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  • View in gallery

    (a) The current operational 80-level global variable-resolution grid used by the REG analysis system. The horizontal resolution inside the blue line is 15 km. The poles of this grid are placed over the Pacific and Atlantic Oceans. (b) The blue line shows the limit of the uniform 15-km resolution of the 58-level global variable-resolution model used operationally prior to the current operational version. The green line shows the second REG-LAM3D-BF test grid. (c) The orange line shows the third REG-LAM3D-BF test grid, and the red line the final REG-LAM3D grid. The dashed lines for the LAM grids represent the inner limit of the boundary condition zone (typically 12 grid points).

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    Current GEM-LAM experimental forecasting windows operated by CMC at 2.5-km horizontal resolution to provide 24-h forecasts.

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    Analysis/model levels: (left) 58 levels of the last operational REG system and (right) 80 levels of the current REG system (operational since 22 Jun 2009). The same levels are also used in the 58- and 80-level versions of the REG-LAM3D system.

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    The structure of the REG-LAM3D data assimilation system. The LAM and driving global model and analysis types are indicated.

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    The 48-h forecast verifications against North American radiosondes. The analysis and forecast models have 58 vertical levels (see Fig. 3, left). The fields considered are the (a) u component of the horizontal wind, (b) modulus of the horizontal wind vector, (c) geopotential, (d) temperature, and (e) dewpoint depression. Biases and standard deviation errors appear in the legend. The REG-LAM3D-BF analysis is used here with the horizontal grid shown in green in Fig. 1b.

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    As in Fig. 5, but 1) the analysis and forecast models have 80 vertical levels (with more of the stratosphere included) rather than 58 vertical levels and 2) the REG-LAM3D-BF analysis is used here with the LAM grid shown by the orange lines in Fig. 1c.

  • View in gallery

    (top) The 48-h REG control forecast with the global variable-resolution model; (bottom) the REG-LAM3D-BF 48-h forecast. The verifying analysis, valid at 0000 UTC 13 Jan 2007, appears as black contours. Shaded regions represent the 48-h accumulated precipitation.

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    Vertical background error cross correlations between the spectral streamfunction and temperature fields for a given horizontal total wavenumber. (top) The BF is at wavenumber 5 (domain size is approximately 10 000 km wide in each direction). (bottom) The spherical-harmonics representation at total wavenumber 20. NMC error samples have been symmetrized about the equator so as to focus on the NH correlation structure in a manner similar to the BF approach. The model top is at level 1 and the contour interval is 0.1.

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    Vertically localized spectral correlation blocks as used by the REG-LAM3D-HEM analysis at global horizontal wavenumber 20. The contour interval is 0.1.

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    (a) Temperature increment (analysis − background) at 500 hPa (contour interval is 0.05 K) resulting from a single 500-hPa temperature observation at 45°N, 90°W with a 1-K residual and observation error for the (top) GLOBAL and (bottom) REG-LAM3D-HEM analyses. Winter season background error statistics were used for both analysis systems. (b) (top) As in (a) but, for the vertical profile of temperature increment at the observation location. Shown are the REG (dashed line) and REG-LAM3D-HEM (solid line) analyses but the background-error standard deviations are those of the REG analysis for intercomparison purposes. (bottom) As in the top panel, but the dashed line is for the REG-LAM3D-BF analysis. (c) Same simulated data as in (a), but for the (top) REG-LAM3D-HEM and (bottom) REG-LAM3D-BF analyses. Each of the analyses used its own background-error standard deviation fields. (d) Same type of simulated data as in (a) but located at 45°N, 163°W. Shown are the REG-LAM3D-BF (solid line) and REG-LAM3D-HEM (dotted line) analysis increments. (e) Normalized deviation from Charney’s nonlinear balance over different latitudinal bands for the control REG operational analysis and the analysis fields resulting from the REG-LAM3D-HEM system valid at 1200 UTC 1 Jan 2007.

  • View in gallery

    The 48-h forecast verifications against North American radiosondes. The analysis and forecast models have 80 vertical levels (see Fig. 3, right). The REG-LAM3D-HEM analysis is used.

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    Winter verifications of 0–24-h accumulated precipitation forecasts against SHEF data. Shown are the REG system (solid blue line), GLOBAL analysis (red dashed line), REG-LAM3D-BF (yellow dashed line), and REG-LAM3D-HEM (purple dotted line). The REG-LAM3D-BF analysis approach shows the worst scores.

  • View in gallery

    Summer verifications of 24–48-h accumulated precipitation forecasts against SHEF data. Shown are the REG system (solid blue line), REG-LAM3D-HEM analysis system (yellow dashed line), and REG-LAM system, but the analysis step is performed with the GLOBAL analysis (red dotted line).

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    As in Fig. 13, but for verification against SYNOP data.

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    Vertical variation of the horizontal background error correlation scales “L” for the Helmholtz’s functions, the temperature, and the logarithm of the specific humidity computed with the NMC 48–24-h approach. The horizontal resolution of the operational GEM model is 33 km and its vertical grid has 80 levels (Fig. 3).

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The Canadian Regional Data Assimilation and Forecasting System

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  • 1 * Data Assimilation and Satellite Meteorology Section, Canadian Meteorological Center, Dorval, Québec, Canada
  • | 2 Recherche en Prévision Numérique, Canadian Meteorological Center, Dorval, Québec, Canada
  • | 3 Development Division, Canadian Meteorological Center, Dorval, Québec, Canada
  • | 4 Operational Division, Canadian Meteorological Center, Dorval, Québec, Canada
  • | 5 Center for Earth Observation Science, University of Manitoba, Winnipeg, Manitoba, Canada
  • | 6 ** Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, Toulouse, France
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Abstract

This paper describes the recent changes to the regional data assimilation and forecasting system at the Canadian Meteorological Center. A major aspect is the replacement of the currently operational global variable resolution forecasting approach by a limited-area nested approach. In addition, the variational analysis code has been upgraded to allow limited-area three- and four-dimensional variational data assimilation (3D- and 4DVAR) analysis approaches. As a first implementation step, the constraints were to impose similar background error correlation modeling assumptions, equal computer resources, and the use of the same assimilated data. Both bi-Fourier and spherical-harmonics spectral representations of background error correlations were extensively tested for the large horizontal domain considered for the Canadian regional system. Under such conditions, it is shown that the new regional data assimilation and forecasting system performs as well as the current operational system and it produces slightly better 24-h accumulated precipitation scores as judged from an ensemble of winter and summer cases. Because of the large horizontal extent of the regional domain considered, a spherical-harmonics spectral representation of background error correlations was shown to perform better than the bi-Fourier representation, considering all evaluation scores examined in this study. The latter is more suitable for smaller domains and will be kept for the upcoming use in the kilometric-scale local analysis domains in order to support the Canadian Meteorological Center’s (CMC’s) operations using multiple domains over Canada. The CMC’s new regional system [i.e., a regional limited-area 3DVAR data assimilation system coupled to a limited-area model (REG-LAM3D)] is now undergoing its final evaluations before operational transfer. Important model and data assimilation upgrades are currently under development to fully exploit this new system and are briefly presented.

Corresponding author address: Dr. Luc Fillion, Direction de la Recherche en Météorologie, Environment Canada, 2121 Route Trans-Canadienne, Dorval QC H9P 1J3, Canada. Email: luc.fillion@ec.gc.ca

Abstract

This paper describes the recent changes to the regional data assimilation and forecasting system at the Canadian Meteorological Center. A major aspect is the replacement of the currently operational global variable resolution forecasting approach by a limited-area nested approach. In addition, the variational analysis code has been upgraded to allow limited-area three- and four-dimensional variational data assimilation (3D- and 4DVAR) analysis approaches. As a first implementation step, the constraints were to impose similar background error correlation modeling assumptions, equal computer resources, and the use of the same assimilated data. Both bi-Fourier and spherical-harmonics spectral representations of background error correlations were extensively tested for the large horizontal domain considered for the Canadian regional system. Under such conditions, it is shown that the new regional data assimilation and forecasting system performs as well as the current operational system and it produces slightly better 24-h accumulated precipitation scores as judged from an ensemble of winter and summer cases. Because of the large horizontal extent of the regional domain considered, a spherical-harmonics spectral representation of background error correlations was shown to perform better than the bi-Fourier representation, considering all evaluation scores examined in this study. The latter is more suitable for smaller domains and will be kept for the upcoming use in the kilometric-scale local analysis domains in order to support the Canadian Meteorological Center’s (CMC’s) operations using multiple domains over Canada. The CMC’s new regional system [i.e., a regional limited-area 3DVAR data assimilation system coupled to a limited-area model (REG-LAM3D)] is now undergoing its final evaluations before operational transfer. Important model and data assimilation upgrades are currently under development to fully exploit this new system and are briefly presented.

Corresponding author address: Dr. Luc Fillion, Direction de la Recherche en Météorologie, Environment Canada, 2121 Route Trans-Canadienne, Dorval QC H9P 1J3, Canada. Email: luc.fillion@ec.gc.ca

1. Introduction

Since 24 September 1997, the Canadian Meteorological Center (CMC) has operated a regional three-dimensional variational data assimilation system (hereafter REG). The reader can find useful information on this system in the original paper by Laroche et al. (1999). Since then, many upgrades have been implemented operationally and the details can be found at Environment Canada’s (EC’s) external Web site address (http://www.msc-smc.ec.gc.ca/cmc/op_systems/recent_e.html). This analysis system is based on the same global variational analysis approach and one can find further information in Gauthier et al. (2007, hereafter GAL07), which describes the basic aspects of the global analysis structure. Very recently, the model (and analysis) lid of the REG system was raised from 10 to 0.1 hPa. This enabled better treatment of stratospheric circulations and necessitated model and data assimilation adjustments. This version became operational at CMC on 22 June 2009 simultaneously for the global and regional data assimilation systems.

In the Global Environmental Multiscale (GEM) model, it is possible to use either a “uniform resolution” (GEM-Global, as in Bélair et al. 2009), a limited-area (GEM-LAM, as in Mailhot et al. 2010), or a variable resolution (GEM-Regional, as in Mailhot et al. 2006) option of the model (Fig. 1a). The variable-resolution scheme uses a rotated-coordinate system with a high-resolution domain that can be located over any portion of the globe.

To further improve Canadian regional forecasts of weather systems and their associated precipitation (up to 2 days over the North American continent), an increase in spatial resolution is needed. This is, for instance, a first step toward the assimilation of North American radar data. In addition to the global and REG systems, CMC routinely runs experimental high-resolution 24-h forecasts once a day for specific local areas over targeted regions in Canada (see Fig. 2). An example of this capability is the 2.5-km resolution forecasts performed for the 2010 Winter Olympics in the Whistler region, of British Columbia, Canada (Mailhot et al. 2010). The data assimilation requirements for such domains imply a huge increase in analysis horizontal resolution (typically going from 180-km analysis increment resolution for the REG system down to 1–10 km). Relying on the global variational analysis code to achieve this over such limited domains is inappropriate in terms of analysis strategy and impractical in terms of computer resources implied.

Despite the very robust, accurate, and flexible global variable resolution modeling concept (see Côté et al. 1993) that justified this approach in numerical weather prediction (NWP) at EC, there are basic practical reasons to opt for a limited-area strategy. A possible avenue for the limited-area variational analysis approach could be like the ones described in ALADIN Consortium–HIRLAM Consortium (2010) or other operational centers such as the Met Office limited-area four-dimensional variational data assimilation (4DVAR), the National Centers for Environmental Prediction (NCEP), or the Japan Meteorological Agency (JMA).

In the following, we will describe the development of a regional limited-area 3DVAR data assimilation system coupled to a limited-area model (LAM; i.e., GEM-LAM). This system will be referred to as the REG-LAM3D system and is especially designed for Canadian requirements in terms of regional weather forecasting. Key aspects that need to be considered are, first, the importance for CMC to maintain a state-of-the-art global data assimilation system, which is at the center of many other applications, such as the regional and local-scale systems. On the other hand, CMC must also maintain GEM-LAM, which is used experimentally at CMC to perform limited-area weather forecasting at very small scales (e.g., 2.5-km resolution over specific Canadian regions; see Fig. 2). This forces two basically different configurations of model and data assimilation systems (using advantageously the same numerics from the GEM-unified model. Second, in terms of data assimilation, the traditional procedure at CMC has been to use the same variational (VAR) analysis code but using innovations data produced with the global variable-resolution model. Background error statistics are kept the same as in the global system. To improve weather forecast accuracy in the 2-day range (especially improving precipitation forecasts) research and development efforts need to be directed toward the production of a more suitable high-resolution mesoscale analysis–forecasting system.

Currently, the REG system uses a global variable-resolution computational grid, which creates an additional configuration to be maintained at CMC. To avoid the variable-resolution approach, it was decided to configure a LAM system where the limited-area region covered the same area as the core of the uniform-resolution portion of the REG grid. The zone of the boundary conditions (which consists of 12 points outside the computational grid) is generated from the current well-maintained 55-km-resolution CMC global model.

In the following, we will go into the details of the development work done in order to achieve a first (transitional) version of the new limited-area 3DVAR analysis and its associated limited-area GEM model configuration. We will emphasize the fact that it is a transitional version, meaning that no attempt has been made to take advantage of new data types or background-error modeling approaches. The constraint of this system is the large spatial size of the forecasting domain specific to Canadian weather forecasts, which is also similar to the regional analysis system at NCEP.

2. The operational regional variable resolution

We review in this section the basic aspects of the current Canadian operational REG system.

a. Model aspects

The Global Environmental Multiscale (GEM) operational model is a two-time-step implicit, semi-Lagrangian, gridpoint model, with a latitude–longitude C grid that is staggered in the horizontal direction and an unstaggered sigma-pressure hybrid vertical grid. A complete description of the formulation of the GEM model is available in Côté et al. (1998). The current Canadian operational GEM-REG forecast system has a global variable resolution with a 15-km uniform core over North America. On 22 June 2009, the operational global and regional model lids were elevated from 10 to 0.1 hPa with an increase of from 58 to 80 vertical levels (see Fig. 3).

b. Analysis and observations

In the REG system, the same operational analysis code as for the global 4DVAR scheme is used but is applied in 3DVAR mode. Innovations are computed from the REG trial fields and the reader is referred to Laroche et al. (1999) for the initial implementation of the REG 3DVAR system at CMC. Table 1 lists all the observations assimilated by the REG. Note also that the REG-LAM3D analysis system and the global operational 4DVAR analysis system use the same observation types.

3. The REG-LAM3D system design and evaluation

Our GEM-LAM runs at 15-km horizontal resolution and uses the same configuration as in GEM-Regional for the vertical levels and the physics schemes. Over the development phase of the REG-LAM3D system, both vertical grids (the old 58- and the new 80-level versions) were used (see Fig. 3). The boundary conditions used to drive the REG-LAM3D model are prescribed by the GEM-Global driving model and can be imposed at a time interval as frequent as the time step of the driving model. An hourly update was chosen. The specifications of the lateral boundary conditions for the LAM configurations of the GEM model are taken from Thomas et al. (1998). Essentially, the nature of the spatial discretization of the basic equations on an Arakawa-C grid combined with the implicit formulation requires that only the normal wind components be specified in order to properly close the mathematical problem.

a. Analysis approaches

We now present the designs and an examination of three basic data assimilation configurations and their associated forecast verifications in the range 0–48 h. We assume that the reader is familiar with the standard formulation aspects of variational analysis systems and the definitions of the analysis variables [see, e.g., Gauthier et al. (1999, 2007), referred to as GAL99 and GAL07, respectively, for global models, and Berre (2000) for limited-area models].

The REG-LAM3D analysis component can presently be run under three different options. The first is to perform the 3DVAR analysis with the standard global spectral approach for the background error statistics and the choice of control variables. This option is referred as GLOBAL. The second option is to run the REG-LAM3D analysis component on a limited-area horizontal analysis grid with bi-Fourier (BF) spectral representation of background error statistics. This option is referred as REG-LAM3D-BF in the rest of the paper. The third option is to run the REG-LAM3D analysis component under a global Gaussian grid with spherical-harmonics representation of Northern Hemisphere background error statistics together with new control variables. This option is referred as the Hemispheric version (REG-LAM3D-HEM). The performance of both REG-LAM3D-BF and REG-LAM3D-HEM will be systematically evaluated for winter and summer cases. In the REG system mode (as for the HEM version), analysis increments are produced on the global Gaussian grid and interpolated to the LAM analysis grid at the end of the analysis step. This option referred to as GLOBAL, to emphasize the fact that the background error statistics and control variables are those of the currently global operational system, has been used systematically during the development work in order to gauge the impacts of analysis–model components on 48-h forecasts. An explicit example of this will be shown later when we discuss the 24-h accumulated precipitation forecasts accuracies and it will justify the choice of the HEM approach in the final form of REG-LAM3D proposed for CMC operations.

In the following, the BF, HEM, and GLOBAL options are explicitly mentioned in order to avoid confusions. The BF option uses a limited-area uniform Arakawa-C grid (as is the case for the GEM-LAM model grid) and, in general, a more traditional application on the kilometric-scale 3DVAR analysis over spatial domains such as the ones shown in Fig. 2. The regional nonlinear forecast model’s horizontal resolution is 15 km in all experiments reported in this study. Among the various 3DVAR code adjustments required for BF, the spatial discretization of the transformation from Helmholtz’s functions to wind components is given in appendix A. Except for the grid’s geometry, this is the first difference in the numerics between the BF approach and the global variational analysis code where the latter uses a global linear Gaussian grid and spherical-harmonics spectral representation to perform spatial derivatives.

The reader is referred to section 2 of the paper by GAL07 for a description of the basic formulation of the minimization problem and the incremental variational approach used in the current operational Global-4DVAR and REG analysis systems. The REG-LAM3D (BF and HEM modes) analysis system is also incremental. In this study, different choices of control variables and background error statistics are compared to the control GLOBAL analysis approach. All systems described are based on the assumption of homogeneous and isotropic nonseparable background error correlations.

The operational GEM global 33-km forecast model provides the source of the forecast error samples to build background error covariances by using the “NMC-24/48” approach. See appendix B for further details. The same error sample files are also used in the HEM version but with further treatment [we expand on this in section 3d(1)]. Following the nomenclature in GAL07, the same basic objects and operators involved in the preconditioning and incremental formulation are being used for the REG-LAM3D (see Tables 1 and 2 in GAL07). One important distinction though is the use of a computational analysis grid, which is (in general) a given rotated uniform latitude–longitude grid on the sphere of limited extent where the poles are no longer at their real geographical locations.

For the REG-LAM3D analysis itself, all observations located outside the LAM domain are discarded. Two possibilities to treat the observations are 1) to perform a preliminary rotation of all the observations to be assimilated in a preprocessing step and 2) to include this rotation as an additional operator within the analysis step. To avoid confusion in the code regarding the true nature of the observations at an arbitrary point of the minimization/postprocessing, it was decided to favor option 2. Performing such rotations concerns only the wind field components since the other scalar components of the analysis increments only need their precomputed “tile index.” This tile index is their associated reference point of the computational grid required to perform a local bilinear interpolation to the observation location. An analogous referencing is done for the global system, but here we refer to the Arakawa C grid. Since tangent wind rotations from “sphere to sphere” are performed in the vicinity of each observation, and this process involves a sequence of steps where trigonometric factors appear in terms of real geographical latitudes and lonitudes, these factors are precomputed before the minimization and stored for subsequent use. This local wind rotation is thus very fast. This is an additional operator to consider within Table 2 of GAL07, together with its associated adjoint operator. Depending on the user-defined computational grid of the REG-LAM3D experiment performed (i.e., rotated or not), this rotation operator is activated or bypassed.

b. Domain size and boundary condition aspects

Starting from the global variable resolution grid of the REG system, a sequence of decisions was made in order to define an appropriate limited-area grid having a uniform 15-km horizontal resolution comparable to the uniform domain of the global variable-resolution grid. For the first configuration test, the limited-area uniform-resolution domain was defined as the uniform resolution part of the original REG domain, where it ran with 58 vertical levels (blue lines in Fig. 1b). The current REG domain represented by blue lines in Fig. 1a is the same horizontal domain as the original with only an extension northward to meet “International Polar Year (IPY) requirements. REG is running with 80 vertical levels and we note the dashed lines that represent the inner limit of the boundary condition zone (typically set to 12 points). Details on the IPY operational modifications can be found at EC’s external Web site (http://www.msc-smc.ec.gc.ca/cmc/op_systems/recent_e.html). Figure 4 shows the data assimilation components involved prior to the launch of the 48-h regional forecasts. A global 4DVAR analysis (e.g., G206 in Fig. 4; G606 is the separate surface analysis) is routinely performed every 6 h, from which the initial conditions are produced (R206) for the LAM-15km integrated for 9 h to provide a background trajectory necessary for the REG-LAM3D-BF analysis (R112 in Fig. 4). Once this analysis is ready, a 48-h LAM forecast is performed. In this first experimental testing of the new analysis–forecasting system, the initial conditions (G206) served as the initial conditions for a global forecast run at operational resolution (33 km), thus providing boundary conditions to the LAM-15km for the 48-h LAM forecast. After extensive testing (40 winter and 40 summer cases) of this data assimilation configuration against the REG system (operational version of the time), it became obvious that the availability of global data at the regional analysis time within the control REG system allows a more beneficial refreshment of large scales and subsequent 48-h forecasts compared to the 54-h global run initiated from the G206 4DVAR analysis. This was judged by conventional scores against radiosonde data (results not shown).

This “large scale” aspect became the motivation for introducing a parallel 3DVAR global analysis step (at T-108 as for the control REG system) valid at the same analysis time as the REG-LAM3D analysis but for the driving global model indicated in Fig. 4 as P206. No special adjustments are required to perform this global 3DVAR analysis for the driving global model since this option is already being used in the REG system. The 9-h background trajectory for this analysis step and the subsequent 48-h forecast were performed at 100-km resolution (i.e., a 400 × 200 uniform latitude–longitude grid). Note that for this analysis, which provides the initial conditions for the driving global model, data from all over the globe are used.

We will now discuss the many phases of the second test configuration that has been fully evaluated. Figure 1b shows the original regular 15-km portion of the REG system (blue lines). Our first attempt was to extend the southern boundary to about 4° latitude north of the final grid shown in Fig. 1b, and this revealed a serious problem. Evaluation results for the wintertime showed a degradation in mass information (geopotential and temperature scores) after only 6 h of time integration. Also, the LAM solution over the southeastern domain differed significantly (a few decameters after 48 h) from the 500-hPa control geopotential from the REG forecast (initiated with the same initial conditions) and the forecast difference had the isotropic signature of fast-propagating gravity waves. The damage was even worse considering the fact that after 48 h another component of the forecast difference had propagated slowly from the southeastern edge of the LAM domain, penetrating into the domain gradually and degrading the forecast. This suggests that not only were erroneous gravity–inertia waves generated but also a slower Rossby-type error emerged from the same source region and gradually diminished the accuracy of the 48-h forecasts. The origins of this triggering mechanism were related to the strong Sierra Madre topography present within the blending zone of the LAM. This type of potential problem is documented in the meteorological literature and is pointed out, for instance, in Warner et al. (1997). The simple cure for this problem in our context was to extend slightly the southern edge of the LAM grid farther south, away from strong topography. The results (not shown) completely removed that problem and they are the justification for the southern extent of the LAM grids (green line) appearing in Fig. 1b. The northern part of the grid was also extended in order to have the same IPY coverage as the REG grid (Fig. 1a). This would better encompass the wintertime southern jets, which can move quite far south and significantly challenge the accuracy and robustness of schemes used to refresh the boundary conditions. This corrected second configuration was run with 58 vertical levels. In Fig. 5, the status of the forecast verifications at this point (against the REG operational at the time) was less accurate for 48-h jet-level winds and there was also a degradation of mass information (e.g., temperature and geopotential). The humidity forecasts were essentially as good as the control run. At this point in the experiments, the REG-LAM3D-BF horizontal grid is the domain limited by the green lines in Fig. 1b and has 58 vertical levels.

c. Stratospheric extension

Development work performed by the CMC global data assimilation and modeling groups in Dorval during 2007 using 80 vertical levels (which includes more of the stratosphere) instead of the 58-level version yielded significantly improved 5-day forecasts. REG also adopted the same vertical level structure. Similar improvements in the stratosphere were observed in the regional system but with less impact on the tropospheric conventional scores and precipitation scores for the 2-day-range regional forecasts. This modification into the regional system was adopted to simplify the maintenance of both global and regional data assimilation systems at CMC. These new global and regional systems then became operational on 22 June 2009 at CMC. The current version of the REG system had been used as the control system in the following upgrades of the REG-LAM3D system. To achieve a clean demonstration of the enhanced performance of the REG-LAM3D system against the REG system, it was decided to adopt the same vertical structure (80 levels) in the REG-LAM3D system. Unless otherwise stated, REG-LAM3D from now on means the 80 vertical levels version. This “stratospheric” exercise led us to, inadvertently, introduce another modification to the computational grid. The latter was motivated by the advancement achieved by our colleagues in the numerical modeling group (RPN). The GEM-LAM model had improved its boundary conditions strategy considerably by allowing a more coherent transfer of physical information from the driving model and by also introducing the use of “hollow cubes.” The latter concept allows the driving global model to write out only the boundary condition zones of each field rather than the entire plane of each field. In addition, the grid rotation geometry of the driving model is constrained to be the same as in the driven LAM model in order to simplify and speed up the spatial interpolations from the grid of the boundary conditions to the grid of the LAM domain. Reducing the extent of the input/output (I/O) and spatial interpolations in such a manner resulted in a huge reduction in computer time.

This upgrade was implemented in the REG-LAM3D system, which allowed a redefinition of the computational grid, as shown in Fig. 1c (orange lines). It is noted that the northern and southern extensions have been maintained (as compared to the green domain in Fig. 1b) but the computational poles are now placed over the oceans. Note, however, that the global 3DVAR analysis of the driving model is still performed with the standard nonrotated Gaussian grid. Finally, it was noted that the GEM-100km model took very little time to perform a 48-h forecast; so, it was decided to examine an increase of the horizontal spatial resolution from 100 to 55 km (i.e., 720 × 360 grid points). The last two modifications of the driving global model, which included 1) rotation of the computational grid and 2) an increase in the horizontal resolution to 55 km, were added in sequence to all of our test cases (40 winter and 40 summer cases) in order to carefully measure the impacts of each modification. Figure 6 shows the evaluation (against radiosondes) at 48 h of the 80-level model using the combined upgrades of the two modifications mentioned above for the winter cases. Similar conclusions were reached for the summer cases but the results are not shown. It is seen from this diagnostic measure that the REG-LAM3D-BF system now is as accurate as the control REG system. Based on these scores, both systems are now equivalent except for temperature and geopotential error biases appearing in the new system.

Further examination of 48-h winter forecasts (40) against analyses (generated from the operational REG system) over the northwest part of the domain over the ocean showed statistically slightly better accuracy in favor of the control REG system. To give an example of this, Fig. 7 shows the 48-h mean sea level pressure forecast maps for the REG and REG-LAM3D-BF systems, where the REG system positions are better for the two low centers southeast of the Alaskan Peninsula. Elsewhere in the domain, the REG-LAM3D-BF forecasts show no signs of inferior performance based on the location and intensity of the pressure systems. After a systematic examination of the cyclogenesis of these weather systems (many of which were originating east of Japan), it was concluded that, as they evolve, a larger high-resolution coverage of these systems was desirable. It was also computationally feasible considering the total computer requirements of the new REG system, which is comparable to the current REG system. Thus, the computational grid was extended farther west over the Pacific Ocean (red domain in Fig. 1c). All the winter and summer cases were rerun and the 48-h forecasts were reevaluated against radiosondes and analyses, and it was demonstrated that this grid extension produced forecasts as accurate as the control REG system statistically. All of these tests were done using computer resources equivalent to those of the REG system.

d. Preoperational version

We now turn to the final REG-LAM3D configuration that has been accepted for parallel testing at CMC on 7 July 2009, and has shown the best results in terms of the all verification statistics and particularly in the accumulated precipitation scores. It is clear at this point that the “LAM” concept has been stretched considerably, that is, the size of the horizontal grid (Fig. 1c, red lines) being now about 10 000 km wide in each direction (i.e., ¼ of the whole globe). Particularly for the limited-area analysis step, the spherical-harmonics representation becomes more relevant for spectral correlation modeling than do the bi-Fourier representations in the north–south direction. Signs of the limitations of the bi-Fourier representation in the current context will be given in the following together with the final strategy adopted.

1) Changes to control variables

Note that to perform the above-mentioned experiments in Figs. 6 and 7, it was necessary to modify our BF analysis; that is, background error statistics needed to be recomputed for this extended vertical domain while still using the National Meteorological Center (NMC, currently known as NCEP) method with the forecasts coming from the 80-level global GEM model. However, a new approach was used concerning the balance-splitting aspects of the analysis increments. Before describing the motivations behind this approach, we give first some details on how we adapted the control variables and background-error correlations of the current operational global variational analysis system to solve an important problem encountered when increasing the number of levels from 58 to 80, as shown in Fig. 3. Once the nonseparable spectral background error correlations were recomputed (using the standard NMC method with 48–24-h lagged forecasts as error samples) and used in the new stratospheric configuration, one specific problem became apparent. The combination of small (e.g., <0.1) correlations and very large variances (especially in the winter hemisphere) caused large spurious increments in the stratosphere. With the standard formulation of control variables, where a linearly balanced component is defined from regression operators, it is difficult to filter small correlations of “balanced temperature Tb,” for instance, between the troposphere and stratosphere. For that purpose, we decided to explicitly compute the vertical correlations for Tb so that they can then be localized with a Schur product (together with cross correlations). The control vector of the minimization was augmented to include Tb and the background-error standard deviations of the balanced and unbalanced part of the temperature were recalibrated so as to maintain the initial total error variance.

Again, the basic motivation for this new definition of control variables was to control small vertical correlations between the troposphere and stratosphere. It is important to realize that stratospheric considerations for the regional data assimilation and forecasting system are less important due to its short 48-h-range forecasts. In addition, work is in progress at EC in designing the REG-LAM3D system to operate strictly in the troposphere with high spatial resolution and to be driven efficiently with boundary conditions from the global forecast model. For the REG-LAM3D system (BF and HEM options), a different approach was developed that allowed us to vertically localize the troposphere–stratosphere vertical correlations in a direct way.

Using the REG-LAM3D domain (red lines in Fig. 1c), cross correlations between the streamfunction and temperature errors resulting from the nonseparable homogeneous/isotropic bi-Fourier spectral approach come out clearly and are shown in Fig. 8 (under BF). Note that the model level 1 is at the top and the total horizontal wavenumber is 5. Since the domain extension is about 10 000 km wide in both directions, this corresponds to a wavelength of about 2000 km. The vertical structure of the cross correlation between the streamfunction and temperature is (to leading order) the result of quasigeostrophic horizontal coupling and hydrostatic balance in the vertical. A monotonic tendency toward sharper vertical correlations is observed as we go to higher wavenumbers. Based on this direct coupling, the approach taken in REG-LAM3D-BF is to directly use this “off-diagonal block” (for each total wavenumber) rather than using the standard regression approach to construct a so-called balanced temperature and surface pressure [e.g., see Berre (2000, section 4) in the context of a limited-area model]. This means that in the new system no balanced–unbalanced part is used in the definition of the control variables, so that the control vector is simply (omitting the spectral form)
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that is, Helmholtz’s functions, temperature, logarithm of specific humidity, and surface pressure. Note that the same preconditioning is used as the standard definition of the control variables except that the variables involved do not refer to any unbalanced components. A vertical localization to each correlation and the cross-correlation blocks is then applied based on the aforementioned need for such filtering.

As mentioned previously, the horizontal extent of the REG-LAM3D grid is very large, thus, it was decided to include a further option of using spherical-harmonics spectral representations rather than bi-Fourier representation of the background error spectral covariances. To capture a Northern Hemisphere (NH) cross correlation similar to the one shown in BF mode (Fig. 8, top), we have symmetrized the ensemble of the forecast error samples about the equator to use as input into the statistics package (global mode). The NH structure (Fig. 8, bottom) resembles the structure captured by the BF statistics approach (Fig. 8, top). Note that the former is used for a horizontal total wavenumber of 20 (∼2000-km length scale), which closely corresponds to the BF in the top panel of Fig. 8, which is at wavenumber 5 (also 2000-km length scale). With this simple modification to the input error samples, the same basic approach as discussed above in the REG-LAM3D-BF mode can be applied here to get the direct spectral correlation blocks. As for the BF mode, no balance components [using the standard balance splitting; see Derber and Bouttier (1999, section 2) for the original exposition of the method in a global context] and regression matrices of any kind are used. Again, it is a simple task to apply vertical localization to these spectral correlation blocks, such as those appearing for instance in Fig. 9 (still for total wavenumber 20), plus the cross-correlation blocks between the streamfunction and temperature just discussed; that is, we only retain the “Ekman pumping” term shown in top-right panel of Fig. 9. Note that the temperature autocorrelations are sharper in the vertical than the corresponding (i.e., same total wavenumber) balanced temperature normally built by the standard regression approach. The latter may induce a large horizontal and vertical scale analysis increment (as compared to the residual “unbalanced” part) depending on the relative weighting by the standard deviations actually used in the minimization.

In the following sections we will evaluate the final REG-LAM3D-HEM system. The analysis approach is different than that of the standard global method but we had also set the horizontal resolution of the analysis and the background-error statistics to T200 on a 400 × 200 Gaussian grid. This detail is important when comparing the experiments discussed below with the standard global analysis code that uses a T108, 240 × 120 Gaussian analysis grid (i.e., the GLOBAL configuration introduced earlier).

2) Single-observation experiments

It is useful at this point to perform “1-obs” (see definition below) experiments to examine the actual response of this direct covariance modeling and the GLOBAL analysis in terms of effective correlation structures implied in the horizontal and vertical analysis increments. We use a simulated temperature innovation at 500 hPa of 1° with a standard deviation error of 1° in the GLOBAL analysis (3DVAR) with its own specification of the background error standard deviation fields.

The top panel in Fig. 10a shows the temperature analysis increment for the GLOBAL analysis while the bottom panel shows the results for the REG-LAM3D-HEM analysis. Induced wind increments, one with the explicit use of balance operators and the other imposing its linear balance couplings directly through the cross correlations, are also superimposed in the figures. It is clear that the GLOBAL system has a stronger projection onto the large horizontal scales, whereas the REG-LAM3D-HEM has more compact horizontal correlations for temperature. Note that the REG-LAM3D-HEM analysis was run with the effective total background error variances coming from the GLOBAL system, where the balanced and unbalanced parts have been used to compute the total variance. In such a case, we expect the exact same amplitude of the temperature increment at the observation point. This latter point is located at 45°N and 90°W (in fact the nearest grid point to this location) and was borrowed from the study by Kleist et al. (2009, hereafter referred to as KAL09) where such 1-obs experiments were performed with the newly implemented Gridpoint Statistical Interpolation (GSI) analysis system at NCEP (cf. their Fig. 2).

Since KAL09 had used their own analysis approach and background error variances (and covariance modeling using recursive filters), a further comparative experiment was performed as above where we now use the REG-LAM3D-HEM’s own error variances. The same simulated observation details were kept as discussed above for the other observation parameters. The result of this experiment is shown in the top panel of Fig. 10c. It is seen that the two systems produce very similar analysis increments even though REG-LAM3D-HEM is based on a homogeneous and isotropic correlation model and its maximum temperature increment is 0.26°, which is about the same amplitude as KAL09’s result (as can be judged from their figure). Looking again at the first experiment (i.e., GLOBAL and REG-LAM3D-HEM in Fig. 10a), the vertical structures of the analysis increments at the observation locations are shown for both experiments in the top panel of Fig. 10b. We can see that both have the same maximum at the observation location (as expected), with the GLOBAL experiment shown with the dashed line and the REG-LAM3D-HEM experiment with the solid line. The significantly different vertical structure probably reflects again the fact that the GLOBAL system has a stronger projection on global scales introduced via the coupling with the streamfunction error structures. We stress that the actual specification of the background error standard deviations in the GLOBAL system is the result of specific adjustments to the global 4DVAR analysis system in order to optimize the global long-range forecasts up to day 5. The results shown here tend to indicate that a different partitioning between the balanced and unbalanced contributions to the analysis increment in the REG system may be desirable but, as we see in the overall evaluations of the REG system, such a modification may be beneficial only to the precipitation scores.

As can be expected from the vertical correlations introduced in Fig. 8, a similar 1-obs experiment in BF mode (Fig. 10b, bottom) shows more resemblance to the REG-LAM3D-HEM experiment (the dashed line is for the REG-LAM3D-BF experiment). To complement the above results, the REG-LAM3D-BF experiment was also performed with its own set of background error variance fields and the 1-obs result is shown in the bottom panel of Fig. 10c. The maximum amplitudes for REG-LAM3D-HEM and REG-LAM3D-BF in each case are very close; the local horizontal correlation structure is very similar but we can note a different behavior in the long tail correlations for BF.

It is also useful to indicate the advantage of using the REG-LAM3D-HEM basis functions (spherical harmonics), which is appropriate for this large horizontal domain on the sphere as opposed to BF. When the bi-Fourier spectral representation of the correlations is performed on a large computational domain (a limited-area uniform latitude–longitude grid) on a sphere, the analysis increments over the western and eastern edges of the REG-LAM3D grid (Fig. 1c, red domain) are distorted. We can expect, for instance, that a 1-obs experiment as above performed over the western edge of the domain would show a deformation of the resulting horizontal analysis increment structure. This is apparent in Fig. 10d, where the REG-LAM3D-HEM results are shown by the dotted lines and the REG-LAM3D-BF results by the solid lines.

3) Examination of the incremental balance

The “direct correlation” approach used here (i.e., no balance–unbalance splitting), as in the standard approach, makes full use of the spectral variability of the direct cross-correlation coupling between the mass and winds and is thus influenced by the north–south (real sphere) variability of the Coriolis parameter. The resulting correlation model for the REG-LAM3D domain reflects mostly extratropical multivariate couplings, although parts of the tropics are present in our southern part of the grid (southern midedge being at 15°N). The standard splitting method does not have this limitation due to the explicit use of a linear balance coupling imposed in the definition of the control variables [e.g., getting a balanced mass variable as a simple local multiplication of the Coriolis parameter times the streamfunction analysis increments at a given iteration of the minimization; see section 3d of Gauthier et al. (1999)]. It is thus important to demonstrate that 1) dynamical balance diagnostics reveal comparable levels of balance in the resulting analysis and 2) no significant damage is done to the tropical part of the domain nor, especially, to the incoming tropical cyclones. The balance aspects of these two points were examined using the nonlinear balance equation, as in Caron et al. (2007) and Caron and Fillion (2010). The level of imbalance was judged by computing the quantity UNBAL defined as a normalized deviation from nonlinear balance [i.e., the mean rms of the residual (mass − wind) divided by the mean amplitude of the mass and the wind components]:
i1520-0434-25-6-1645-e32
where the overbar represents a mean over the vertical levels from 850 to 250 hPa and the rms are computed at a given vertical level over the ensemble of points within a given latitudinal band in the shared domain of the REG and REG-LAM3D analyses. It is important to note that the quantity UNBAL can, in theory, reach values much higher than 100% [e.g., in the cases of a wind (mass) field with a null mass (wind) field, it can easily be shown that values can reach 200%]. Figure 10e shows the typical degree of imbalance resulting from the REG and REG-LAM3D-HEM analyses, where both systems use the same observation sets. According to this measure, the REG-LAM3D-HEM analysis has a slightly higher level of imbalance. Concerning the second point mentioned above, the accuracy of the tropical cyclone forecasts in the REG-LAM3D-HEM system was examined via a set of tropical cyclones cases. Results (not shown) indicate that the REG-LAM3D-HEM system did not perform worse than the current REG operational system.

4) Objective evaluations

Extensive evaluations of the REG-LAM3D analysis system (in BF and HEM mode) have been performed. Figure 11 shows evaluations of 48-h forecasts for the 40-winter cases from the REG-LAM3D-HEM system against the REG control. It is clear that the new system, REG-LAM3D-HEM, performs as well as the control although a remaining bias on the mass field above about 150 hPa is still present. Similar results are obtained for the 40-summer test cases. Also, the REG-LAM3D-BF analysis system on all of these cases performs as well as the REG-LAM3D-HEM system for those diagnostic scores. These results are not shown.

Before discussing the final diagnostics related to the precipitation forecasts, we make a final remark concerning the longer-term forecast accuracy of the REG-LAM3D system. Although operational regional weather forecasts were integrated to 48 h, we extended the forecast and evaluations up to 5 days. As expected, the global variable strategy upon which the REG system is built starts to deviate noticeably from day 3 and drastically at day 5 (results not shown), whereas the REG-LAM3D-HEM remains very accurate, benefiting from the driving uniform 55-km global grid GEM forecast. It is interesting to note that this “disconnection time frame” had already been observed in the early development formulation of the GEM global variable grid configuration within a shallow-water context [see the discussion of the “Andre Robert” criteria on p. 240 of Côté et al. (1993)].

At this stage of the evaluation of the REG, REG-LAM3D-BF, and REG-LAM3D-HEM analysis systems, all three are judged to be equivalent. We now examine the precipitation scores for each system and, as well, the GLOBAL system, which uses the standard global variational analysis code in 3DVAR mode. In Fig. 12 (winter cases), the blue line is for the REG control experiment and the red dashed line is for the GLOBAL experiment, which shows a pattern of behavior that is very similar to the REG. The yellow dashed–dotted line is for the REG-LAM3D-BF experiment. The BF precipitation scores are noticeably bad and show the worst precipitation bias. Similar results were obtained for the summer cases (synoptic reports were also evaluated with similar conclusions). The REG-LAM3D-HEM and GLOBAL analyses perform quite similarly. When comparing against the Cooperative Observer Program (COOP) precipitation dataset (hereafter referred to as SHEF data, information online at http://www.weather.gov/oh/hrl/shef/README_shef_general.htm) in Fig. 13, the latter two analysis systems still produce similar results but consistently beat the control REG analyses (bias and threat scores are better).

At this point, we have discarded the REG-LAM3D-BF approach. We have noted similar levels of performance between REG-LAM3D-HEM and the standard GLOBAL approach. Evaluations against the surface synoptic observations (SYNOP) data (Fig. 14), however, show slightly better performance for the REG-LAM3D-HEM approach against the GLOBAL approach in terms of the threat scores and bias scores. Further testing (not shown here) has been done on the REG-LAM3D-HEM, GLOBAL, and REG analysis systems with other sets of winter 2007 cases and summer 2008 cases with similar conclusions; in other words, the results are quite robust. It is relevant to mention that since the REG-LAM3D-HEM analysis is performed at T200 and the GLOBAL analysis at T108, it may be argued that this explains the improvement seen in precipitation scores just mentioned. The fact is that we purposely reran all 80 test cases with REG-LAM3D-HEM analysis at T108 and reevaluated all the scores. The conclusion is that the “resolution effect” accounts for about half of the improvement seen here for the REG-LAM3D-HEM system, but it does not explain the full benefit observed with the REG-LAM3D-HEM approach. To further demonstrate that the REG-LAM3D-HEM approach is the source of the improvements in the precipitation forecasts observed compared to the REG system, we reran the original winter and summer test cases and purposely substituted the REG trial fields for the REG-LAM3D-HEM analysis. Improvements similar to those reported previously were observed. This latter improvement does not come from the modelization (i.e., LAM) itself since we also reran all cases with the LAM model integrated with the REG analyses without noticeable improvement on all scores. The slightly better precipitation scores (in summer particularly) from REG-LAM3D as compared to the GLOBAL experiments indicate that the REG-LAM3D-HEM analysis approach performs very well by implementing a more direct use of background error correlations without explicit partitioning of the balanced and unbalanced parts of the errors.

5) Computational requirements

A major requirement imposed on this new REG-LAM3D-HEM system is its computational efficiency. One may wonder about the computational cost of running a driving global model and an LAM as compared to the global variable-resolution model. The latter system performs a 48-h forecast in 23 min with 544 CPUs. We can run an LAM forecast in the same time with only 384 CPUs. The driving model (which runs at 55-km horizontal resolution on a global, uniform latitude–longitude grid) can complete a run with 256 CPUs in 12 min. If we compare the sum of the CPUs multiplied by the minutes of the variable resolution grid (12 512) and the LAM + driving global (8832 + 3072 = 11 904), we arrive at a ratio of 95% of the resources needed. If we take into account the computational cost for the analysis part, which takes exactly the same time for the LAM and driving global, we obtain a final ratio of 102% of the computational resources compared to those of the operational system with a global variable-resolution grid. In other words, the REG and REG-LAM3D-HEM are computationally equivalent.

4. Summary and conclusions

In this study, we have gradually introduced a sequence of modifications to the current regional analysis and forecasting system. The current operational regional system (REG) is based on the global variable-resolution approach, uses a minimal change to the existing global variational analysis code, and runs in 3DVAR mode.

Further progress toward a fully flexible mesoscale regional analysis and forecasting system requires a significant increase in the spatial resolution of both the data assimilation and forecasting components. The main interest of this regional system is the tropospheric 48-h forecasts over the North American continent. The new regional data assimilation and forecasting system has been built by merging the two basic approaches fully supported at CMC, that is, a global system and a limited-area system, thus eliminating the need to further maintain the global variable-resolution strategy. The resulting analysis and forecasting system is called the REG-LAM3D system. The first crucial step was to demonstrate that using data available to both REG and REG-LAM3D systems, the same background error correlation assumptions, and the same computer resources, the new system performs equally to as the control system. In the course of this demonstration, the following results were obtained. 1) The bi-Fourier spectral representation of background error statistics performed as well as the control system for 48-h forecast scores against radiosonde data but failed to produce accurate precipitation forecasts. Due to the large horizontal extension of the REG-LAM3D grid, the bi-Fourier representation was replaced by spherical harmonics in a “hemispheric type” (HEM) representation of background error statistics. 2) A new analysis approach has been designed to eliminate the traditional balanced–unbalanced splitting in the definition of the analysis control variables in favor of direct basic analysis variables and background error correlations. The strong performance of the new REG-LAM3D-HEM system has been demonstrated here through the use of 40 winter and 40 summer cases. Further testing of this new system (not presented in this study) has continued and currently a total of 240 test cases confirm all of the conclusions mentioned here without exceptions. This REG-LAM3D-HEM analysis and forecasting system was officially approved for parallel testing at CMC in July 2009 and it will be transferred to operations pending positive evaluations from the parallel runs.

For future developments of this REG-LAM system, the following important upgrades are in progress (some are already known to perform very well): 1) transition to REG-LAM-4DVAR, 2) inclusion of additional data such as ground-based GPS, 3) inclusion of a flow-dependent background error covariance matrix deduced from the global ensemble forecast system following Buehner’s method (Buehner et al. 2010), 4) activation of the implicit normal mode tangent-linear approach described in Fillion et al. (2007) and Kleist et al. (2009), and 5) transition to a higher horizontal and vertical spatial resolution tropospheric configuration of REG-LAM where the boundary conditions for the model top are provided by a driving global model.

In addition to publishing the results of these upgrades, an article will be written on specific applications of the bi-Fourier formulation within the limited-area configuration in our variational analysis code as this is better suited for smaller horizontal domains at the kilometric scale.

Acknowledgments

The authors would like to acknowledge our colleagues at Dorval—Simon Pellerin, Bin He, Cecilien Charette, and Stephen Macpherson—for their support with the variational analysis system and model upgrades involved along the way. Thanks are also due to our colleagues, Drs. Bernard Dugas and Jean Côté, for the discrete Fourier transform package used in this study. A special thank you goes to our colleagues Michel Valin, Yves Chartier, and Djamel Bouhemhem from the scientific computing division for their kind, expert, and constant support since the beginning of this development work.

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APPENDIX A

From Helmholtz’s Functions to Wind Components on the Limited-Area Arakawa-C Analysis Grid

In the following, we give the spatial discretization used to go from Helmholtz’s functions to wind field components required during the minimization step. From the streamfunction and velocity potential (ψ, χ), we have (see Haltiner and Williams 1980, their Eqs. (6-74) and (6-75)]
i1520-0434-25-6-1645-eqa1
Note that the computational grid is described in λ and θ coordinates. We use the variable μ ≡ sinθ instead of θ. For example, finite differences are computed using increments of Δμ. In the LAM configuration used here, the computational grid used in this study has a constant value of Δθ.
The essential information on the Arakawa C grid, spatial differentiation operators, and terminology used here can be found in appendix A of Côté et al. (1998). We start with a given χ on the scalar grid and ψ at the center of each scalar-grid tile. In such a case, the centered derivatives [similar to (A.10) and (A.11) in Côté et al. (1998)] become simple. For instance, a U value on the u grid requires from the above equations the following derivatives to be computed:
i1520-0434-25-6-1645-eqa2
We get
i1520-0434-25-6-1645-eqa3
The adjoint of the above computations is also required by the minimization.

APPENDIX B

Computation of Bi-Fourier Spectral Background Error Correlations

In the following, we give the basic formulas used to compute the spectral densities and spectral correlation matrices involved in the representation of the homogeneous and isotropic part of background error correlations. For the fundamental theory, the reader can find details in standard textbooks such as Yaglom (1987) and Daley (1991, section 3.3) for simple applications to specific correlation modeling in two dimensions. The paper by Berre (2000), especially in the appendixes, gives a concrete, detailed, and useful description of the bi-Fourier spectral computation of forecast error correlations. We note in passing the following two simple typographic errors in his text: Eq. (2) should have D rather than Ns, and in the last line on p. 664, s rather than s* should be used. The same steps as described in his appendix B were used here except that we use 24-/48-h forecasts rather than 12-/36-h forecasts. Using the same binning approach (also done with dk* = 1.5), the number of spectral components falling into each total horizontal wavenumber bin is represented in the array mband.

a. Spectral correlations

The nonseparable background error covariance matrix consists of a vertical covariance matrix for each total wavenumber kt and is noted . Given a three-dimensional error field with spectral components at vertical level m and noted ãktm, the spectral covariance matrix is given by
i1520-0434-25-6-1645-eqb1
The overbar denotes an ensemble mean and the value α is used to avoid double counting when summing over all spectral components contained in the full spectral annulus (i.e., positive and negative xy wavenumbers under the reality condition). The resulting correlation matrix actually used in the analysis step is thus given by
i1520-0434-25-6-1645-eqb2
The vertical variation of the horizontal correlation scales is given in section 2c of Berre (2000). The specific formula after the homogeneous and isotropic assumption used in his Eq. (2) is also used here:
i1520-0434-25-6-1645-eqb3
For spectral representation, we use the two-dimensional discrete Fourier transform (2D-DFT) to 1) represent the horizontal part of the forecast error fields on the REG-LAM3D analysis grid when preparing background error correlations discussed above and 2) to do (undo) the preconditioning of the minimization in the 3DVAR analysis. The reader will find useful theory on the DFT and its applications in Briggs and Henson (1995, especially chapter 5). For a given two-dimensional field on a grid with uniform grid spacing in the x direction (M points) and uniform grid spacing in the y direction (N points) (but not necessarily the same spacing as the x direction) with grid coefficients fmn, the forward and inverse two-dimensional DFTs are given, respectively, by
i1520-0434-25-6-1645-eqb4
This two-dimensional transform can be efficiently computed using a 1D DFT sequentially in each direction (cf. Briggs and Henson 1995).

The error-sample fields prior to spectral decomposition were made biperiodic on an extended domain. This “biperiodicization” is defined as in Haugen and Machenhauer (1993). It should be noted that, due to the large horizontal correlation scales at levels between 250 hPa and the model top, an initial value of 20% grid extension had to be increased to 40%. This is to avoid symmetric reflection of analysis increments due to observation assimilated close to the lateral walls of the inner analysis domain.

A word should be said about spectral array allocation with the global and LAM analysis configurations. Since nothing particularly special is involved in manipulating spectral fields in both representations, a unified allocation structure was adopted so as to minimally impact the global variational analysis code. This means that the spectral arrays are kept the same and the typical declaration, for instance, for temperature, is sptt(nla,2,nflev), where nflev is the total number of vertical levels on the analysis grid and nla is the total number of complex spectral components. The latter is correctly set in the beginning when the dimensions of the minimization problem are established. Manipulations of spectral arrays within parts of the minimization code are done in the same way as in the LAM or global systems.

b. Construction of error samples

The NMC approach was used for all configurations of REG-LAM3D using 48–24-h forecast differences for the summer and winter cases tested. Typically 120 (sometimes up to 150) cases per season were used. The GEM global 33-km forecast model was used together with global 4DVAR analyses. Due to the 55-km horizontal resolution of the REG-LAM3D analysis grid, such 33-km forecast differences were sufficient for estimating the spectral error covariances at the scales of the analysis. Note that the same vertical grid and very similar physics schemes were used for the global and REG-LAM analysis and forecast systems. At the beginning of the development project, the 58-level vertical grid was used, as shown in the left panel of Fig. 3; subsequently during 2007, it was replaced by the “stratospheric” version with 80 vertical levels, as shown in the right panel of Fig. 3. The forecast variables considered from these forecasts are the 1) zonal and meridional wind components, 2) temperature, 3) geopotential (useful for the background check), 4) specific humidity, 5) surface pressure, and 6) ground temperature.

The REG-LAM3D analysis operates with Helmholtz’s streamfunction and velocity potential as in the global analysis system. In BF mode, a specific subroutine was constructed to prepare the Helmholtz’s functions for the Arakawa C limited-area analysis grid. Global forecast wind differences were then used over the REG-LAM3D domain. Directly computing the Helmholtz’s on the sphere (using spherical harmonics) before interpolating to the REG-LAM3D domain was, however, found to be more accurate.

c. Some statistics

Using the above-mentioned procedure, Fig. B1 shows the computed vertical variation of the horizontal correlation scales for the 1) streamfunction, 2) velocity potential, 3) temperature, and 4) logarithm of the specific humidity. The large increase in the horizontal correlation scale from the troposphere to the stratosphere is clearly visible. As mentioned in section 3, this puts a severe constraint on the minimal extension zone to be used for the biperiodicization of the fields in the bi-Fourier approach when the stratosphere is included, especially for Helmholtz’s functions. It should be noted that even if one chooses to use vorticity–divergence rather than Helmholtz’s functions (based on the fact that the former has much smaller horizontal correlation scales), the problem reappears somewhere else. Indeed, by inverting the vorticity/divergence increments during the minimization in the process of deriving the wind component analysis increments (to be compared with the wind innovations), one has to use a large extension zone. This is in order to avoid the emergence of the computed Helmholtz’s function increments on the opposite domain boundary for observations close to the boundaries of the analysis domain. These aspects are also considered in the vorticity–divergence local kilometric analysis that comes with the current bi-Fourier approach and is now available in the variational analysis code. Vorticity and divergence analysis variables are used in the local kilometric analysis since the previous global approach is not applicable. This option is viable for such small domains on the sphere since Laplace inversion with Fourier basis functions becomes as trivial as inverting a Laplacian on the sphere using spherical harmonics.

Fig. 1.
Fig. 1.

(a) The current operational 80-level global variable-resolution grid used by the REG analysis system. The horizontal resolution inside the blue line is 15 km. The poles of this grid are placed over the Pacific and Atlantic Oceans. (b) The blue line shows the limit of the uniform 15-km resolution of the 58-level global variable-resolution model used operationally prior to the current operational version. The green line shows the second REG-LAM3D-BF test grid. (c) The orange line shows the third REG-LAM3D-BF test grid, and the red line the final REG-LAM3D grid. The dashed lines for the LAM grids represent the inner limit of the boundary condition zone (typically 12 grid points).

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 2.
Fig. 2.

Current GEM-LAM experimental forecasting windows operated by CMC at 2.5-km horizontal resolution to provide 24-h forecasts.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 3.
Fig. 3.

Analysis/model levels: (left) 58 levels of the last operational REG system and (right) 80 levels of the current REG system (operational since 22 Jun 2009). The same levels are also used in the 58- and 80-level versions of the REG-LAM3D system.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 4.
Fig. 4.

The structure of the REG-LAM3D data assimilation system. The LAM and driving global model and analysis types are indicated.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 5.
Fig. 5.

The 48-h forecast verifications against North American radiosondes. The analysis and forecast models have 58 vertical levels (see Fig. 3, left). The fields considered are the (a) u component of the horizontal wind, (b) modulus of the horizontal wind vector, (c) geopotential, (d) temperature, and (e) dewpoint depression. Biases and standard deviation errors appear in the legend. The REG-LAM3D-BF analysis is used here with the horizontal grid shown in green in Fig. 1b.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 6.
Fig. 6.

As in Fig. 5, but 1) the analysis and forecast models have 80 vertical levels (with more of the stratosphere included) rather than 58 vertical levels and 2) the REG-LAM3D-BF analysis is used here with the LAM grid shown by the orange lines in Fig. 1c.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 7.
Fig. 7.

(top) The 48-h REG control forecast with the global variable-resolution model; (bottom) the REG-LAM3D-BF 48-h forecast. The verifying analysis, valid at 0000 UTC 13 Jan 2007, appears as black contours. Shaded regions represent the 48-h accumulated precipitation.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 8.
Fig. 8.

Vertical background error cross correlations between the spectral streamfunction and temperature fields for a given horizontal total wavenumber. (top) The BF is at wavenumber 5 (domain size is approximately 10 000 km wide in each direction). (bottom) The spherical-harmonics representation at total wavenumber 20. NMC error samples have been symmetrized about the equator so as to focus on the NH correlation structure in a manner similar to the BF approach. The model top is at level 1 and the contour interval is 0.1.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 9.
Fig. 9.

Vertically localized spectral correlation blocks as used by the REG-LAM3D-HEM analysis at global horizontal wavenumber 20. The contour interval is 0.1.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 10.
Fig. 10.

(a) Temperature increment (analysis − background) at 500 hPa (contour interval is 0.05 K) resulting from a single 500-hPa temperature observation at 45°N, 90°W with a 1-K residual and observation error for the (top) GLOBAL and (bottom) REG-LAM3D-HEM analyses. Winter season background error statistics were used for both analysis systems. (b) (top) As in (a) but, for the vertical profile of temperature increment at the observation location. Shown are the REG (dashed line) and REG-LAM3D-HEM (solid line) analyses but the background-error standard deviations are those of the REG analysis for intercomparison purposes. (bottom) As in the top panel, but the dashed line is for the REG-LAM3D-BF analysis. (c) Same simulated data as in (a), but for the (top) REG-LAM3D-HEM and (bottom) REG-LAM3D-BF analyses. Each of the analyses used its own background-error standard deviation fields. (d) Same type of simulated data as in (a) but located at 45°N, 163°W. Shown are the REG-LAM3D-BF (solid line) and REG-LAM3D-HEM (dotted line) analysis increments. (e) Normalized deviation from Charney’s nonlinear balance over different latitudinal bands for the control REG operational analysis and the analysis fields resulting from the REG-LAM3D-HEM system valid at 1200 UTC 1 Jan 2007.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 11.
Fig. 11.

The 48-h forecast verifications against North American radiosondes. The analysis and forecast models have 80 vertical levels (see Fig. 3, right). The REG-LAM3D-HEM analysis is used.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 12.
Fig. 12.

Winter verifications of 0–24-h accumulated precipitation forecasts against SHEF data. Shown are the REG system (solid blue line), GLOBAL analysis (red dashed line), REG-LAM3D-BF (yellow dashed line), and REG-LAM3D-HEM (purple dotted line). The REG-LAM3D-BF analysis approach shows the worst scores.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 13.
Fig. 13.

Summer verifications of 24–48-h accumulated precipitation forecasts against SHEF data. Shown are the REG system (solid blue line), REG-LAM3D-HEM analysis system (yellow dashed line), and REG-LAM system, but the analysis step is performed with the GLOBAL analysis (red dotted line).

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. 14.
Fig. 14.

As in Fig. 13, but for verification against SYNOP data.

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Fig. B1.
Fig. B1.

Vertical variation of the horizontal background error correlation scales “L” for the Helmholtz’s functions, the temperature, and the logarithm of the specific humidity computed with the NMC 48–24-h approach. The horizontal resolution of the operational GEM model is 33 km and its vertical grid has 80 levels (Fig. 3).

Citation: Weather and Forecasting 25, 6; 10.1175/2010WAF2222401.1

Table 1.

List of observations assimilated by all data assimilation systems (i.e., the operational REG, the operational global 4DVAR, and the new REG-LAM3D system).

Table 1.
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