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  • View in gallery

    Domain used for the estimation of the “NMC lagged forecast differences” data using latitude–longitude projection with numerical poles shifted from the geographical poles. The resolution is 0.49° (∼55 km) and the domain extent is roughly 8000 km by 8000 km. Topography (shades) is shown for visualization purposes only.

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    Number of times that each surface grid point in the 100 members of lagged forecast are (a) precipitating, (b) in the light precipitation class, (c) in the moderate precipitation class, and (d) in the heavy precipitation class. Note the different scale in (d).

  • View in gallery

    Vertical profile of average (over the 100 members) statistical measure of the degree of linear balance in the lagged forecast differences computed over dry areas (boldface, ○) and over light (□), moderate (⋄), and heavy (▵) precipitation areas. (a) Vertical profiles of average horizontal correlation between mass and wind component and (b) ratio of average horizontal rms value of mass and wind component (wind rms divided by mass rms). The gray dotted line in (b) indicates a ratio of 1.

  • View in gallery

    Normalized deviation from linear balance for dry areas and for light, moderate, and heavy precipitation areas in the ensemble of lagged forecast differences. See text for the details on the computation of this measure.

  • View in gallery

    Vertical profile of average (over the 100 members) statistics of explained temperature by Ns and Nl over light precipitation areas in the ensemble of lagged forecast differences. (a) δTbδT correlation for δTb computed with Ns (boldface, ○) and for δTb computed with Nl (□). (b) The rms of δTb computed with Ns (boldface, ○), of δTb computed with Nl (□) and of δT (dotted, ⋄). (c) The rms of δTu when δTb is computed with Ns (boldface, ○) and when δTb is computed with Nl (□). The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa.

  • View in gallery

    As in Fig. 5, but for Ns and Nm over moderate precipitation areas.

  • View in gallery

    As in Fig. 5, but for Ns and Nh over heavy precipitation areas.

  • View in gallery

    Average (over the 100 members) statistics of explained surface pressure by Ns and the diabatic operator associated to each precipitation areas in the ensemble of lagged forecast differences. (a) δps,bδps correlation for δps,b computed with Ns (black bars) and for δps,b computed with the corresponding diabatic operator (gray bars). (b) The rms of δps,b computed with Ns (black bars), of δps,b computed with the corresponding diabatic operator (gray bars) and of δps (dotted bars). (c) The rms of δps,u when δps,b is computed with Ns (black bars) and when δps,b is computed with the corresponding diabatic operator (gray bars).

  • View in gallery

    Wind and mass analysis increments in response to the assimilation of a single observation of zonal wind at 500 hPa using Ns. (left) Wind analysis increments (arrows, kt) at the closest model level from 500 hPa. (right) Vertical cross section of temperature (thin lines, contour interval: 0.02 K) and streamfunction (thick lines, contour interval: 0.1 m2 s−1) analysis increments. The points N and S show the limits of the cross section. Positive (negative) values are in solid (dashed) lines and zero contours are omitted. The long-dashed line shows the location of the vertical profiles in Fig. 10. The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa.

  • View in gallery

    Analysis increments in response to the assimilation of a single observation of zonal wind at (a),(b) 250, (c),(d) 500, and (e),(f) 850 hPa using standard and diabatic balance operators. (left) Vertical profile of temperature analysis increments using Ns (bold, ○), Nl (□), Nm (⋄), and Nh (▵). The gray doted line indicates a value of zero. The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa. (right) Surface pressure analysis increments at the location of the profile in the left panel using Ns (std), Nl (light), Nm (mod), and Nh (heavy).

  • View in gallery

    Wind and mass analysis increments in response to the assimilation of a single observation of temperature at 500 hPa using Ns. (a) Temperature analysis increments (contour interval: 0.2 K) at the closest model level from 500 hPa. (b) Vertical cross section of vorticity (thin lines, contour interval: 0.1 × 10−5 s−1) and temperature (thick lines, contour interval: 0.2 K) analysis increments. The point N and S show the limits of the cross section. Positive (negative) values are in solid (dashed) lines and zero contours are omitted. The long dashed line shows the location of the vertical profiles in Fig. 12. The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa.

  • View in gallery

    As in Fig. 10, but in response to the assimilation of a single observation of temperature with (left) vertical profile of vorticity analysis increments.

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An Examination of Background Error Correlations between Mass and Rotational Wind over Precipitation Regions

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  • 1 Meteorological Research Division, Environment Canada, Dorval, Québec, Canada
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Abstract

The differences in the balance characteristics between dry and precipitation areas in estimated short-term forecast error fields are investigated. The motivation is to see if dry and precipitation areas need to be treated differently in atmospheric data assimilation systems. Using an ensemble of lagged forecast differences, it is shown that perturbations are, on average, farther away from geostrophic balance over precipitation areas than over dry areas and that the deviation from geostrophic balance is proportional to the intensity of precipitation. Following these results, the authors investigate whether some improvements in the coupling between mass and rotational wind increments over precipitation areas can be achieved by using only the precipitation points within an ensemble of estimated forecast errors to construct a so-called diabatic balance operator by linear regression. Comparisons with a traditional approach to construct balance operators by linear regression show that the new approach leads to a gradually significant improvement (related to the intensity of the diabatic processes) of the accuracy of the coupling over precipitation areas as judged from an ensemble of lagged forecast differences. Results from a series of simplified data assimilation experiments show that the new balance operators can produce analysis increments that are substantially different from those associated with the traditional balance operator, particularly for observations located in the lower atmosphere. Issues concerning the implementation of this new approach in a full-fledged analysis system are briefly discussed but their investigations are left for a following study.

Corresponding author address: Jean-François Caron, Meteorological Research Division, Environment Canada, 2121 route Transcanadienne, Dorval, QC H9P 1J3, Canada. Email: jean-francois.caron@ec.gc.ca

This article included in the Intercomparisons of 4D-Variational Assimilation and the Ensemble Kalman Filter special collection.

Abstract

The differences in the balance characteristics between dry and precipitation areas in estimated short-term forecast error fields are investigated. The motivation is to see if dry and precipitation areas need to be treated differently in atmospheric data assimilation systems. Using an ensemble of lagged forecast differences, it is shown that perturbations are, on average, farther away from geostrophic balance over precipitation areas than over dry areas and that the deviation from geostrophic balance is proportional to the intensity of precipitation. Following these results, the authors investigate whether some improvements in the coupling between mass and rotational wind increments over precipitation areas can be achieved by using only the precipitation points within an ensemble of estimated forecast errors to construct a so-called diabatic balance operator by linear regression. Comparisons with a traditional approach to construct balance operators by linear regression show that the new approach leads to a gradually significant improvement (related to the intensity of the diabatic processes) of the accuracy of the coupling over precipitation areas as judged from an ensemble of lagged forecast differences. Results from a series of simplified data assimilation experiments show that the new balance operators can produce analysis increments that are substantially different from those associated with the traditional balance operator, particularly for observations located in the lower atmosphere. Issues concerning the implementation of this new approach in a full-fledged analysis system are briefly discussed but their investigations are left for a following study.

Corresponding author address: Jean-François Caron, Meteorological Research Division, Environment Canada, 2121 route Transcanadienne, Dorval, QC H9P 1J3, Canada. Email: jean-francois.caron@ec.gc.ca

This article included in the Intercomparisons of 4D-Variational Assimilation and the Ensemble Kalman Filter special collection.

1. Introduction

For numerical weather prediction (NWP) forecasts at mesoscale and very-short-range time scales (e.g., nowcasting), the forecast of precipitation is of major interest but also poses the greatest challenge. A large part of the quality of the forecast relies on the quality of the initial conditions (the so-called analysis). The mesoscale analysis must contain the necessary information to allow the NWP model to start with precipitation areas at the right location and to correctly evolve these precipitation areas during the time integration. Because of the importance of moist physical processes at these spatial and time scales, it is necessary for instance to impose an adequate balance on mass and wind analysis increments over precipitation areas in data assimilation systems. The benefits are 1) to maximize the amount of information extracted from observations (i.e., information on the mass field from wind observations and vice versa), and 2) to minimize the excitation of spurious fast gravity waves in the early stage of the forecast. The latter being particularly important for optimal assimilation of precipitation-related data (Errico et al. 2007).

To improve the representation of the divergent part of the wind over precipitation areas in variational data assimilation (Var) systems, Fillion et al. (2005) extended the idea of Fisher (2003) to use the quasigeostrophic (QG) omega equation to relate mass and divergent wind increments by introducing a coupling between diabatic forcing and divergence. Recently, Pagé et al. (2007) demonstrated the ability of another form of the omega equation to diagnose summertime mesoscale convective systems with a significant accuracy and envisage its utility as a balance constraint in a mesoscale Var system. These new approaches are currently being examined in the Environment Canada (EC) limited-area Var system (Fillion et al. 2005).

In terms of rotational wind balance, EC’s limited-area Var system uses, like other mesoscale systems [e.g., Aire Limitée Adaptation Dynamique Développement International (ALADIN; Sadiki et al. 2000) and the High-Resolution Limited-Area Model (HIRLAM; Lindskog et al. 2006)], a balance operator that is obtained, as proposed by Derber and Bouttier (1999) for a global system, by linear regression between the mass field and the rotational component of the flow (transformed into a mass variable using a given balance assumption) from an ensemble of estimated forecast errors. Traditionally, when constructing the mass–rotational wind operator, all points (from both dry and precipitation areas) of the ensemble of estimated forecast errors are mixed together. However, balance properties can differ significantly between dry and precipitation areas since, for example, latent heating associated with condensation processes tends to destroy balance at certain scales (Errico 1989). Therefore, this form of rotational balance representation in Var systems might not optimally represent the balance characteristics over precipitation areas. In this paper, we first investigate if the balance properties of the rotational part of the flow differ between dry and precipitation areas within an ensemble of lagged forecast differences. Following the results obtained, we investigate if some improvements in the coupling between mass and rotational wind increments over precipitation areas can be achieved by using only the precipitation points within an ensemble of lagged forecast difference errors to construct the so-called diabatic operator by linear regression. Our aim is to suggest that future data assimilation strategies, particularly those for mesoscale forecasting, should benefit by treating mass and rotational wind forecast error correlations differently over precipitation regions.

The organization of this paper is as follows. In the next section, we first describe in detail the current and tested representation of mass–rotational wind coupling in EC’s Var system. Section 3 then presents the data used to construct the ensemble of lagged forecast differences and the methodology adopted to define precipitation areas in this study. The differences in the balance properties between dry and precipitation areas in the ensemble of lagged forecast differences form the subject of section 4. In section 5, the accuracy of the standard and the diabatic balance operators to explain the mass field from information on the rotational wind field over precipitation areas is compared and their different behavior in a three-dimensional (3D) Var system is inspected in a series of simplified data assimilation experiments. Section 6 presents a summary of the results and a discussion of future investigations.

2. Treatment of balance in Var systems

In this section we give the essentials about the representation of balance in EC’s Var system, which is inspired by the Derber and Bouttier (1999) approach. After defining the general representation of balance, we detail the operators involved in the representation of the balance between mass and rotational wind increments. We conclude by presenting the new approach tested to improve this latter balance over precipitation areas.

a. Definition of the analysis variables

In the variational analysis procedure, we are seeking analysis increments (δxa), that is, model state variable corrections to an a priori estimate. In our case, the analysis increment vector is represented by
i1520-0493-138-2-563-e1
where u and υ represent, respectively, zonal and meridional component of the horizontal wind; T is the temperature, ps is the surface pressure; and q is the specific humidity. Using Helmholtz’s theorem, the horizontal flow is expressed in terms of streamfunction (Ψ; representing the rotational component) and velocity potential (χ; representing the divergent component):
i1520-0493-138-2-563-e2
The last transformation involves a splitting of the analysis increment vector in terms of the so-called balanced (subscript b) and unbalanced (subscript u) components:
i1520-0493-138-2-563-e3
In the balanced part of the analysis increments [first parenthesis on the right-hand side of (3)], any changes in one of the variable during the assimilation process (i.e., technically: during the minimization of the cost function) will impact the other balanced variables through some a priori defined balance operators. In the unbalanced component [second parenthesis on the right-hand side of (3)], a change in one of those variables will have no impact on the other unbalanced variables. It is the background error statistics that control the partitioning between the balanced and the unbalanced components. To construct these statistics, the same balance operators involved in the coupling of the balanced component of the analysis increments are used to partition an ensemble of estimated forecast errors into balanced and unbalanced (i.e., residual) components from which statistical information is extracted. Therefore, the choice of the balance operators influences the resulting analysis increments in two ways: 1) by controlling the relations between the variables of the balanced components and 2) by controlling the partition between balanced (coupled) and unbalanced (uncoupled) components. Equation (3) forms the control vector in EC’s Var system. We note that (3) does not take into account an unbalanced component in the rotational part of the wind increments [although some methods exist to include this latter component, e.g., see Fillion et al. (2007); Roulstone et al. (2007)] nor for a balanced component in the humidity increments.

b. Standard mass–rotational wind coupling

The coupling between the mass and the rotational wind components of the analysis increments is done using
i1520-0493-138-2-563-e4
where H is a horizontal balance operator that transforms the streamfunction increments into mass variable increments and N is a vertical balance operator. Here linear balance is adopted as the horizontal balance operator, which simply means that H = f, where f is the Coriolis parameter.1 The N operator is obtained through a linear regression between the streamfunction (transformed into a mass variable using H) and both temperature and surface pressure variables within an ensemble of estimated forecast errors (see appendix A for complete details on how the regression matrix N is constructed). The role of N is thus to modify the balance imposed by H toward an “observed” linear balance and to act as a hydrostatic operator that transforms mass profiles into temperature and surface pressure increments. We remark that here N is defined in gridpoint space and not in spectral space as in Derber and Bouttier (1999). Nevertheless, this gridpoint space representation of the balance operator does not prevent our Var system from using nonseparable background error correlations. Finally, we also remark that our mass–wind coupling does not take into account the contribution from unbalanced divergence as in Derber and Bouttier [1999; see their Eq. (8)].

c. Diabatic balance operators

The standard way to construct N (hereinafter referred as Ns) is to regress using all grid points of an ensemble of estimated forecast error regardless of whether these grid points are in a dry or in a precipitation area. Here we define four new balance operators: Nl, Nm, and Nh, obtained from linear regression using, respectively, only forecast error grid points in light, moderate, and heavy precipitation areas, and, finally, Nd, constructed using only forecast errors at grid points in dry areas. The methodology to partition the forecast error grid points among the different areas is presented in the following section.

3. Data and precipitation classes

In this study, the so-called NMC method (Parrish and Derber 1992) is used to generate estimates of forecast error. The NMC method consists of computing differences of lagged forecasts valid at the same time. Although the NMC method represents a crude estimate of the true forecast error covariances, it is believed to represent a good approximation to the true dynamical balances between the variables since current NWP models represent these balances well.

Using data from the Canadian operational global forecasting system (Bélair et al. 2009), an ensemble of lagged forecast differences was constructed using 24- and 48-h forecasts. The ensemble consists of 100 members of forecast differences (48 minus 24 h) at 12-h intervals over 50 consecutive days over the months of December 2006 and January 2007. The data, originally on a global latitude–longitude grid of 0.3° (∼33 km) horizontal resolution, were interpolated to a regional rotated latitude–longitude grid of 0.49° (∼55 km) horizontal resolution covering the North American continent and its adjacent oceans (Fig. 1). The characteristics of the data are summarized in Table 1. The domain and the resolution adopted for the lagged forecast difference fields correspond to the three-dimensional variational data assimilation (3D-Var) grid configuration of the next generation analysis system of EC’s regional forecasting system [see Fillion et al. (2005) for details].

Dry and precipitation areas were defined using the instantaneous precipitation rate (P) from 24- and 48-h forecasts. The information was combined by computing the mean precipitation rate [P = (P24 + P48)/2] at each grid point at each (i.e., 100) verification time. Values of P greater than 0 but less than 0.1 mm h−1 were associated with dry areas in order to avoid including regions of marginal moist processes in the precipitation areas. Figure 2a presents the geographical distribution of the number of precipitation grid points (P ≥ 0.1 mm h−1) in the ensemble of lagged forecasts. Most of the precipitation occurs in the intense baroclinic regions over the northeast Pacific, the eastern portion of the continent, and the North Atlantic. Our domain also captures some orographically enhanced precipitation in Central America. Combining all precipitation grid points reveals that only 14.7% of the total grid points in the ensemble are precipitation grid points (Table 2). The rather rare occurrence of precipitation suggests that Ns, computed using all the grid points is, as expected, essentially representative of the dry dynamics.

Precipitation areas were split into three classes (light, moderate, and heavy) based on the intensity of P. The values of P associated with each class as well as the distribution of the number of grid points in the ensemble of lagged forecasts associated with each class are presented in Table 2. The ranges of P for the three classes of precipitation were chosen to obtain a three class distribution that looks similar to the full precipitation distribution (not shown); that is, a rapidly decaying distribution. The geographical distribution of the occurrence of each precipitation class is presented in Figs. 2b–d. It can be seen that each class is rather well distributed over the domain with the exception that moderate and heavy precipitation classes are essentially confined to oceanic and coastal areas since the central and northern parts of the continent, in a cold and dry air regime at this time of the year, can only experience light precipitation.

Finally, we remark that the choice of the number of precipitation classes (i.e., 3) adopted here is somewhat arbitrary. However, increasing the number of classes would lower the number of grid points associated with each class and, without increasing the size of the ensemble, would increase the risk to generate an “underfitted” linear regression operator (N), especially for the classes in the tail of the precipitation distribution. With the three class partitioning adopted here, even the rather rarely occurring heavy precipitation class (i.e., 1.3% of ∼1.4 × 108 grid points, see Table 2) contains enough data to construct N, which has a dimension of 58 × 59 (i.e., the number of vertical levels of streamfunction times the number of vertical levels of temperature plus the surface pressure).

4. Degree of balance over precipitation areas

In this section we investigate by how much the balance properties in the ensemble of lagged forecast differences differ between dry and precipitation areas. The degree of balance between the mass and the rotational wind component of the perturbations is estimated using the linear balance equation since the latter is used in the current formulation of the mass–rotational wind coupling in our Var system [see Eq. (4)]. However, we use here a linear balance defined in terms of the second derivative of mass and streamfunction fields in order to enhance the local differences:
i1520-0493-138-2-563-e5
where δΦ represents the geopotential difference. If the (rotational) wind field (right-hand side) and the mass field (left-hand side) of the lagged forecast differences are perfectly in geostrophic balance (neglecting the df/dy effect, i.e., the latitudinal variation of f ), the two sides of (5) will be identical. Since (5) is usually defined on pressure levels, we interpolated the lagged forecast differences, originally on 58 terrain-following levels, onto 26 pressure levels2 to avoid the generation of any imbalance due to the vertical coordinate.

To evaluate the balance, the correlation and root-mean-square (rms) values were computed for each member of the ensemble.3 The statistics were computed, respectively, over each of the four areas defined in section 3. Each grid point, from 1000 to 10 hPa, was assigned to a class of precipitation based on the precipitation classes valid at its corresponding surface grid point. The different statistics were then averaged over the 100 members. Figures 3a,b indicate that the balance in the lagged forecast differences exhibits significant differences between dry and precipitation areas, especially in the mid and upper troposphere. At every level, the correlation between mass and wind is lower over precipitation areas than in dry areas, with the exception of the lowest 200 hPa for light and moderate precipitation classes (Fig. 3a). We also note that, over precipitation areas, the correlation is inversely proportional to the precipitation intensity. Over precipitation areas, the ratio between the wind and the mass component amplitude (wind rms divided by mass rms) is also lower than in dry areas (Fig. 3b), indicating that the mass component is relatively more important over precipitation areas. At many levels (mainly in the upper troposphere) the observed differences are again proportional to the precipitation intensity.

The level of imbalance in the dry and each of the precipitation areas was measured directly by computing a normalized deviation from linear balance (U) defined as the mean rms of the residual (mass minus wind) divided by the mean amplitude of the mass and the wind components:
i1520-0493-138-2-563-e6
The mean represents here an average over the 100 members and over the vertical levels between 925 and 200 hPa. The results, depicted in Fig. 4, confirm that the lagged forecast differences are, on average, further away from geostrophic balance over precipitation areas than in dry areas and that the deviation from the geostrophic balance is proportional to the intensity of the precipitation. We take these results as a strong motivation to improve the standard mass–rotational wind coupling in our Var system.

We also remark that the lagged forecast differences used here come from a winter period when precipitation is dominated by stratiform processes. The greater imbalance observed over precipitation areas might not only be due to condensational processes. Dynamical processes generating the (stratiform) precipitation could also be an important source of deviation from geostrophic balance. Further investigations would be advisable, but are outside the scope of this paper.

5. Standard VS diabatic balance operators

a. Explained mass field

We now investigate if constructing diabatic balance operators, as presented in section 2c, can improve the coupling between mass and rotational wind increments over precipitation areas. Each balance operator (Nd, Nl, Nm, and Nh) was constructed using grid points attributed to each class based on the precipitation class valid at its corresponding surface grid point. Here we examine whether these new balance operators can explain more of the mass field than the standard operator (Ns) based on information on the wind field in the ensemble of lagged forecast differences. Precisely, we evaluate if the balanced temperature (δTb) and balanced surface pressure (δps,b) increments are more similar to the temperature (δT) and surface pressure (δps) differences from the lagged forecast differences when δTb and δps,b are computed using the diabatic operators in (4), over their respective areas, instead of Ns.

To judge the accuracy of the different balance operators, correlations between balanced and total fields as well as rms values for each field were computed, respectively, over each of the four domains (dry, light, moderate, and heavy) in the 100 members of the ensemble of lagged forecast differences. The different statistics were then averaged over the 100 members. We chose to omit the results for the dry areas since no noticeable changes were observed between Nd and Ns. This result was foreseeable since Ns is constructed using a large majority of grid points from dry areas (see Table 2 and text in section 3).

1) Explained temperature

The diabatic balance operators lead, on average, to modest but significant improvements, especially in the lower troposphere, of the amount of temperature differences explained from information on the rotational wind over each precipitation area. For the light precipitation areas (Fig. 5), the changes are relatively small. The correlation between δTb and δT is slightly improved below 800 hPa when using Nl (Fig. 5a). However, the amplitude of δTb, much smaller than δT at most model levels when computed with Ns, is modestly increased with Nl at every level in the troposphere (Fig. 5b). This leads to a very small reduction in the amount of unbalanced temperature (δTu i.e., δT minus δTb) when using Nl compared to Nd. In moderate precipitation areas, the changes caused by using the diabatic operators become more significant (Fig. 6). The Nm improved the correlation between δTb and δT especially below 600 hPa (Fig. 6a) and an improvement in the amplitude of δTb similar to the one observed over light precipitation areas is also noticeable throughout the troposphere (Fig. 6b). As a consequence, a clear reduction of δTu in the lower troposphere is obtained from Nm (Fig. 6c). Finally, it is over heavy precipitation areas that the largest improvements are observed (Fig. 7). Improvements in terms of correlation by Nh are concentrated again under 600 hPa but reach as much as 0.1 around 850 hPa (Fig. 7a), while the improvement in the amplitude of δTb is relatively similar to that observed in the other precipitation areas (Fig. 7b). Due mainly to the large changes in the correlation, Nh significantly reduces δTu in the lower troposphere (Fig. 7c).

At first glance, it might seem counterintuitive to find that the diabatic balance operators explained more temperature difference than the standard balance operator, whereas it was shown in section 4 that the perturbation flow is farther away from linear balance in precipitation regions. It is important to recall that one major role of N is to modify the linear balance imposed by H [see Eq. (4)] toward an observed linear balance. Therefore, with the current balance representation, less geostrophic coupling does not necessary imply less mass–rotational wind coupling. The ability of the diabatic balance operators to explain more temperature differences than the standard balance operator is simply an indication that the mass–rotational wind coupling in precipitation regions can be improved by using a different linear relationship, which is farther away from linear balance in these areas.

2) Explained surface pressure

For the explained surface pressure, the diabatic balance operators also lead, on average, to modest but significant improvements. In this case the improvement is essentially due to a better representation of the amplitude of δps,b by the diabatic balance operators and very little from changes in the pattern of δps,b (Fig. 8). Indeed, each diabatic balance operator improves only marginally the correlation between δps,b and δps compared to Ns over each precipitation area (Fig. 8a). We remark that, in terms of δps,b, Ns produces correlation with δps much higher that what is observed in terms of temperature (cf. Figs. 8a, 5a, 6a, and 7a). Therefore less room is available for improvement in the surface pressure components over precipitation areas. However, the improvements in the amplitude of δps,b by the diabatic balance operators are considerable: each diabatic balance operator reduces the departure by at least ∼50% in terms of amplitude between δps,b computed with Ns and δps (Fig. 8b). These changes lead to a small reduction of the amplitude of unbalanced surface pressure (δps,u, i.e., δps minus δps,b; Fig. 8c).

3) A test on the impact of reducing the number of grid points used in the experiment

The fact that the number of grid points used in the linear regression for the diabatic balance operators is considerably lower than for the standard balance operator and the fact that the previous comparison was made on the same small number of data used to construct these operators raise the following question: Are the improvements by the diabatic balance operators observed here due to 1) a particular relationship between mass and rotational wind over precipitation areas that is better represented by the new operators or 2) the fact that the number of grid points used to construct these new operators is relatively small? To answer this question, we have constructed three new classes of data by choosing grid points randomly4 in the ensemble of lagged forecast differences. We have ensured that each of these three “random” classes includes both dry and precipitation (of various intensity) points. The only similarity with the three precipitation classes is that the three random classes were constructed using respectively 9.4%, 4.0%, and 1.3% of the total number of grid points in the ensemble of lagged forecast differences, which is the exact same amount of grid points included in the three precipitation classes (see Table 2). New balance operators (N) were constructed for these random classes of data and the same comparison shown in the previous subsections was performed. If reducing the size of the data impacted the diabatic operators, then we should observe improvement when using the new random operators in the random areas that is inversely proportional to the number of grid points contained in each class. However, no significant change was observed (not shown) for each random class, which suggests that the improvements by the diabatic balance operators observed previously are actually due to a better representation of a particular coupling between mass and rotational wind over precipitation areas.

b. Single observation experiments

A series of simplified data assimilation experiments were performed to shed some light on the degree of modification brought by the diabatic balance operators in terms of analysis increments. Our point here is to investigate only the sensitivity of the analysis increments to the change of the balance operator, not to judge of the accuracy of the resulting analysis increment. Therefore, despite changing the balance operators in our assimilation experiments, the same background error statistics derived using Ns and computed over the entire domain were used in every experiment. The change in the background error correlations between mass and rotational wind over precipitation regions in the following assimilation experiments will thus only be partially measured. When testing the diabatic balance operator approach in a full-fledged Var system, a new set of background error statistics fully coherent with the new balance operators and with a partitioned domain will need to be developed. However, this is outside the scope of this paper.

The experiment consists of using each balance operator to assimilate single observations of zonal wind or temperature placed successively at different heights (250, 500, and 850 hPa). The limited-area version of EC’s 3D-Var analysis system is used for this purpose in the grid configuration shown in Fig. 1 and described in Table 1. This Var system uses a bi-Fourier spectral representation with nonseparable background error correlations. The analysis control variables are as shown in section 2a with the exception that the coupling between balanced variables and divergent wind is neglected here (i.e., all of the divergent wind analysis increments are considered as unbalanced). Therefore, wind (mass) observations can only impact mass (wind) analysis increments through the operator N in (4). The innovation (d) is set to 1 m s−1 (1 K) and an observational error (ε) of 1 m s−1 (1 K) is imposed for the zonal wind (temperature) observation experiments. The background error statistics are derived from the ensemble of lagged forecast differences presented in section 3.

1) Wind observation

Figure 9 presents the mass and wind analysis increments resulting from the assimilation of a zonal wind observation placed at the center of the domain (i.e., in the southwestern part of Manitoba) at 500 hPa and using Ns. The horizontal wind analysis increment distribution at the closest model level to 500 hPa (Fig. 9a) follows a typical nondivergent wind isotropic correlation model (e.g., see Fig. 5.4 in Daley 1991): the wind analysis increment is maximum at the location of the observation and a cyclonic (anticyclonic) circulation is located north (south) of the observation. A north–south vertical cross section shows that the region of cyclonic and anticyclonic circulation extends in the vertical with opposite sign temperature analysis increments located below and above 400 hPa and north and south of the observation (Fig. 9b). These temperature and streamfunction distributions agree with thermal wind balance and are an indication of the geostrophic nature of the coupling imposed by Ns.

To evaluate the modifications brought by the diabatic balance operators, we present a series of vertical profiles of temperature increments as well as surface pressure increments south of the zonal wind observations (see the long dashed line in Fig. 9b). When the assimilated zonal wind observation is placed at 250 hPa (Figs. 10a,b) or 500 hPa Figs. 10c,d) the differences in both the temperature and surface pressure fields are generally proportional to the intensity of the precipitation but remain relatively small. However, with a zonal wind observation placed at 850 hPa, the differences become important in the lower troposphere (Figs. 10e,f). A significant shift toward negative increment values, proportional to the precipitation intensity, is made by the diabatic operators below 500 hPa (Fig. 10e). The small positive temperature increment below 850 hPa obtained with Ns is even replaced by relatively large negative values with the use of Nm and Nh. In terms of surface pressure increments, each diabatic balance operator generates a larger positive increment than Ns and the differences reach a maximum of ∼30% with Nh (Fig. 10f).

In terms of coherent analysis increments required over regions where moist physics is important, the results shown here strongly suggest a significant impact on the lower atmosphere vertical temperature stratification. This is expected to impact the moist physics in the resulting analysis. Before using this new technique in an assimilation system based on lagged forecast error samples, a careful examination should be made to ensure that the mean changes observed here in the temperature stratification are systematic in the day to day background errors.

2) Temperature observation

We repeat the single observation experiments this time using a temperature observation. Figure 11 presents the mass and wind analysis increments resulting from the assimilation of the temperature observation when placed at 500 hPa and using Ns. The temperature analysis increment is at a maximum at the location of the observation and decreases in the horizontal following the combined balanced plus unbalanced horizontal background error correlation specified in the analysis system (Fig. 11a). A north–south vertical cross section shows that above the region of positive temperature increments, maximum at 500 hPa, is a region of negative temperature increments centered at 200 hPa (Fig. 11b). The existence of this area of negative temperature increments is due to the vertical correlations in the background error statistics and represents the typical cold (warm) anomaly at the tropopause level associated with a warm (cold) anomaly in the troposphere. The rotational wind increments generated by Ns shows a strong anticyclonic circulation centered in the upper troposphere between the positive and the negative temperature increments and a weak cyclonic circulation in the lower troposphere under the positive temperature increment (Fig. 11b). These temperature and wind distributions imposed by Ns are again coherent with geostrophic balance.

To compare the behavior of the various balance operators, vertical profiles of vorticity increments as well as surface pressure increment values were examined at the location of the observation (see the long dashed line in Fig. 11b). With the temperature observations placed at 250 hPa, only Nh made a modest change in the vorticity profile by reducing the amplitude of the positive increments in the upper levels (Fig. 12a). In terms of surface pressure increments, no significant changes are observed (Fig. 12b). While the vorticity increments from the diabatic balance operators show only small changes when the observation is lowered to 500 hPa (Fig. 12c), the amplitude of the surface pressure increments are considerably increased by all diabatic balance operators and the differences reach as high as ∼100% when Nh is used instead of Ns. Finally, with a temperature observation at 850 hPa, the surface pressure increments from the diabatic balance operators shows changes that are as large as those observed with the temperature observation at 500 hPa (Fig. 12f). In terms of vorticity increments, the diabatic balance operators produce a small but significant shift, proportional to the precipitation intensity, toward positive vorticity values below 600 hPa (Fig. 12e).

The experiments conducted here with zonal wind and temperature observations showed that the diabatic balances can produce analysis increments significantly different than when using the standard balance operator especially when the observations are located in the low levels.

6. Summary and discussion

Improving the balance between mass and wind analysis increments imposed by data assimilation schemes over precipitation areas is important in general and a key ingredient in the advancement of mesoscale forecasting. A standard method for the coupling of mass and rotational wind increments in variational analysis systems is to construct an operator using linear regression between the mass field and the rotational wind components of the flow from an ensemble of estimated forecast errors. Traditionally, when constructing the mass–rotational wind operator, all points (from both dry and precipitation areas) of the ensemble of estimated forecast errors are treated in the same way, thus preventing the ability to represent the particular balance over precipitation areas.

Using an ensemble of 48–24-h lagged forecasts (NMC method) to generate estimates of forecast errors, it was shown that indeed the rotational balance characteristics differs significantly between dry and precipitation areas. The lagged forecast differences are, on average, further away from geostrophic balance over precipitation areas than in dry areas and the deviation from the geostrophic balance is proportional to the intensity of precipitation. We take these results as a strong motivation to improve the current mass–rotational wind coupling in Var systems.

We investigated if some improvements in the coupling between mass and rotational wind increments over precipitation areas can be achieved by using only the precipitation points within the ensemble of lagged forecast differences to construct diabatic operators by linear regression. The ability of the standard and the diabatic balance operators to explain the mass field from information on the wind field was compared in the precipitation areas of the ensemble of lagged forecast differences. The comparison revealed that the diabatic balance operators can lead to significant improvements over the standard balance operator in the lower troposphere. The improvements are generally proportional to the intensity of precipitation. Differences in the behavior of the standard and the diabatic balance operators were then exemplified in a series of simplified single observation experiments. Results from the assimilation of a zonal-wind or a temperature observation placed at various heights, revealed that the diabatic balance operators can produce analysis increments substantially different from the standard balance operator. In general, the differences tend to increase when the observation height is decreased and they are proportional to the intensity of the precipitation prevailing in the data used to construct the diabatic balance operator. We take these results as a good incentive to pursue the evaluation of the relevance and feasibility of this new approach.

The next step in the evaluation of the diabatic balance operator approach is its implementation in a full-fledged analysis system. However, before proceeding to the implementation of the diabatic balance operator approach, several issues will need to be addressed from in-depth investigations. For instance, background error statistics derived from the same area as the diabatic balance operators were derived from will need to be developed. While the computation of standard deviations for precipitation regions is relatively easy, computing the corresponding vertical and horizontal error correlations in spectral space is a great challenge since we must deal with irregular domains. Also, a method is needed to determine the horizontal variation of the type of diabatic balance operator (dry, light, moderate, or heavy) to use during the minimization. Another issue is the risk of generating horizontal discontinuities in the analysis increments with a multiple operator approach. These investigations are left for a future study.

Aside from the issues related to the implementation of this new approach, the results presented here raise many other opportunities for further investigations:

  • It would be advisable to do a similar investigation with another dataset of lagged forecast differences (e.g., data from a summertime convective regime and with a finer horizontal resolution).
  • With the growing interest in using ensemble-based methods to replace the NMC method to generate forecast error statistics in Var systems (e.g., Buehner 2007; Berre et al. 2007), it would be relevant to investigate if balance properties of perturbations from, for example, the ensemble Kalman filter technique (e.g., Houtekamer et al. 2009) show similar differences between dry and precipitation areas. Finding similar differences would represent a further incentive to use such flow-dependent forecast error estimates in Var systems.
  • This study and the current formulation of EC’s limited-area Var system does not take into account the correlation between mass and divergence as in Derber and Bouttier (1999). It would be of interest to investigate in a future study the mass and divergence balance in dry and precipitation areas and if further improvement of the mass–wind coupling in precipitation regions could be obtained by including a mass–divergence operator defined for precipitation regions only. This would complement the approach suggested by Fillion et al. (2007).
  • The precipitation rate was adopted here as the criteria to discriminate the different balance properties over precipitation areas. It would be desirable to investigate if other dynamical variables (e.g., Rossby number) could be used as a complement in order to better discriminate the different balance particularities over precipitation areas.
  • Since balance depends on scale, it would also be relevant to investigate to what extent vertical and horizontal scales differ between dry and precipitation regions in forecast error estimates.
  • Finally, it would be interesting to compare the diabatic balance operator approach with the recently proposed tangent linear normal mode technique (Fillion et al. 2007; Kleist et al. 2009) to improve the representation of balance in Var systems.

Acknowledgments

The authors thank Dr. Mark Buehner for an internal review of this paper for the EC Meteorological Research Division.

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APPENDIX

Constructing N

As a first step, the forecast error estimates of streamfunction are transformed into a mass variable using linear balance:
i1520-0493-138-2-563-ea1
Then a linear regression between δ𝗣 and both the forecast error estimates of temperature and of surface pressure (δ𝗧, δ𝗽s) is sought. For instance, for temperature we pose
i1520-0493-138-2-563-ea2
which is equivalent to
i1520-0493-138-2-563-ea3
Isolating N leads to
i1520-0493-138-2-563-ea4
The same approach is used for surface pressure.

Fig. 1.
Fig. 1.

Domain used for the estimation of the “NMC lagged forecast differences” data using latitude–longitude projection with numerical poles shifted from the geographical poles. The resolution is 0.49° (∼55 km) and the domain extent is roughly 8000 km by 8000 km. Topography (shades) is shown for visualization purposes only.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 2.
Fig. 2.

Number of times that each surface grid point in the 100 members of lagged forecast are (a) precipitating, (b) in the light precipitation class, (c) in the moderate precipitation class, and (d) in the heavy precipitation class. Note the different scale in (d).

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 3.
Fig. 3.

Vertical profile of average (over the 100 members) statistical measure of the degree of linear balance in the lagged forecast differences computed over dry areas (boldface, ○) and over light (□), moderate (⋄), and heavy (▵) precipitation areas. (a) Vertical profiles of average horizontal correlation between mass and wind component and (b) ratio of average horizontal rms value of mass and wind component (wind rms divided by mass rms). The gray dotted line in (b) indicates a ratio of 1.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 4.
Fig. 4.

Normalized deviation from linear balance for dry areas and for light, moderate, and heavy precipitation areas in the ensemble of lagged forecast differences. See text for the details on the computation of this measure.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 5.
Fig. 5.

Vertical profile of average (over the 100 members) statistics of explained temperature by Ns and Nl over light precipitation areas in the ensemble of lagged forecast differences. (a) δTbδT correlation for δTb computed with Ns (boldface, ○) and for δTb computed with Nl (□). (b) The rms of δTb computed with Ns (boldface, ○), of δTb computed with Nl (□) and of δT (dotted, ⋄). (c) The rms of δTu when δTb is computed with Ns (boldface, ○) and when δTb is computed with Nl (□). The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for Ns and Nm over moderate precipitation areas.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 7.
Fig. 7.

As in Fig. 5, but for Ns and Nh over heavy precipitation areas.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 8.
Fig. 8.

Average (over the 100 members) statistics of explained surface pressure by Ns and the diabatic operator associated to each precipitation areas in the ensemble of lagged forecast differences. (a) δps,bδps correlation for δps,b computed with Ns (black bars) and for δps,b computed with the corresponding diabatic operator (gray bars). (b) The rms of δps,b computed with Ns (black bars), of δps,b computed with the corresponding diabatic operator (gray bars) and of δps (dotted bars). (c) The rms of δps,u when δps,b is computed with Ns (black bars) and when δps,b is computed with the corresponding diabatic operator (gray bars).

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 9.
Fig. 9.

Wind and mass analysis increments in response to the assimilation of a single observation of zonal wind at 500 hPa using Ns. (left) Wind analysis increments (arrows, kt) at the closest model level from 500 hPa. (right) Vertical cross section of temperature (thin lines, contour interval: 0.02 K) and streamfunction (thick lines, contour interval: 0.1 m2 s−1) analysis increments. The points N and S show the limits of the cross section. Positive (negative) values are in solid (dashed) lines and zero contours are omitted. The long-dashed line shows the location of the vertical profiles in Fig. 10. The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 10.
Fig. 10.

Analysis increments in response to the assimilation of a single observation of zonal wind at (a),(b) 250, (c),(d) 500, and (e),(f) 850 hPa using standard and diabatic balance operators. (left) Vertical profile of temperature analysis increments using Ns (bold, ○), Nl (□), Nm (⋄), and Nh (▵). The gray doted line indicates a value of zero. The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa. (right) Surface pressure analysis increments at the location of the profile in the left panel using Ns (std), Nl (light), Nm (mod), and Nh (heavy).

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 11.
Fig. 11.

Wind and mass analysis increments in response to the assimilation of a single observation of temperature at 500 hPa using Ns. (a) Temperature analysis increments (contour interval: 0.2 K) at the closest model level from 500 hPa. (b) Vertical cross section of vorticity (thin lines, contour interval: 0.1 × 10−5 s−1) and temperature (thick lines, contour interval: 0.2 K) analysis increments. The point N and S show the limits of the cross section. Positive (negative) values are in solid (dashed) lines and zero contours are omitted. The long dashed line shows the location of the vertical profiles in Fig. 12. The vertical axis shows pressure values for model vertical levels estimated using a surface pressure value of 1000 hPa.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Fig. 12.
Fig. 12.

As in Fig. 10, but in response to the assimilation of a single observation of temperature with (left) vertical profile of vorticity analysis increments.

Citation: Monthly Weather Review 138, 2; 10.1175/2009MWR2998.1

Table 1.

Characteristics of the NMC method based on forecast differences data.

Table 1.
Table 2.

Precipitation class and its distribution in the ensemble of lagged forecasts. The total number of points is equal to the number of model data grid points (158 × 158 × 58) times the ensemble size (100).

Table 2.

1

Although the term involving the latitudinal variation of f is neglected in our linear balance definition, f is allowed to vary.

2

These levels are 1000, 975, 950, 925, 900, 850, 800, 750, 700, 650, 600, 550, 500, 450, 400, 350, 300, 250, 200, 150, 100, 70, 50, 30, 20, and 10 hPa.

3

In the computation of the statistics, data on pressure levels below the surface were excluded.

4

In short, specific rows of data, arbitrarily chosen, were selected in each sample of lagged forecast differences.

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