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    The sensitivity of the time mean RMSE (106 m2 s−1) of the atmospheric streamfunction with respect to the GC localization half-width (km), where the solid circles and dotted triangles represent the results of the standard EnKF and EnKF–MSA algorithms. Here, seven half-widths including 125, 250, 500, 1000, 1500, 2000, and 2500 km are examined.

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    A snapshot of truth minus analysis (posterior) field (106 m2 s−1) of the atmospheric streamfunction for the GC half-width values as (a) 125, (b) 500, (c) 1000, and (d) 2000 km in the standard EnKF, where the standard deviation of observation error is 106 m2 s−1. Note that the solid and dashed curves represent the −106 and 106 contours, respectively.

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    The flowchart of the EnKF–MSA hybrid method for a data assimilation cycle, where N is the ensemble size. The steps from 1 to 5 represent the sequential procedures as follows: step 1 adjusts the ensemble member using the observation with the standard EnKF, step 2 subtracts the interpolated ensemble mean produced by EnKF from the initial observation to generate the observational residual, step 3 retrieves the multiscale information from observational residual with the MSA method, step 4 adds the analysis field generated by MSA to the ensemble mean produced by EnKF to obtain the new ensemble mean, and step 5 adds the new ensemble mean to the ensemble perturbations to generate the final ensemble members.

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    Model grids (plus sign) and observational locations (solid circle) in the twin experiment. In the Northern Hemisphere, observations on all model grids are assumed to be available. In the Southern Hemisphere, observations on all odd x- and y-index grids are available.

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    Time series of RMSE (106 m2 s−1) of the atmospheric streamfunction for the GC half-width values as 125 (red curve), 500 (black curve), 2000 (blue curve), and 2500 km (green curve) in the EnKF–MSA hybrid method.

  • View in gallery

    The spatial RMSEs (106 m2 s−1) of the atmospheric streamfunction for the GC half-width values as (a),(b) 250; (c),(d) 1000; and (e),(f) 2500 km in (right) the standard EnKF and (left) the hybrid method. Note that (a),(b); (c),(d); and (e),(f) employ the same shade scales, respectively.

  • View in gallery

    Time series of RMSEhybrid minus RMSEEnKF in hybrid for a 2500-km GC half-width in the EnKF of the hybrid method, where the dashed line represents no difference. RMSEhybrid and RMSEEnKF in hybrid represent the RMSEs of the atmospheric streamfunction for the hybrid method and the EnKF in the first step of the hybrid method, respectively.

  • View in gallery

    (a) The sensitivity of the RMSE (106 m2 s−1) of the atmospheric streamfunction with respect to the number of scale levels of MSA in the hybrid method. (b) Time series of the RMSE of the atmospheric streamfunction for 0 (red curve), 1 (black curve), and 3 (blue curve) scale levels in the hybrid method. Note that the hybrid method reduces to the standard EnKF when the number of scale levels is 0. The 1 and 3 scale levels correspond to 250-km and 250-, 500-, and 1000-km levels in the MSA, respectively.

  • View in gallery

    The time series of the RMSE (106 m2 s−1) of the atmospheric streamfunction for (a) 250- and (b) 2500-km GC half-width values, where the black, blue, and red curves represent the results of the standard EnKF, the EnKF–MSA hybrid method, and the hyrbid method without smoothing term (i.e., EnKF–SCM method) in the cost function of MSA, respectively. Here the green curve stands for the results of a redesigned hybrid method that changes two scale levels (2500 and 4000 km) to five levels (250, 500, 1000, 2000, and 4000 km) in the 2500-km a-value experiment. The red curve in (b) stands for the results of the redesigned hybrid method when the smoothing term is dropped in the cost function of MSA.

  • View in gallery

    The Southern Hemisphere results (106 m2 s−1) of the hybrid method at the first data assimilation cycle (that is the 0600 on the 20th day) for a 250-km a value. (a) The true difference defined in the text. (b) Observational residuals. (c) Analysis result of the MSA. (d) The relative difference defined in the text. Note that (a)–(c) adopt the upper shade scale, while (d) employs the lower shade scale. Additionally, the black curve in (d) indicates the zero contour.

  • View in gallery

    Results (106 m2 s−1) of the MSA in the hybrid method at the first data assimilation cycle (i.e., the 0600 on the 20th day) for a 250-km a value. (a) Result of the first level. (b) Result of the sum of first two levels. (c) Result of the sum of first three levels. (d) Result of the sum of all five scale levels. Note that all panels adopt the same shade scale.

  • View in gallery

    As in Fig. 10, but for the 2500-km a value and the following five scale levels of MSA in the hybrid method: 250, 500, 1000, 2000, and 4000 km.

  • View in gallery

    The time series of the RMSE (106 m2 s−1) of the atmospheric streamfunction for (a) 125- and (b) 2500-km a values, where the black and blue curves represent the results of the EnKF–MSA hybrid method with and without MSA applied to the NH observations. Here the experiment with 2500-km a value corresponds to the redesigned experiment in section 4d.

  • View in gallery

    The spatial RMSEs (106 m2 s−1) of the atmospheric streamfunction for (a),(b) 125- and (c),(d) 2500-km a values in the hybrid method with MSA applied to (a),(c) SH observations and (b),(d) global observations. Note that (a),(b) and (c),(d) employ the same shade scales, respectively. Here the experiment with 2500-km a value corresponds to the redesigned experiment in section 4d.

  • View in gallery

    The variation of normalized values of cost function of MSA in the hybrid method with the 500-km a value for the first (solid dot), second (hollow dot), third (solid diamond), and fourth (hollow diamond) levels. The x axis is the iterate step. Here the normalized factor for each scale level is the value of cost function at the first iterate step.

  • View in gallery

    Variations of (a),(c) ACC and (b),(d) the normalized RMSE with the lead time of the forecasted ensemble means of the atmospheric streamfunction started from the analysis fields of (a),(b) 125- and (c),(d) 2500-km a values for the hybrid method (blue curve) and the standard EnKF (black curve). Note that the dashed line stands for the 0.6 ACC value.

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A Compensatory Approach of the Fixed Localization in EnKF

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  • 1 Key Laboratory of Marine Environmental Information Technology, State Oceanic Administration, National Marine Data and Information Service, Tianjin, China
  • | 2 NOAA/Geophysical Fluid Dynamics Laboratory, Princeton University, Princeton, New Jersey
  • | 3 Key Laboratory of Marine Environmental Information Technology, State Oceanic Administration, National Marine Data and Information Service, Tianjin, China
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Abstract

While fixed covariance localization can greatly increase the reliability of the background error covariance in filtering by suppressing the long-distance spurious correlations evaluated by a finite ensemble, it may degrade the assimilation quality in an ensemble Kalman filter (EnKF) as a result of restricted longwave information. Tuning an optimal cutoff distance is usually very expensive and time consuming, especially for a general circulation model (GCM). Here the authors present an approach to compensate the demerit in fixed localization. At each analysis step, after the standard EnKF is done, a multiple-scale analysis technique is used to extract longwave information from the observational residual (referred to the EnKF ensemble mean). Within a biased twin-experiment framework consisting of a global barotropical spectral model and an idealized observing system, the performance of the new method is examined. Compared to a standard EnKF, the hybrid method is superior when an overly small/large cutoff distance is used, and it has less dependence on cutoff distance. The new scheme is also able to improve short-term weather forecasts, especially when an overly large cutoff distance is used. Sensitivity studies show that caution should be taken when the new scheme is applied to a dense observing system with an overly small cutoff distance in filtering. In addition, the new scheme has a nearly equivalent computational cost to the standard EnKF; thus, it is particularly suitable for GCM applications.

Corresponding author address: Xinrong Wu, 93 Liuwei Rd., Hedong District, Tianjin, 300171, China. E-mail: xinrong_wu@yahoo.com

Abstract

While fixed covariance localization can greatly increase the reliability of the background error covariance in filtering by suppressing the long-distance spurious correlations evaluated by a finite ensemble, it may degrade the assimilation quality in an ensemble Kalman filter (EnKF) as a result of restricted longwave information. Tuning an optimal cutoff distance is usually very expensive and time consuming, especially for a general circulation model (GCM). Here the authors present an approach to compensate the demerit in fixed localization. At each analysis step, after the standard EnKF is done, a multiple-scale analysis technique is used to extract longwave information from the observational residual (referred to the EnKF ensemble mean). Within a biased twin-experiment framework consisting of a global barotropical spectral model and an idealized observing system, the performance of the new method is examined. Compared to a standard EnKF, the hybrid method is superior when an overly small/large cutoff distance is used, and it has less dependence on cutoff distance. The new scheme is also able to improve short-term weather forecasts, especially when an overly large cutoff distance is used. Sensitivity studies show that caution should be taken when the new scheme is applied to a dense observing system with an overly small cutoff distance in filtering. In addition, the new scheme has a nearly equivalent computational cost to the standard EnKF; thus, it is particularly suitable for GCM applications.

Corresponding author address: Xinrong Wu, 93 Liuwei Rd., Hedong District, Tianjin, 300171, China. E-mail: xinrong_wu@yahoo.com

1. Introduction

Because of its easy implementation, especially in a massively parallelized computing environment, ensemble Kalman filters (EnKFs; Evensen 1994, 2007; Hamill and Snyder 2000; Anderson and Anderson 1999) are becoming operational data assimilation methods in the weather and climate community. Another advantage of EnKFs versus variational analysis methods is the flow-dependent background error covariance that is evaluated by a model ensemble. However, because of the sampling error from a finite ensemble size, the ensemble-evaluated background variance is usually underestimated, and spurious correlations exist between a state variable and remote observations. Various static additive (e.g., F. Zhang et al. 2004; Whitaker et al. 2008; Houtekamer et al. 2009), multiplicative (e.g., Anderson and Anderson 1999) variance inflation schemes, and adaptive inflation methods (e.g., Anderson 2007b; Anderson 2009; Li et al. 2009; Miyoshi 2011) have been developed to address the first issue. To remove the long-distance spurious correlations and increase the reliability of ensemble-evaluated background covariance, the localization technique was introduced into ensemble-based filters (Houtekamer and Mitchell 1998). Note that although localization can also be applied to the background error covariance in the observation space (e.g., Houtekamer and Mitchell 1998; Ott et al. 2004; Hunt et al. 2007; Greybush et al. 2011), we focus on background error covariance localization in the state space in this study.

Originally, Houtekamer and Mitchell (1998) investigated the impact of the accuracy of background error covariance on the analysis quality, which brought about the fixed (parametric or static) localization methods (e.g., Hamill et al. 2001; Houtekamer and Mitchell 2001; Anderson 2001; Szunyogh et al. 2005, 2008). The fixed localization is usually realized by a Schur product (an element-by-element multiplication) of the ensemble-estimated covariance with an analytic localization operator. A widely used fixed localization function is the compactly supported fifth-order polynomial approximation (Gaspari and Cohn 1999, hereafter the GC function) of a normal probability distribution. Other parametric models include the exponential function, the Matérn function, and so on. Zhang et al. (2009) adopted several different localization distances to account for different physical scales. Zhu et al. (2011) sampled a fixed localization function by a set of local correlation function ensemble members so that the filter can assimilate nonlocal observations. All fixed covariance models need to determine a cutoff distance (impact radius) that defines the maximum impact range of observations. The impact radius significantly influences the analysis quality in EnKF, and the optimal cutoff radius is related to ensemble size as well as the properties of observing system and numerical model (Houtekamer and Mitchell 1998; Mitchell et al. 2002). However, it is expensive to tune the optimal cutoff distance given a specific ensemble size, model, and observing system. Therefore, many efforts have been made toward the nonparametric (adaptive) localization algorithms. Anderson (2007a) used a hierarchical filter to estimate the localization function using a group of ensembles. Bishop and Hodyss (2007) proposed the flow-dependent moderation (localization) functions that are built from powers of smoothed ensemble correlations. Then the moderation function is advanced to the ensemble correlations raised to a power (ECO-RAP method; Bishop and Hodyss 2009a,b) that propagates and adjusts the width of the localization function by computing powers of raw ensemble correlations. Bishop and Hodyss (2011) extended the ECO-RAP method to a global ensemble four-dimensional variational data assimilation scheme and demonstrated that the covariance function can adapt to anisotropic aspects of the flow. As a follow-up study, Bishop et al. (2011) proposed a computationally efficient algorithm for incorporating fixed localized ensemble covariance into variational data assimilation schemes. Jun et al. (2011) presented a kernel smoothing method with variable bandwidth to adaptively localize the covariance, and their results demonstrate that the nonparametric method provides a more accurate estimate of background covariance than the GC function. Anderson and Lei (2013) developed an empirical localization technique that computes localization from an observing system simulation experiment. Results in a low-order model show that the proposed method produces lower root-mean-square errors in most cases compared to assimilations using tuned localizations. Recently, Lei and Anderson (2014) investigated this localization algorithm in the Community Atmosphere Model, version 5, and obtained promising results. Although the adaptive models are promising, a certain limitation of the adaptive methods is the high computational cost. Besides the nonadaptive and adaptive correlation models, wavelet approaches (e.g., Deckmyn and Berre 2005; Pannekoucke et al. 2007), recursive filters (e.g., Wu et al. 2002; Purser et al. 2003), diffusion-based models (e.g., Weaver and Courtier 2001; Pannekoucke and Massart 2008; Weaver and Mirouze 2013; Yaremchuk and Nechaev 2013), and the wavelet and diffusion hybrid method (Pannekoucke 2009) are also developed to localize the covariances. These methods are also computationally complex. A more detailed review of localization methods is documented by Berre and Desroziers (2010). Note that most of the current localization techniques have pros and cons. For the parametric methods, one advantage is the low computational cost while one deficiency is the strong dependence on the impact radius. In this study, we present an approach to compensate for the following two disadvantages of the fixed localization models if the cutoff distance is not optimal: loss of longwave observational information due to a small cutoff distance, and contamination of the analysis model states by noises caused by the long-distance spurious correlation if a large cutoff distance is used. The compensatory approach uses a multiple-scale analysis (MSA) technique to retrieve multiple-scale information from the observational residuals (the differences between observations and the interpolated analysis ensemble means produced by EnKF) and adds the analysis fields to the ensemble mean of EnKF. The hybrid method proposed in this study is inspired from the previous studies that deal with the different spatial scales in filtering. Hamill and Snyder (2000), Lorenc (2003), as well as Rainwater and Hunt (2013) combined static (low-resolution ensemble evaluated) background error covariance and (high resolution) ensemble-evaluated background error covariance for the variational (ensemble filter) method. Buehner (2012) put forward a spatial/spectral localization approach that separately accounts for different-scale error covariances through a bandpass filter. Results of their data assimilation experiment justify that this method can reduce the error in spatial correlation estimates. Motivated by the work of Buehner (2012), Miyoshi and Kondo (2013) proposed a dual-localization method. Results of a perfect twin experiment show great advantage of their method over the single localization method. Afterward, Kondo et al. (2013) investigated the sensitivities of the parameters in the dual-localization approach, including the smoothing function and two localization scales. While these multiscale localization methods attempt to use ensemble-based flow-dependent covariance in longer-range covariances, the proposed EnKF–MSA hybrid method addresses the issue of losing longwave information (contaminating the analysis) caused by overly small (overly large) impact radiuses. With a global barotropical spectral model and a biased twin experiment as well as an idealized observing system, the performance of the proposed algorithm is deeply investigated.

After the introduction, section 2 briefly describes the global barotropical spectral model, an EnKF algorithm [ensemble adjustment Kalman filter (EAKF); Anderson (2003)], the idea of MSA, as well as the implementation of the hybrid method. Section 3 introduces a biased twin experiment. Section 4 thoroughly investigates the performance of the proposed method. Impact of the compensatory scheme on the weather forecast is presented in section 5 while a summary and a general discussion are given in section 6.

2. Methodology

a. The model

To clearly address the issue raised in the introduction, we employ a global barotropical spectral model based on the equation of potential vorticity conservation (Haltiner and Williams 1980):
e1
where and f represent the relative vorticity and planetary vorticity, respectively (i.e., Coriolis parameter); and H is the depth of the atmospheric layer. After introducing the geostrophic streamfunction , the relative vorticity equals to , where is the second-order Laplacian. With the assumption of β plane (i.e., f = f0 + βy), the absolute vorticity (i.e., ) can be approximated by the quasigeostrophic potential vorticity:
e2
where represents the effect of topography; , y represents the northward meridional distance from equator; and indicates the Cressman parameter. Subsequently, Eq. (1) can be rewritten as
e3
where denotes the Jacobian operator. Equation (3) can be further expanded as
e4

A rhomboidal 21 truncation is applied for the transformation between spectral coefficients and grid values. The state variables are spectral coefficients [the atmospheric streamfunction at the 64 (longitude) × 54 (latitude) Gaussian grid points] for the time stepping (the data assimilation). The integration step size is a half-hour. A leapfrog time step is used to integrate the model and a Robert–Asselin time filter (Robert 1969; Asselin 1972) is applied to damp the spurious computational modes.

b. The EAKF algorithm in Anderson (2001)

In the standard Kalman filter (Jazwinski 1970), the analysis solution can be explicitly derived as
e5
where the Kalman-gain matrix is formulated as
e6
Here, xb represents the background of state vector x with the dimension of M × 1; is the observation operator, which is usually nonlinear; yo is the observation vector with the dimension of K × 1; is the linearized matrix of ; and and b are the observation error covariance matrix and the background error covariance matrix, respectively. Table 1 lists several notations used in this study.
Table 1.

Glossary of notations in this study.

Table 1.
For an EnKF algorithm, Eq. (5) is used to update the ensemble mean of state variable as follows:
e7
where and are the analysis and background of ensemble mean of state vector. After the ensemble mean is updated by Eq. (7), the ensemble perturbations are adjusted with a specific method depending on the version of EnKF. In the EnKF context, the background error covariance matrix b is estimated by N forecasted dynamical ensemble members:
e8
Here, is the nth ensemble of background perturbation, which is defined as
e9
where is the nth realization of background field. As the ensemble size is usually much smaller than the model dimension for typical atmospheric and oceanic applications, may suffer from sampling error. Namely, a small ensemble size may significantly reduce the rank of and even change the positive definiteness of (e.g., Greybush et al. 2011).
To resolve this issue, some EnKF experts introduce localization schemes into through a Schur product:
e10
where ρ is an M × M local support correlation matrix whose ith-row and jth-column element ρi,j represents the compactly supported correlation coefficient between the ith model grid and the jth model grid. Then, the Kalman-gain matrix can be estimated by
e11
Substituting Eq. (11) and the linearized observational operator into Eq. (7) gives the updated formulation of ensemble mean in EnKF frame as
e12
In this study, we use one of the standard EnKFs (i.e., the EAKF; Anderson 2001), to perform the data assimilation experiments. When observational error is assumed to be independent, EAKF can sequentially assimilate observations. For a single observation yo, the adjustments of ensemble mean and ensemble perturbation1 of yo are first computed by (see Anderson 2001, 2003):
e13
and
e14
respectively. The posterior and prior ensemble means of yo are denoted by and . Here R and denote the standard deviation of observational error and the prior standard deviation of yo. The ith prior ensemble of yo, , is usually obtained through applying a linear interpolation to the prior ensemble of state variable.
Since the same linear relationship2 exists between and and between and as follows:
e15
where and represent the adjustments of ensemble mean and the ith ensemble perturbation of the jth state variable xj, respectively. is the prior error covariance between xj and yo. Therefore, a quantity, namely, the increment, is defined to combine the adjustments of ensemble mean and ensemble perturbation (Anderson 2003):
e16
where and represent the observational increment of yo and the state increment of xj for the ith ensemble, respectively.
Thus, the implementation of EAKF can be compressed to the following two steps: first compute the observational increment using Eqs. (13), (14), and (16); and second use Eq. (15) to project the observational increment to state increment. During these two steps, the background error covariance between yo and xj only appears in the linear regression formula in Eq. (15). Therefore, when the covariance localization is introduced into EAKF, the local support correlation is imposed in the numerator of the coefficient of linear regression as
e17
where ρj,y represents the localization factor between yo and xj.
Although various fixed localization models exist, we focus on a widely used GC function (Gaspari and Cohn 1999) in this study, that is
e18
where b denotes the physical distance between yo and xj, a represents the half-width of the GC function (that is half of the impact radius).

c. Some limitations in the fixed covariance localization methods

For fixed cutoff distance covariance localization, on the one hand, tuning the optimal cutoff distance is time-consuming and expensive for a real atmospheric or oceanic model; on the other hand, the analysis quality of EnKF is sensitive to the impact radius. Here, we give an example for the application of the GC function in EnKF. Figure 1 displays the dependence of the time mean root-mean-square error (RMSE) of the ensemble mean on the parameter a.3 Note that the time mean ensemble mean RMSE in this study is defined as
e19
where S represents the number of analysis steps, s indexes the analysis step while RMSEs denotes the RMSE at the sth analysis step; im and jm are the dimensions of zonal and meridional model grids (that is 64 and 54), respectively. The ensemble mean of the atmospheric streamfunction are represented by . The superscripts “prior” and “t” denote the prior and the truth values, respectively. The optimal value of a is about 1500 km and the results of EnKF are sensitive to a. Figure 2 displays a snapshot of minus the posterior (denoted by ) for as of 125 (Fig. 2a), 500 (Fig. 2b), 1000 (Fig. 2c), and 2000 km (Fig. 2d). For a small a (like 125 km), many longwaves are lost, which is especially significant for the place where the observations are sparsely distributed (see solid circles in Fig. 4). For 50- and 1000-km values of a, some longwave signals are still lost by EnKF. When a exceeds a critical value, the localization cannot effectively suppress the long-distance spurious correlations which conversely contaminates the analysis solution of EnKF (Fig. 2d). Thus, for extreme a values (like 125 and 2000 km here), multiscale information, which is even stronger than the observational error (the solid and dashed curves in Fig. 2), is left in the truth residuals. Here, the truth residual is the difference between truth and the analysis of EnKF:
e20
with xt representing the truth of x. However, only the observational residual defined as
e21
is available in practice. For overly large and overly small a values, yres still contains some multiscale information of to a certain extent. Thus, it is promising to develop a compensatory approach to retrieve the multiple scale information from the observational residuals for extreme cutoff distances.
Fig. 1.
Fig. 1.

The sensitivity of the time mean RMSE (106 m2 s−1) of the atmospheric streamfunction with respect to the GC localization half-width (km), where the solid circles and dotted triangles represent the results of the standard EnKF and EnKF–MSA algorithms. Here, seven half-widths including 125, 250, 500, 1000, 1500, 2000, and 2500 km are examined.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

Fig. 2.
Fig. 2.

A snapshot of truth minus analysis (posterior) field (106 m2 s−1) of the atmospheric streamfunction for the GC half-width values as (a) 125, (b) 500, (c) 1000, and (d) 2000 km in the standard EnKF, where the standard deviation of observation error is 106 m2 s−1. Note that the solid and dashed curves represent the −106 and 106 contours, respectively.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

d. Multiple scale analysis

Based on the analyses above, in this section, we introduce an MSA method to address the described issue. The MSA approach is a variant of a multigrid method that was initially suggested for solving differential equations (Briggs et al. 2000) and later introduced into data assimilation community (e.g., Li et al. 2008, 2010; Xie et al. 2011).

Under the framework of three-dimensional variational analysis, MSA sequentially extracts observational signals from longwave to shortwave through refining the correlation scale from a large value to a small value. The cost function for the lth-scale level is formulated as
e22
where L is the number of the scale levels; δx(l), (l), (l), , and d(l) represent the increment of the state vector x, the linear projection operator from the observation space to the state space, the background error covariance matrix, the observational error covariance matrix in MSA, and the observational innovation vector for the lth-scale level, respectively. Note that the observation term in Eq. (22) is slightly different from that in the cost function of traditional variational algorithm that projects the model state to observation space. MSA here conversely maps the observation to state space through the operator . Thus, the observation error covariance matrix here is also different from the traditional one that is usually denoted as . If we set and MSA to the pseudoinverse of and T−1, respectively, the observation term in Eq. (22) will be equivalent to the classical one. The last term in Eq. (22) is the smoothing term where (l) is the smoothing matrix.4 The dimensions of the above five matrices are M × 1, M × K, M × M, M × M and K × 1, respectively. For each level, d(l) is defined as
e23
where (l) is the bilinear interpolation operator from the state space to the observation space for the lth-scale level, and is the background field of x in MSA; δxMSA(l–1)represents the analysis result of the (l − 1)th scale level [see the following Eq. (25)].
The gradient of the cost function J(l) with respect to the control vector δx(l) can be derived as
e24
Thus, the analytic solution of Eq. (22) is
e25
With Eqs. (22) and (24) as well as an optimization algorithm [like the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method, e.g., Liu and Nocedal 1989], the numerical approximation of Eq. (25) can be obtained.
Then the total adjustment (increment) of x produced by MSA is
e26
Note that each scale level of MSA also has a correlation scale (localization factor) that is implied in (l). To be consistent with the GC localization in EnKF, the element of (l), Lij(l) which denotes the weight of the jth observation on the ith state variable, also employs the GC function
e27
where the a(l) represents the GC localization half-width for the lth-scale level, and bij is the physical distance between the jth observation and the ith state variable. The denominator is a normalization factor.

e. An EnKF–MSA hybrid method

To break the limitations of the fixed covariance localization in EnKF described in section 2c, we present an EnKF–MSA hybrid method in this section. Figure 3 shows the flowchart of the hybrid method for a data assimilation cycle. The sequential implementation of the hybrid method is as follows:

  • Step 1: Adjust the ensemble members of state variable using the observation with the standard EnKF algorithm with a GC half-width.
  • Step 2: Project linearly the analysis ensemble mean produced by EnKF to the observation positions to get the EnKF-estimated posterior observation values. Then Eq. (21) is used to compute the observational residuals.
  • Step 3: Apply MSA to observational residuals to extract multiscale information from longwave to shortwave. Under this circumstance, the localization factor (here is the GC half-width) for the ith-scale level in MSA [i.e., a(i)] should decrease monotonously from a(1) to a(L) as i increases from 1 to L. In addition, to keep consecutive with the localization of EnKF, the GC half-width in MSA for the last scale level [i.e., a(L)], which has the smallest scale is set to a.
  • Step 4: Add the analysis field generated by MSA to the ensemble mean produced by EnKF to obtain the final ensemble mean.
  • Step 5: Add the new ensemble mean to the ensemble perturbations to generate the final ensemble members.
Fig. 3.
Fig. 3.

The flowchart of the EnKF–MSA hybrid method for a data assimilation cycle, where N is the ensemble size. The steps from 1 to 5 represent the sequential procedures as follows: step 1 adjusts the ensemble member using the observation with the standard EnKF, step 2 subtracts the interpolated ensemble mean produced by EnKF from the initial observation to generate the observational residual, step 3 retrieves the multiscale information from observational residual with the MSA method, step 4 adds the analysis field generated by MSA to the ensemble mean produced by EnKF to obtain the new ensemble mean, and step 5 adds the new ensemble mean to the ensemble perturbations to generate the final ensemble members.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

In addition to the five-step procedure, two more important points are worth being mentioned as follows. First, to inherit the advantage of the standard EnKFs over the variational methods (i.e., the flow-dependent background error covariance), the MSA only adjusts the ensemble mean. This also minimizes the computational cost without losing the advantage if all ensemble members are adjusted. According to Eq. (21), observations at each data assimilation cycle can be divided into two parts: and yres. While the standard EnKF can extract observational information implied in the first term with a fixed localization factor, it cannot deal with the second term that is the observational residual. For extreme values of impact factors, the observational residual may contain multiscale information (see Figs. 2a,d). Under this circumstance, the MSA in the hybrid method is used to retrieve the multiscale signals from observational residuals [Eq. (21)] to compensate the loss of longwave information (the contamination of the analysis) in the standard EnKF caused by an overly small (large) cutoff distance. Thus, in the hybrid method, the analysis solution of ensemble mean of model state is plus δxMSA [Eq. (26)], which is retrieved by MSA from yres. In other words, the observations are assimilated in two steps but the two analysis increments are completely compensatory. Second, since MSA is applied to the observational residuals, leading the prior information (e.g., the background error covariance) of truth residual [Eq. (20)] unknown, the background term in the cost function [Eq. (22)] of MSA is neglected. is simply set to the bilinear interpolation operator . For the first scale level, the observational innovation vector y(1) is actually the observational residual yres in Eq. (21). Thus, the background field of MSA is actually the analysis field of EnKF (i.e., ). The smoothing matrix is set to a static two-order smoothing operator . If the 1D vector δx(l) is reshaped by a 2D matrix (l) with (im, jm) dimension, the expression of the smoothing term in Eq. (22) can be formulated by
e28
where is the (i, j)th element of (l), and the four coefficients are
e29
where lon(i) and lat(j) represent the longitude and latitude of the (i, j)th model grid, respectively. It is easy to infer that the smoothing matrix is a quasi-diagonal matrix whose magnitude is O(), where is the identity matrix. With the absence of the background term in Eq. (22), MSA attempts to obtain the optimal solution of δx(l) through balancing the smoothing term and the observation term. Although the classical observation error covariance matrix is usually set to a constant multiplying the identity matrix, we set in MSA framework for simplicity. With the assumptions above, Eq. (22) can be simplified as
e30
Combining Eqs. (12), (25), and (26) gives the updated ensemble mean produced by the hybrid method:
e31
Note that with ignoring the background term, Eq. (30) is not the cost function in variational approaches anymore. If the smoothing term is further dropped, MSA will reduce to the successive correction method (denoted as SCM, e.g., Bratseth 1986) and the solution in Eq. (31) becomes the SCM starting from the EnKF analysis. According to the formula of the smoothing matrix [Eqs. (28) and (29)], it is easy to infer that the main role of is to remove the local small-scale noises (such as “bull’s-eye”) in the analysis of MSA. The net effect of introducing the smoothing term into SCM is to extract the strong signals from observational residuals. Thus, the MSA here is different from SCM. Because of the existence of the observational error in the observational residuals, MSA may benefit from the smoothing term while SCM attempts to approximate the observational residual as much as possible.

The final ensemble members can be obtained through adding the above ensemble mean to the ensemble perturbations updated by the standard EnKF.

3. Biased twin-experiment setup

A biased twin-experimental framework is designed to investigate the performances of two assimilation schemes. The sole source of model error is assumed to arise from the uncertainty of the time filter coefficient. We set the time filter coefficient value as 0.02 in the assimilation model, which produces an apparent bias with the truth model that uses 0.01 as its time filter coefficient value.

Started from the streamfunction at 1200 UTC 1 January 1991 derived from the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis 500-hPa u and υ data, both the truth model and assimilation model are spun up for 30 days to derive their own initial conditions. Then the truth model is integrated for another 240 days to generate “observations” which sample the “truth” model states. The observational interval is set to 6 h (12 time steps). A Gaussian noise with the standard deviation of 106 m2 s−1 is imposed to the truth streamfunction to simulate the “observational” error. Furthermore, to simply reflect the spatial structure of the observing system, observations for all model grids in the Northern Hemisphere (NH) are assumed to be available. In the Southern Hemisphere (SH), observations on odd x-index and y-index grids are assumed to be available. Namely, only ¼ of the grid points are observed in the SH. Figure 4 displays the observing system (solid circle) and the model grids (plus sign). The numerical values of K and M in this experiment are 2176 and 3456, respectively.

Fig. 4.
Fig. 4.

Model grids (plus sign) and observational locations (solid circle) in the twin experiment. In the Northern Hemisphere, observations on all model grids are assumed to be available. In the Southern Hemisphere, observations on all odd x- and y-index grids are available.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

Initial ensemble perturbations of the atmospheric streamfunction for the assimilation are generated by adding a Gaussian white noise with the same standard deviation of observational error to the biased initial condition generated by the assimilation model. The ensemble size is set to a typical value of 20. Additionally, because the leapfrog scheme is used to integrate the model, a two time-level adjustment method (S. Zhang et al. 2004) is applied to the data assimilation. That says observations at time t are used to adjust the model states at time t and t − 1.

Three experiments are conducted to evaluate the performances of two assimilation algorithms. The first one is the ensemble control run (without observational constraint), serving as the reference experiment, denoted as CTL; the second one is the standard EnKF; and the third one is the EnKF with MSA. To simply examine the validity of MSA, it is only applied to observational residuals in the SH in this study. Additionally, the MSA in this study is activated after 20 model days, which is roughly the length of the spinup of the standard EnKF. The goal of this setting is to investigate whether MSA can further enhance the accuracy of the ensemble mean after the standard EnKF reaches its equilibrium. In fact, extra experiments that apply the MSA from the first model day obtain similar equilibriums as the experiments in this study with a shorter spinup (not shown). Two data assimilation algorithms use the same observing system and ensemble initial conditions as that in the first experiment. Discarding the assimilation results in the first 140 days as the spinup, the results of the last 100 days are used to conduct error statistics and analysis. The time mean RMSE of defined by Eq. (19) for CTL is about 2.7 × 107 m2 s−1. Because of the overly large error of , the results of CTL are not plotted in the later figures.

To check the dependence of two assimilation methods on a, seven values of a, 125, 250, 500, 1000, 1500, 2000, and 2500 km, are used. The default relationship5 between a and {a(i), i = 1, …, L} as well as L is defined in Table 2, where a(i) can be formulated as
e32
Thus, a(1) and a(L) are constantly set to 4000 km and a, respectively.
Table 2.

The relationship between the half-width of GC localization (denoted as a, km) in the EnKF of the hybrid method and the half-widths of GC localization [denoted as a(i), km] in the MSA of the hybrid method in this study.

Table 2.

Additionally, trial-and-error tests justify that assimilation experiments with the optimal static inflation factors draw the same conclusions as that with no inflation. Therefore, to simplify the issues and separate the effect of variance inflation, we did not employ the variance inflation in the data assimilation experiments in this study.

4. Results of the hybrid method

The dependences of the hybrid method on the GC half-width in EnKF and the number of scale levels in MSA are first investigated in this section. Then, the comparisons between the hybrid method and the standard EnKF as well as the EnKF–SCM method are conducted. Afterward, the sensitivity study of the hybrid algorithm with respect to observing system is performed while a simple analysis of the computational cost is presented at last.

a. Dependence on the GC localization half-width

For different a values in the EnKF of the hybrid method, the number of scale levels and the GC half-widths in MSA are listed in Table 2. The dashed triangle curve in Fig. 1 gives the time-averaged ensemble mean RMSE [Eq. (19)] of when the parameter a takes various values in the hybrid method. Apparently, the optimal a for the hybrid method is about 1500 km. To further understand the performance of the hybrid method, we examine the time series of the RMSE and the spatial distribution of RMSE. Here the first RMSE is defined as the RMSEs in Eq. (19) while the second RMSE for the (i, j)th grid is computed by
e33

Figure 5 shows the time series of RMSE of for four a values: 125 (red curve), 500 (black curve), 2000 (blue curve), and 2500 km (green curve). As a increases from 125 to 2000 km, the RMSE is significantly reduced. For a 2500-km a value, the spinup period of the hybrid method is much longer than other cases. After the spinup period (here is about 100 days), the RMSE of with a 2500-km a value is the worst among seven cases.

Fig. 5.
Fig. 5.

Time series of RMSE (106 m2 s−1) of the atmospheric streamfunction for the GC half-width values as 125 (red curve), 500 (black curve), 2000 (blue curve), and 2500 km (green curve) in the EnKF–MSA hybrid method.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

Figure 6 displays the spatial RMSEs of for a values as 250 (Figs. 6a,b), 1000 (Figs. 6c,d), and 2500 km (Figs. 6e,f) in the standard EnKF (Figs. 6b,d,f) and the hybrid method (Figs. 6a,c,e). For a small a value (like 250 km) or a large a value (like 2500 km), compared to the standard EnKF, the hybrid method can greatly reduce the error of in the SH where the observations are sparsely distributed. For a nearly optimal a value (like 1500 km), despite the error in the SH produced by the hybrid method is a little larger than that generated by the standard EnKF, the difference is much less than that for GC half-width values of 250 and 2500 km.

Fig. 6.
Fig. 6.

The spatial RMSEs (106 m2 s−1) of the atmospheric streamfunction for the GC half-width values as (a),(b) 250; (c),(d) 1000; and (e),(f) 2500 km in (right) the standard EnKF and (left) the hybrid method. Note that (a),(b); (c),(d); and (e),(f) employ the same shade scales, respectively.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

To check the validity of the MSA, we first define three error statistics: first is the RMSE of produced by the hybrid method (denoted by RMSEhybrid), second is the RMSE of generated by the EnKF in the step 1 (see section 2e) of the hybrid method (denoted by RMSEEnKF in hybrid), and third is the difference between RMSEhybrid and RMSEEnKF in hybrid (i.e., RMSEhybrid − RMSEEnKF in hybrid). Here, the first two quantities are respectively calculated by
e34
and
e35
where the superscript “hybrid posterior” (“posterior of EnKF in hybrid”) represents the posterior field produced by the hybrid method (the EnKF in the hybrid method). This definition can directly examine the validity of the MSA. Obviously, a negative difference means that the MSA is valid. Figure 7 plots the time series of the difference for the 2500-km a value. Apparently, during 20–100 days, most of the differences are less than zero, demonstrating that the MSA in the hybrid method can continuously refine the quality of the analysis solution of the EnKF. After the spinup period of the hybrid method, the error of reaches an equilibrium state.
Fig. 7.
Fig. 7.

Time series of RMSEhybrid minus RMSEEnKF in hybrid for a 2500-km GC half-width in the EnKF of the hybrid method, where the dashed line represents no difference. RMSEhybrid and RMSEEnKF in hybrid represent the RMSEs of the atmospheric streamfunction for the hybrid method and the EnKF in the first step of the hybrid method, respectively.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

b. Dependence on the number of scale levels

To investigate the dependence of the hybrid method on the number of scale levels in MSA, we choose a relative small value (here is 250 km) of a in EnKF. The default five scale levels in MSA are first compressed to the first one, two, three and four levels, and then extended to include 8000 km and (8000 and 16 000 km) levels to generate seven configurations of a. Thus, eight experiments, including the experiment with 0 level in MSA, which reduces to the standard EnKF, are conducted in sum.

Figure 8a shows the sensitivity of the time mean RMSE [Eq. (19)] of with respect to the number of scale levels of the MSA. Apparently, the saturate number is about 3. Compared with the standard EnKF, the hybrid method with three scale levels reduces the time mean RMSE of by 22%. Note that even with one scale level that has the same GC half-width (250 km) as the standard EnKF, the hybrid method still can remarkably reduce the error of . This justifies that the EnKF with the 250-km a value cannot completely retrieve the observational signals whose spatial scales are within 250 km. Results of the time series of the RMSE [i.e., the RMSEs in Eq. (19)] of for the standard EnKF and the hybrid method (Fig. 8b) also confirm this point. It is worth mentioning that one may think that the increasing tendencies of the RMSEs in Fig. 8b and Fig. 9 for the standard EnKF are caused by the absence of variance inflation. However, the EnKF experiments with optimal static multiplicative inflation factors still present the similar rising trends (not shown). Thus, the increasing tendencies of RMSEs in the standard EnKF experiments are not caused by the lack of variance inflation.

Fig. 8.
Fig. 8.

(a) The sensitivity of the RMSE (106 m2 s−1) of the atmospheric streamfunction with respect to the number of scale levels of MSA in the hybrid method. (b) Time series of the RMSE of the atmospheric streamfunction for 0 (red curve), 1 (black curve), and 3 (blue curve) scale levels in the hybrid method. Note that the hybrid method reduces to the standard EnKF when the number of scale levels is 0. The 1 and 3 scale levels correspond to 250-km and 250-, 500-, and 1000-km levels in the MSA, respectively.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

Fig. 9.
Fig. 9.

The time series of the RMSE (106 m2 s−1) of the atmospheric streamfunction for (a) 250- and (b) 2500-km GC half-width values, where the black, blue, and red curves represent the results of the standard EnKF, the EnKF–MSA hybrid method, and the hyrbid method without smoothing term (i.e., EnKF–SCM method) in the cost function of MSA, respectively. Here the green curve stands for the results of a redesigned hybrid method that changes two scale levels (2500 and 4000 km) to five levels (250, 500, 1000, 2000, and 4000 km) in the 2500-km a-value experiment. The red curve in (b) stands for the results of the redesigned hybrid method when the smoothing term is dropped in the cost function of MSA.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

c. Comparison with the standard EnKF

In this section, we first quantitatively compare the hybrid method with the standard EnKF. Then the error analysis of the time series and spatial distributions of RMSEs for two methods are conducted. Note that the two RMSEs here are computed the same as those in Figs. 5 and 6.

From Fig. 1, compared with the standard EnKF (the solid circle curve), the hybrid method has much weaker dependence on the cutoff distance. For relatively small or large a values, the time mean RMSE of produced by the hybrid method is much smaller than that produced by the standard EnKF. For a nearly optimal a values (such as 1000 and 1500 km here), the hybrid algorithm is a little worse than the standard EnKF. From Fig. 2c, the signal in the SH implied in the truth residual [Eq. (20)] is much weaker than the standard deviation of observational error for 1000-km a value. Thus, under this circumstance, the observational residual [Eq. (21)] is noise dominant. Although Fig. 2 is a snapshot result, substantive examinations draw the same conclusion. Without background term in the cost function, it is very hard for MSA to extract useful information from the observational residuals.

Totally speaking, on the one hand, if we roughly average the time mean RMSEs of for different a value cases, the hybrid method reduces the error of by 43% (from 1.0 × 106 m2 s−1 of the standard EnKF to 5.7 × 105 m2 s−1). On the other hand, if we simply define the sensitivity of the data assimilation scheme with respect to a as
e36
where is defined as in Eq. (19) and i indexes different value of a, num equals to 7 in this study, and the sensitivities of the standard EnKF and the hybrid method are 1.2 × 106 m2 s−1 and 1.2 × 105 m2 s−1, respectively. Thus, relative to the standard EnKF, the sensitivity of the hybrid method is reduced by 90%.

Figure 9 displays the time series of the RMSE of for a values as 250 (Fig. 9a) and 2500 km (Fig. 9b) in the standard EnKF (black curve) and the hybrid method (blue curve). We discuss the advantages of the hybrid method relative to the standard EnKF from the following two aspects.

For a small a (like 250 km), the hybrid method can significantly reduce the error of with a short spinup period (see Fig. 9a). Comparison of the spatial RMSE of (Figs. 6a,b) also demonstrates that the hybrid method can substantially improve the quality of in the SH. To look into the detailed performances of these two schemes, we first define the following two quantities: One is the true difference defined by
e37
where represents the posterior (analysis) ensemble mean of produced by the EnKF in the hybrid method. The other is the relative difference defined by
e38
where represents the posterior ensemble mean of produced by the hybrid method. Obviously, a negative (positive) at a grid point means that the MSA is valid (invalid) there.

Then, we examine the SH6 results of the hybrid method at the first data assimilation cycle (i.e., 0600 on the 20th day). Figure 10 shows the SH results of (Fig. 10a), observational residuals (Fig. 10b) that locate at the observational positions, the analysis solution of the MSA (Fig. 10c), and (Fig. 10d) when a is set to 250 km. Since the cutoff distance is very small, the signals implied in the observational residuals are expected to be longwave dominant (Fig. 10b). Moreover, comparisons among Figs. 10a–c justify that on the one hand, observational residuals more or less contain true longwave information with strong signals; on the other hand, MSA can reasonably retrieve the multiple scale information from the observational residuals. The hybrid method can effectively reduce the error where the signal is strong, that is the darker blue in Fig. 10d occurs in the same areas as the darker red or blue places in Fig. 10a. This also validates the correctness of the analysis at the end of section 2e.

Fig. 10.
Fig. 10.

The Southern Hemisphere results (106 m2 s−1) of the hybrid method at the first data assimilation cycle (that is the 0600 on the 20th day) for a 250-km a value. (a) The true difference defined in the text. (b) Observational residuals. (c) Analysis result of the MSA. (d) The relative difference defined in the text. Note that (a)–(c) adopt the upper shade scale, while (d) employs the lower shade scale. Additionally, the black curve in (d) indicates the zero contour.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

To reflect the analysis process of MSA, we plot the results of MSA for the first one (Fig. 11a), the sum of first two (Fig. 11b), the sum of first three (Fig. 11c), and the sum of all five scale levels (Fig. 11d).7 It is obvious that MSA can sequentially extract the multiple scale information from longwave to shortwave. And the total analysis of MSA can reflect strong signals implied in the observational residuals.

Fig. 11.
Fig. 11.

Results (106 m2 s−1) of the MSA in the hybrid method at the first data assimilation cycle (i.e., the 0600 on the 20th day) for a 250-km a value. (a) Result of the first level. (b) Result of the sum of first two levels. (c) Result of the sum of first three levels. (d) Result of the sum of all five scale levels. Note that all panels adopt the same shade scale.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

For a large a value (such as 2500 km), even with two scale levels in MSA (see Table 2), the hybrid method can greatly reduce the error (see the blue and black curves in Fig. 9b) of model state after a long spinup period of data assimilation. According to the analysis of Fig. 7, the model state can be gradually refined through adding back the two longwave signals. Additionally, examinations show that the long spinup period of the hybrid method here is mainly caused by too few scale levels and an overly large localization factor. Therefore, the spinup periods are expected to be shortened through including some small scale levels into MSA.

d. Improvement of the performance of the hybrid method for overly large a values

Motivated from the analysis in the section 4c, we redesign the experiment of the hybrid method for 2500-km a value. Two scale levels in MSA are modified to five levels, including 250, 500, 1000, 2000, and 4000 km. The green curve in Fig. 9b presents the time series of the RMSE of generated by the redesigned hybrid method. Compared with the previous experiment (the blue curve in Fig. 9b), the redesigned method not only greatly shortens the spinup period of data assimilation but also notably reduces the error of from 7.9 × 105 to 6.1 × 105 m2 s−1. Thus, the dependence of the hybrid method on a will be further lightened.

Here, we also check the performance of the redesigned experiment at the first data assimilation cycle (Fig. 12). Since the EnKF with an overly large a value contaminates the model state, the observational residuals may include various scale information (Fig. 12b), which is different from the situation (Fig. 10b) with the 250-km a value. Therefore, the MSA should include some small-scale levels. Fig. 12c proves that the MSA here can also extract the strong signals from the observational residuals (Fig. 12b) and subsequently restore the stained model state where the amplitudes of the observational residuals are relatively large (Fig. 12d).

Fig. 12.
Fig. 12.

As in Fig. 10, but for the 2500-km a value and the following five scale levels of MSA in the hybrid method: 250, 500, 1000, 2000, and 4000 km.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

It is worth mentioning that one may argue that MSA with only two levels (i.e., 2500 and 4000 km) in the original design of the hybrid method for 2500-km GC half-width can also trace the shortwave information of the model state (see Figs. 6e,f) and improve the analysis of the standard EnKF. The reason is that although MSA can only retrieve the longwave information from the observational residuals with large scales, it is also only valid in the places (not shown) where the signals are strong. On the one hand, when the MSA analysis is added to the EnKF ensemble mean analysis, some shortwave information can also be improved. On the other hand, the net effect at each data assimilation cycle during the spinup period of the hybrid method is that the RMSE [i.e. the RMSEs in Eq. (19)] produced by the hybrid method is smaller than that produced by the standard EnKF. Because of the too few and too large-scale levels, MSA can only gradually refine the model state with a long spinup period (see the green curve in Figs. 5 and 7 and the blue curve in Fig. 9b).

Combining the above results and the results for small as suggests that in the practical applications, for an overly large a, we should include some scales smaller than a into MSA; for an overly small a, we should fix the smallest scale [i.e., the last scale a(L)] to a and include larger a(i)s in MSA.

e. Comparison with the EnKF–SCM method

As analyzed at the end of section 2e, without the background term, the overall impact of MSA is to move the analysis ensemble mean of EnKF closer to observations than the raw covariances intend to do. When the smoothing term is dropped in the cost function of MSA, the EnKF–MSA hybrid method degrades to EnKF–SCM method. Since the EnKF–MSA method has been somewhat ameliorated in the last section, it is necessary and meaningful to investigate the performances of these two hybrid methods. Because of the limited space, we only analyze the results of experiments with 250- and 2500-km a values here.

The red curve in Fig. 9 shows the time series of the RMSE of the atmospheric streamfunction with 250- (Fig. 9a) and 2500-km (Fig. 9b) a values for the EnKF-SCM scheme. Obviously, for extreme a values, even without smoothing term, the hybrid method still can gradually refine the analysis ensemble mean produced by EnKF, especially for an overly large a value. The reason is that the truth residuals contain signals stronger than the standard deviation of observational error (Figs. 2a,d). With this precondition, even when the model state at the observed model grids is successively corrected from the analysis of EnKF to the observation, the signal-to-noise ratio in the adjustment is high. For the sparse observing system, the net effect may lead the improvement of model state. When the smoothing term is introduced, the assimilation quality is further greatly enhanced, which contributes to the fact that the local small-scale noise (such as “bull’s-eye”) is filtered and the extracted signal from the observational residuals is more deterministic. Thus, the EnKF–MSA method in the practice may benefit from the smoothing term, although the smoothing operator here is relatively simply.

f. Dependence on observing systems

As the foregoing analysis, the quality of MSA in the proposed hybrid method is sensitive to the signal-to-noise ratio of the observational residual. Because of the complex observing network in the real world, the quality of the signal-to-noise ratio is highly geographic dependent. Therefore, the assimilation quality of the presented new method must be sensitive to the observing system. Although the observing system in this study is highly simplified, we still can conceptually evaluate the dependence of the hybrid method on the observing system, and point out whether MSA in the hybrid method should be applied to the dense observing system or not for extreme a values.

Taking 125 and 2500 km for an example of extreme a values, we apply MSA in the hybrid method to global observations. Through comparing the results here with that in section 4a for 125 km and that in section 4d for 2500 km, we can answer the above question. Figure 13 shows the time series of RMSE [see the RMSEs in Eq. (19) for the definition] of the streamfunction for 125- (Fig. 13a) and 2500-km (Fig. 13b) a values, where the blue (black) curve represents the results of the hybrid method without (with) MSA applied to NH observations. Note that here the blue curve in Fig. 13b is the same as the green curve in Fig. 9b. For an overly small a value, application of MSA to dense observed model grids adversely worsens the quality of the model state. The reason is that signals in the truth residual defined as in Eq. (20) in NH are weaker than the standard deviation of observational error (see Fig. 2a). It is difficult for MSA to correctly retrieve the useful information from observational residual. Thus, caution should be taken when the new scheme is applied to dense observations with an overly small cutoff distance. For an overly large a value, however, when MSA is applied to observations in NH, the model state is further refined compared to the results of the redesigned experiment in section 4d. This can also be explained by Fig. 2d, which points that the truth residuals in NH contain some signals stronger than the standard deviation of observational error. Note that the oscillations that exist in the black curves in Fig. 13 are caused by the inconsistency between the analysis of MSA and the model dynamics. According to the analysis process of MSA in this study, no model dynamics is contained, which may cause shocks between the analysis of MSA and the analysis of EnKF. Figure 14, which plots the spatial RMSEs [see Eq. (33) for the definition] of streamfunction for two a-value experiments, also justifies the above conclusions.

Fig. 13.
Fig. 13.

The time series of the RMSE (106 m2 s−1) of the atmospheric streamfunction for (a) 125- and (b) 2500-km a values, where the black and blue curves represent the results of the EnKF–MSA hybrid method with and without MSA applied to the NH observations. Here the experiment with 2500-km a value corresponds to the redesigned experiment in section 4d.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

Fig. 14.
Fig. 14.

The spatial RMSEs (106 m2 s−1) of the atmospheric streamfunction for (a),(b) 125- and (c),(d) 2500-km a values in the hybrid method with MSA applied to (a),(c) SH observations and (b),(d) global observations. Note that (a),(b) and (c),(d) employ the same shade scales, respectively. Here the experiment with 2500-km a value corresponds to the redesigned experiment in section 4d.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

g. Analysis of the computational cost

According to the description of the hybrid algorithm in section 2e, here we roughly analyze the computational cost of the hybrid method. Since the MSA only updates the ensemble mean, the additional computational cost relative to the standard EnKF is caused by the MSA.

Take a 500-km a value for an example, Fig. 15 shows the variation of normalized values of cost function of the MSA in the hybrid method with respect to iterate steps for the first (solid dot), second (hollow dot), third (solid diamond), and fourth (hollow diamond) scale levels. Here, the normalization factor is the value of cost function at the first iterate step. Apparently, the cost function converges fastest for the first scale level. For all cases, the largest number of iterations that is required to make the cost function converge is about 10. Under this precondition, the time consumption caused by MSA is very little relative to that caused by the standard EnKF.

Fig. 15.
Fig. 15.

The variation of normalized values of cost function of MSA in the hybrid method with the 500-km a value for the first (solid dot), second (hollow dot), third (solid diamond), and fourth (hollow diamond) levels. The x axis is the iterate step. Here the normalized factor for each scale level is the value of cost function at the first iterate step.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

5. Impact of the compensatory scheme on weather forecast

An effective way to verify the superiority of the compensatory scheme over the standard EnKF for extreme cutoff distances is to perform forecast experiments. A total of 20 forecast initial conditions are selected every 5 model days apart from analysis fields of 100–195 days for the hybrid method and the standard EnKF with 125- and 2500-km a values. Note that here the analysis results of the hybrid method with 2500-km a value are produced by the redesigned experiment in section 4d. The 20 forecast cases are integrated up to 60 days for two assimilation methods. The global anomaly correlation coefficient (ACC) and RMSE of the forecasted ensemble mean are used to evaluate the pattern correlation and amplitude error relative to the truth. The formulas of these two quantities for the sth lead time are
e39
and
e40
respectively. Here R equals to 20 and r (i and j) indexes the forecast case (model grid). The superscripts “f” and “t” represent the forecasted and truth quantities. Here () denotes the forecasted (true) ensemble mean anomaly of , that is
e41
where () denotes the climatology of at the (i, j)th grid point for the assimilation (truth) model. Here () is the spatial mean of (). The climatological standard deviation of at the (i, j)th grid point, denoted by , is produced by the assimilation model.

Figure 16 shows the variations of ACC (Figs. 16a,c) and RMSE (Figs. 16b,d) with the lead time of the forecasted ensemble means of started from the analysis fields of 125- (Figs. 16a,b) and 2500-km (Figs. 16c,d) a values for the hybrid method (blue curve) and the standard EnKF (black curve). For the 125-km a value case, although the advantage is no so evident, the hybrid method can maintain higher ACCs and smaller RMSEs during the first 4 days of lead time compared to the standard EnKF. That is to say the proposed approach can more or less enhance the short-term weather forecast skill for an overly small cutoff distance. For the 2500-km a value case, the hybrid method can greatly increase the short-term weather forecast skill relative to the standard EnKF (see the blue and black curves in Figs. 16c,d). If an ad hoc value of 0.6 ACC is used to evaluate the valid time scale of forecast (Hollingsworth et al. 1980), the hybrid method can extend the valid weather forecast time scale of the standard EnKF by about 10 days. In addition, results in Fig. 1 justify that improvement from the compensatory method relative to the standard EnKF for an overly large localization factor is much larger than that for an overly small localization factor. Therefore, the forecast results are consistent with the analysis results.

Fig. 16.
Fig. 16.

Variations of (a),(c) ACC and (b),(d) the normalized RMSE with the lead time of the forecasted ensemble means of the atmospheric streamfunction started from the analysis fields of (a),(b) 125- and (c),(d) 2500-km a values for the hybrid method (blue curve) and the standard EnKF (black curve). Note that the dashed line stands for the 0.6 ACC value.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00369.1

6. Summary and discussion

Covariance localization was initially introduced to enhance the reliability of ensemble-evaluated background error covariance by reducing the long-distance spurious correlation resulting from sampling errors of a finite ensemble. Although fixed covariance localization can greatly improve the analysis quality, it has significant limitations: insufficient longwave information with a small cutoff distance or contaminated analysis states if a large cutoff distance is used, while tuning an optimal cutoff distance is always difficult. Under these circumstances, we develop an EnKF and multiple-scale analysis (MSA) hybrid method to break the limitations and improve the performance of the standard EnKF. At each analysis step, after the standard EnKF is done, the MSA is used to extract multiscale information from observational residual (the difference between observations and interpolated analysis ensemble means produced by EnKF). Within a biased twin-experiment framework based on a global barotropical spectral model and an idealized observing system, the performance of the proposed method is examined. Results show that the hybrid method is superior to a standard EnKF for overly small or large cutoff distances and it has less dependence on cutoff distances. Consistently, the compensatory scheme can enhance the short-term weather forecast skill, especially for an overly large cutoff distance. In addition, it is shown that caution should be used in sensitivity studies with respect to observing systems when the new scheme is applied to dense observations with an overly small cutoff distance. Also, the new method has a nearly equivalent computational cost compared to the standard EnKF and thus it is suitable for GCM applications.

Although the compensatory approach presented in this study is promising, there are many challenges before it can be applied to the real weather climate models for state estimation and prediction initialization.

First, the MSA method in this study is actually a spatial smoothing of observational residuals. Given the observing network of Fig. 4 in this study, which is rather homogeneous and sufficiently dense coverage, the spatial averaging is robust and informative. Although additional experiments (not shown) that assume a sparser observing system than that used in this study have also demonstrated the superiority of the hybrid method over the standard EnKF, because of the complexity of the real observing systems that are highly heterogeneous, very sparse and irregular such as ocean in pre-Argo era, or even atmosphere in presatellite era, applying multiple-scale analysis to assimilate instrumental measurements into a realistic atmospheric, oceanic, or atmosphere–ocean coupled general circulation model should be examined to identify the problems and seek out the solutions.

Second, the comparison between the compensatory approach and the adaptive localization model shall be performed to increase our understanding about the multiple-scale analysis.

Third, from the results in this study, MSA can improve the accuracy of the ensemble mean for an overly small or an overly large localization factor. Under this circumstance, the analysis ensemble perturbations must be smaller in response to more accurate analysis of ensemble mean. However, the proposed hybrid method does not apply any changes to ensemble perturbations in response to the corrections made by MSA now. To remove this inconsistency, the presented hybrid method shall also been further ameliorated. Actually, extra experiments that attempt to add the variance inflation to the hybrid method gain larger errors of the model state than that without inflation and the optimal inflation factors are less than 1.0, which further verifies that the squeeze of ensemble perturbations may be more important than the variance inflation for the current version of the hybrid method. Additionally, results of experiments with optimal inflation factors also show similar increasing tendencies as Figs. 5, 8b, 9, and 13, demonstrating that the rising trends of the RMSEs in the hybrid method in this study are also not caused by the absence of variance inflation.

Fourth, when the forward operator is more complex than interpolation (e.g., for satellite radiances), the validity of MSA should also be examined.

Last, since the optimal localization scale would increase when ensemble size is increased, the longwave information loss caused by localization would be reduced. Whether the hybrid method can still outperform the EnKF with large ensemble sizes for extreme localization factors shall also be investigated.

Acknowledgments

The authors thank three anonymous reviewers for their thorough and helpful suggestions on the earlier version of this manuscript. This research is cosponsored by grants from the National Natural Science Foundation (Grants 41030854, 41306006, 41376015, 41376013, 41106005, 41176003, and 41206178).

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1

Note that the prior ensemble of the observation yo is usually projected by the background ensemble of model state through the linearization of the observation operator h.

2

Note that here the linear relationship is actually the local linearization of the nonlinear observation operator h through a least squares regression between the observation and the model state to be adjusted.

3

Please refer to section 3 for the details of the experiment.

4

Note the matrices (l), (l), (l), and (l) will be specified in section 2e.

5

Note that two extra experiments with 1500- and 2500-km values of a do not adhere to Eq. (32).

6

Since the MSA is only applied in the SH, the results there are shown.

7

Note that Fig. 11d is the same as Fig. 10c, but with a different shade scale.

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