentioned earlier, the phase errors for T contain basically large-scale features and cannot be well represented by the small 5-km grid domain. Therefore, for any given τ value, the TES impacts the T field on the 5-km grid less significantly than those on the 15- and 45-km grids. On the other hand, wind and moisture fields are rich in small-scale features and these small-scale features are best represented by the 5-km grid, and their strong sensitivity to the phase errors of the weather systems may explain the enhanced impact of TES. Another possible explanation is the limited amount of observational data available for verification within the 5-km grid domain. Both the 45- and 15-km grid domains contain a large number of rawinsondes stations while the 5-km grid domain has only four stations. However, we found during our verification that, for the RMS error calculation over the 3D domain at each verification time, there were about 80–110 observational data points available for each state variable. Although this number is smaller compared to the amount of observational data (more than 1000) over the 15- or 45-km grid domain, it should be large enough for the calculation of RMS errors. The third speculation is that the weather pattern of the selected storm case may also be a factor that affects the TES impact on different grids. To investigate this, the experiment of the continuous data assimilation–forecast cycles for a 6-day period discussed in section 3b is used again with τ = 3 h. The 6-day averaged RMS errors of T, u, υ, and q for the ensemble means of Ens10 and Ens10-tes on the 15- and 5-km grids are calculated and presented in Fig. 13. For easy comparison, the results for the 45-km grid shown in Fig. 11 are also displayed in Fig. 13 (top). It is found that the TES impacts on all three nested grids look similar in both pattern and magnitude. This tells us that weather patterns can be a major factor that affects TES performance and the impact of observational data number for verification can be basically ruled out. But the effects of domain size and grid resolution on TES impact are still not completely clear since in a continuously cycled data assimilation–forecast experiment forecasts from the high-resolution nested grid are continuously affected by its parent grid through lateral boundary conditions.

Fig. 13.
Fig. 13.

As in Fig. 11, but for the results from COAMPS’s three nested grids: (top) 45-, (middle) 15-, and (bottom) 5-km grid for (from left to right) T (°C), u (m s−1), υ (m s−1), and q (g kg−1).

Citation: Weather and Forecasting 30, 4; 10.1175/WAF-D-14-00108.1

4. Conclusions

The TES technique developed by Xu et al. (2008b) for ensemble-based filters has been tested in this study with real conventional and satellite data. The NRL EnKF was used as the data assimilation system and the U.S. Navy’s COAMPS model was employed as both the target model for data assimilation and the forecast model in the experiments. The purpose of this study is to investigate, when ensemble size is restricted because of limited computational resources, whether and to what extent the degradation in performance of the ensemble analyses and forecasts due to insufficient ensemble size can be compensated by the TES technique without increasing the number of ensemble runs.

The results from the experimental study show that, in dynamically active areas, the correlation among the ensemble forecasts at different time levels is relatively small when the sampling time interval is selected in the mesoscale time range (around 3 h) for the TES. When the number of ensemble members is not large enough to adequately capture the uncertainty of the model forecast errors, the forecast error spread can be better represented by the time-expanded ensemble with two additional ensemble members sampled from each model run at the two time levels that are 3 h before and after the analysis time. This has been verified by the notable enhancement of the ensemble spreads in these areas as well as the remarkably improved ensemble analyses and forecasts from the TES technique. The selection of τ can impact the TES performance, but the impact is not very sensitive to the variation of τ (between 1 and 6 h). Thus, with τ selected between 1 and 6 h, TES can always show a positive impact. We also notice that the results from Ens10-tes overall are close to but no better than those from Ens30. This suggests that the TES technique can be most beneficial when applied to time-critical applications where ensemble analyses and forecasts cannot be conducted with the normally feasible and required ensemble size within the time limits or used in some special situations in which only limited computational resources are available. As the ensemble size increases, the advantages of the TES technique may be reduced. When the ensemble size is sufficiently large [e.g., 64 or larger as shown by Houtekamer and Mitchell (1998) and Zhao et al. (2013)], the benefits from TES may become marginal. Therefore, this technique should not be considered as a replacement for the original ensemble sampling method for ensemble data assimilation with a large number of ensemble members.

Acknowledgments

The authors thank Mr. John Cook of the On-Demand Systems Section of the Marine Meteorology Division, Naval Research Laboratory, for the very helpful discussions and valuable inputs into this study. This research was supported by the Office of Naval Research (ONR) under PE 0602435N, N000140910526, and other ONR-sponsored collaboration projects. Computational resources were provided by the Department of Defense Supercomputing Resource Centers at the Navy Oceanographic Office and the Air Force Research Laboratory.

REFERENCES

  • Anderson, J. L., 2012: Localization and sampling error correction in ensemble Kalman filter data assimilation. Mon. Wea. Rev., 140, 23592371, doi:10.1175/MWR-D-11-00013.1.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., , and Collins N. , 2007: Scalable implementations of ensemble filter algorithms for data assimilation. J. Atmos. Oceanic Technol., 24, 14521463, doi:10.1175/JTECH2049.1.

    • Search Google Scholar
    • Export Citation
  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 14310 162, doi:10.1029/94JC00572.

    • Search Google Scholar
    • Export Citation
  • Greybush, S. J., , Kalnay E. , , Miyoshi T. , , Ide K. , , and Hunt B. R. , 2011: Balance and ensemble Kalman filter localization techniques. Mon. Wea. Rev., 139, 511522, doi:10.1175/2010MWR3328.1.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , Whitaker J. S. , , and Snyder C. , 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 27762790, doi:10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hodur, R. M., 1997: The Naval Research Laboratory’s Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS). Mon. Wea. Rev., 125, 14141430, doi:10.1175/1520-0493(1997)125<1414:TNRLSC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hogan, T. F., , and Rosmond T. E. , 1991: The description of the Navy Operational Global Atmospheric Prediction System’s spectral forecast model. Mon. Wea. Rev., 119, 17861815, doi:10.1175/1520-0493(1991)119<1786:TDOTNO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., , and Mitchell H. L. , 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796811, doi:10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Lu, H. J., , Xu Q. , , Yao M. M. , , and Gao S. T. , 2011: Time-expanded sampling for ensemble-based filters: Assimilation experiments with real radar observations. Adv. Atmos. Sci., 28, 743757, doi:10.1007/s00376-010-0021-4.

    • Search Google Scholar
    • Export Citation
  • McLay, J. G., , Bishop C. H. , , and Reynolds C. A. , 2008: Evaluation of the ensemble transform analysis perturbation scheme at NRL. Mon. Wea. Rev., 136, 10931108, doi:10.1175/2007MWR2010.1.

    • Search Google Scholar
    • Export Citation
  • McLay, J. G., , Bishop C. H. , , and Reynolds C. A. , 2010: A local formulation of the ensemble transform (ET) analysis perturbation scheme. Wea. Forecasting, 25, 985993, doi:10.1175/2010WAF2222359.1.

    • Search Google Scholar
    • Export Citation
  • Meng, Z., , and Zhang F. , 2007: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part II: Imperfect model experiments. Mon. Wea. Rev., 135, 14031423, doi:10.1175/MWR3352.1.

    • Search Google Scholar
    • Export Citation
  • Meng, Z., , and Zhang F. , 2008a: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part III: Comparison with 3DVAR in a real-data case study. Mon. Wea. Rev., 136, 552540.

    • Search Google Scholar
    • Export Citation
  • Meng, Z., , and Zhang F. , 2008b: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part IV: Comparison with 3DVAR in a month-long experiment. Mon. Wea. Rev., 136, 36713682, doi:10.1175/2008MWR2270.1.

    • Search Google Scholar
    • Export Citation
  • Mitchell, H. L., , Houtekamer P. L. , , and Pellerin G. , 2002: Ensemble size, balance, and model-error representation in an ensemble Kalman filter. Mon. Wea. Rev., 130, 27912808, doi:10.1175/1520-0493(2002)130<2791:ESBAME>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Peng, M. S., , Ridout J. A. , , and Hogan T. F. , 2004: Recent modifications of the Emanuel convective scheme in the Navy Operational Global Atmospheric Prediction System. Mon. Wea. Rev., 132, 12541268, doi:10.1175/1520-0493(2004)132<1254:RMOTEC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and Hamill T. M. , 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
  • Xu, Q., , Lu H. , , Gao S. , , Xue M. , , and Tong M. , 2008a: Time-expanded sampling for ensemble Kalman filter: Assimilation experiments with simulated Radar observations. Mon. Wea. Rev., 136, 26512667, doi:10.1175/2007MWR2185.1.

    • Search Google Scholar
    • Export Citation
  • Xu, Q., , Wei L. , , Lu H. , , Qiu C. , , and Zhao Q. , 2008b: Time-expanded sampling for ensemble-based filters: Assimilation experiments with a shallow-water equation model. J. Geophys. Res., 113, D02114, doi:10.1029/2007JD008624.

    • Search Google Scholar
    • Export Citation
  • Xue, M., and et al. , 2001: The Advanced Regional Prediction System (ARPS)—A multiscale nonhydrostatic atmospheric simulation and prediction tool. Part II: Model physics and applications. Meteor. Atmos. Phys., 76, 143165, doi:10.1007/s007030170027.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Meng Z. , , and Aksoy A. , 2006: Tests of an ensemble Kalman filter for mesoscale and regional-scale data assimilation. Part I: Perfect model experiments. Mon. Wea. Rev., 134, 722736, doi:10.1175/MWR3101.1.

    • Search Google Scholar
    • Export Citation
  • Zhang, F., , Weng Y. , , Sippel J. , , Meng Z. , , and Bishop C. , 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
  • Zhao, Q., , Zhang F. , , Holt T. , , Bishop C. , , and Xu Q. , 2013: Development of a mesoscale ensemble data assimilation system at the Naval Research Laboratory. Wea. Forecasting, 28, 13221336, doi:10.1175/WAF-D-13-00015.1.

    • Search Google Scholar
    • Export Citation
1

COAMPS is a trademark of the Naval Research Laboratory.

Save