Integrated Hybrid Data Assimilation for an Ensemble Kalman Filter

Lili Lei aKey Laboratory of Mesoscale Severe Weather, Ministry of Education, Nanjing University, Nanjing, China
bSchool of Atmospheric Sciences, Nanjing University, Nanjing, China
cFrontiers Science Center for Critical Earth Material Cycling, Nanjing University, Nanjing, China

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Zhongrui Wang aKey Laboratory of Mesoscale Severe Weather, Ministry of Education, Nanjing University, Nanjing, China
bSchool of Atmospheric Sciences, Nanjing University, Nanjing, China

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Zhe-Min Tan aKey Laboratory of Mesoscale Severe Weather, Ministry of Education, Nanjing University, Nanjing, China
bSchool of Atmospheric Sciences, Nanjing University, Nanjing, China

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Abstract

Hybrid ensemble–variational assimilation methods that combine static and flow-dependent background error covariances have been widely applied for numerical weather predictions. The commonly used hybrid assimilation methods compute the analysis increment using a variational framework and update the ensemble perturbations by an ensemble Kalman filter (EnKF). To avoid the inconsistencies that result from performing separate variational and EnKF systems, two integrated hybrid EnKFs that update both the ensemble mean and ensemble perturbations by a hybrid background error covariance in the framework of EnKF are proposed here. The integrated hybrid EnKFs approximate the static background error covariance by use of climatological perturbations through augmentation or additive approaches. The integrated hybrid EnKFs are tested in the Lorenz05 model given different magnitudes of model errors. Results show that the static background error covariance can be sufficiently estimated by climatological perturbations with an order of hundreds. The integrated hybrid EnKFs are superior to the traditional hybrid assimilation methods, which demonstrates the benefit to update ensemble perturbations by the hybrid background error covariance. Sensitivity results reveal that the advantages of the integrated hybrid EnKFs over traditional hybrid assimilation methods are maintained with varying ensemble sizes, inflation values, and localization length scales.

Significance Statement

Data assimilation is critical for providing the best possible initial condition for forecast and improving the numerical weather predictions. The hybrid ensemble–variational data assimilation method has been widely adopted and developed by many operational centers. The hybrid ensemble–variational assimilation method combines the advantages of ensemble and variational methods and minimizes the weaknesses of the two methods, and thus it outperforms the stand-alone variational and ensemble assimilation methods. The hybrid ensemble–variational assimilation method often computes the control analysis using a variational solver with hybrid background error covariances, but generates the ensemble perturbations by an ensemble Kalman filter (EnKF) system with pure flow-dependent background error covariances. The inconsistencies that result from performing separate variational and EnKF systems can lead to suboptimality in the hybrid ensemble–variational assimilation method. Therefore, integrated hybrid EnKF methods that utilize the framework of an EnKF to update both the ensemble mean and ensemble perturbations by the hybrid background error covariance, are proposed. The integrated hybrid EnKFs use climatological ensemble perturbations to approximate the static background error covariance. The integrated hybrid EnKFs are superior to the traditional hybrid ensemble–variational assimilation methods by producing smaller errors, and the advantages are persistent with varying assimilation parameters.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Zhe-Min Tan, zmtan@nju.edu.cn

Abstract

Hybrid ensemble–variational assimilation methods that combine static and flow-dependent background error covariances have been widely applied for numerical weather predictions. The commonly used hybrid assimilation methods compute the analysis increment using a variational framework and update the ensemble perturbations by an ensemble Kalman filter (EnKF). To avoid the inconsistencies that result from performing separate variational and EnKF systems, two integrated hybrid EnKFs that update both the ensemble mean and ensemble perturbations by a hybrid background error covariance in the framework of EnKF are proposed here. The integrated hybrid EnKFs approximate the static background error covariance by use of climatological perturbations through augmentation or additive approaches. The integrated hybrid EnKFs are tested in the Lorenz05 model given different magnitudes of model errors. Results show that the static background error covariance can be sufficiently estimated by climatological perturbations with an order of hundreds. The integrated hybrid EnKFs are superior to the traditional hybrid assimilation methods, which demonstrates the benefit to update ensemble perturbations by the hybrid background error covariance. Sensitivity results reveal that the advantages of the integrated hybrid EnKFs over traditional hybrid assimilation methods are maintained with varying ensemble sizes, inflation values, and localization length scales.

Significance Statement

Data assimilation is critical for providing the best possible initial condition for forecast and improving the numerical weather predictions. The hybrid ensemble–variational data assimilation method has been widely adopted and developed by many operational centers. The hybrid ensemble–variational assimilation method combines the advantages of ensemble and variational methods and minimizes the weaknesses of the two methods, and thus it outperforms the stand-alone variational and ensemble assimilation methods. The hybrid ensemble–variational assimilation method often computes the control analysis using a variational solver with hybrid background error covariances, but generates the ensemble perturbations by an ensemble Kalman filter (EnKF) system with pure flow-dependent background error covariances. The inconsistencies that result from performing separate variational and EnKF systems can lead to suboptimality in the hybrid ensemble–variational assimilation method. Therefore, integrated hybrid EnKF methods that utilize the framework of an EnKF to update both the ensemble mean and ensemble perturbations by the hybrid background error covariance, are proposed. The integrated hybrid EnKFs use climatological ensemble perturbations to approximate the static background error covariance. The integrated hybrid EnKFs are superior to the traditional hybrid ensemble–variational assimilation methods by producing smaller errors, and the advantages are persistent with varying assimilation parameters.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Zhe-Min Tan, zmtan@nju.edu.cn

1. Introduction

Data assimilation seeks to find the best estimate of the state of a dynamical system given prior information and observations from the dynamic system (Kalnay 2002). To combine the advantages of ensemble and variational methods while at the same time minimizing the weaknesses of the two methods, hybrid ensemble–variational assimilation methods have been proposed and developed (e.g., Hamill and Snyder 2000; Etherton and Bishop 2004). Previous studies have shown that the hybrid ensemble–variational assimilation methods can outperform stand-alone variational- or ensemble-based assimilation methods, and they have been widely applied for regional (Wang et al. 2008; Liu et al. 2008, 2009; Liu and Xiao 2013; Zhang and Zhang 2012), and global (Buehner et al. 2013; Clayton et al. 2013; Kuhl et al. 2013; Kleist and Ide 2015) numerical weather predictions.

One category of hybrid data assimilation incorporates flow-dependent background error covariances from an ensemble of data assimilations to the background error covariance at the start of a 4D-variational (4D-Var) assimilation window (Bonavita et al. 2012, 2016). The ensemble of data assimilations consists of independent adjoint-based 4D-Var analyses. Another category of hybrid data assimilation supplements the static background error covariance with the flow-dependent background error covariance from an ensemble, using either a directly combined covariance (e.g., Hamill and Snyder 2000) or a variational-based control variable method (e.g., Lorenc 2003). The hybrid analysis increment is typically computed by use of the variational framework, while the ensemble members are often updated by an ensemble Kalman filter (EnKF; Evensen 1994; Burgers et al. 1998) with an option to recenter the ensemble mean around the hybrid solution. Various hybrid ensemble–variational assimilation methods that combines static and flow-dependent background error covariances have been proposed, but many of these hybrid methods with technical differences are theoretically equivalent (Wang et al. 2007). Compared to variational assimilation methods, hybrid assimilation methods take advantages of the background error covariance with flow-dependent and multivariate definitions through the use of the ensemble. Compared to the EnKF, the static background error covariance used by hybrid assimilation methods can help to mitigate sampling errors resulted from limited ensemble sizes (Etherton and Bishop 2004; Wang et al. 2009). EnKF typically performs localization in observation space, either implemented with the Kalman gain matrix of serial Kalman filters (e.g., Anderson 2001; Whitaker and Hamill 2002) or incorporated with the observation error variance matrix of localized ensemble transform Kalman filters (e.g., Hunt et al. 2007; Greybush et al. 2011); while hybrid assimilation methods apply localization in model space. Model-space localization could be more favorable for nonlocal observations, such as satellite radiance measurements (Campbell et al. 2010), although the opposite could be true when negative correlations between radiance observations and state variables exist (Lei and Whitaker 2015). Moreover, model-space localization can be implemented in an EnKF through a modulation approach (Bocquet 2016; Bishop et al. 2017; Lei et al. 2018; Farchi and Bocquet 2019).

Besides the commonly used hybrid assimilation methods that combine the static and flow-dependent background error covariances, Penny (2014) proposed an alternative by averaging the Kalman gains of a variational method and an EnKF. This hybrid formulation is equivalent to average of the analyses from a variational method and an EnKF. While the ensemble mean is calculated by the hybrid formulation, the ensemble perturbations are updated by the EnKF. Penny (2014) applied the hybrid formulation with the Lorenz96 model (Lorenz 1996) and found that this hybrid formulation improves the analysis over the stand-alone variational and EnKF methods. The combination of Kalman gains of variational and EnKF methods can provide dynamically varying background error covariance estimates and improved representation of model uncertainties (Penny et al. 2015). The dynamically defined background error covariances are essential for achieving an accurate state estimate, especially for strongly coupled data assimilation (Penny et al. 2019).

Previous hybrid assimilation methods use a separate EnKF system to generate the posterior ensemble perturbations. The inconsistencies that result from performing separate hybrid assimilation and EnKF systems can lead to suboptimality in the hybrid assimilation method. To remedy the inconsistency, the Ensemble Variational Integrated Localized (EVIL) proposed by Auligné et al. (2016) represents a modification to the hybrid assimilation method. EVIL uses information gained in the variational minimization procedure and then generates the posterior ensemble perturbations, instead of updating ensemble perturbations from a separate EnKF system. The posterior ensemble perturbations are drawn from the analysis error covariance matrix, which is a by-product of the conjugate-gradient based minimization algorithm. To obtain a reasonable representation of the correct eigenspectrum of the analysis error covariance matrix, the number of iterations of the conjugate-gradient procedure needs to be relatively large. Several hundreds were suggested for the cut-down system (Auligné et al. 2016), which may not yet be affordable for practical applications.

Instead of utilizing the variational framework to calculate the analysis increment, integrated hybrid assimilation algorithms that utilize an alternate framework of an EnKF are proposed here. The newly proposed integrated hybrid assimilation algorithms for an EnKF approximate the static background error covariance matrix by using a set of climatological ensemble perturbations. Similar ideas are proposed by Counillon et al. (2009) for ocean data assimilation and Sakov et al. (2018) for ensemble assimilation with additive model error, but mainly focusing on the update of ensemble mean. Compared to the commonly used hybrid assimilation methods that calculate the hybrid analysis increment by either averaging the static and flow-dependent background error covariances or averaging the Kalman gains of a variational method and an EnKF, the integrated hybrid assimilation algorithms can update both the ensemble mean and ensemble perturbations by supplementing information of the approximated static background error covariance with the flow-dependent background error covariance from an ensemble. Compared to EVIL that generates posterior ensemble members from the variational minimization procedure, the integrated hybrid assimilation algorithms do not require large number of iterations, and can be easily implemented for an existing EnKF system.

The structure of this paper is as follows. Section 2 describes various data assimilation algorithms, which include the ensemble square root filter (EnSRF; Whitaker and Hamill 2002), hybrid covariance data assimilation method (HCDA; Hamill and Snyder 2000), hybrid gain data assimilation method (HGDA; Penny 2014), and the two newly proposed integrated hybrid data assimilation methods, integrated hybrid ensemble Kalman filter with augmented perturbations (IHCEnKF) and integrated hybrid ensemble Kalman filter with climate perturbations (IHGEnKF). The one-scale and two-scale models described by Lorenz (2005, hereafter L05) and experimental design are discussed in section 3. Section 4 presents the results and discussion. The conclusions are summarized in section 5.

2. Data assimilation methods

To find the best estimate of state vector x for a dynamic system, ensemble-based data assimilation methods use short-term ensemble forecasts to compute the flow-dependent background error covariance P f , while variational data assimilation methods use the static background error covariance matrix B . Various data assimilation methods that are examined and proposed here are described below. Table 1 summarizes the major differences among the data assimilation methods.

Table 1.

Descriptions of data assimilation methods.


Table 1.

a. Ensemble square root filter (EnSRF)

The ensemble square root filter (EnSRF; Whitaker and Hamill 2002) is a deterministic flavor of the ensemble Kalman filter (EnKF; Evensen 1994). The EnSRF uses the traditional Kalman gain to update the ensemble mean and a reduced Kalman gain to update the ensemble perturbations. Given N ensemble forecast members x i f ( i = 1 , , N ) , x ¯ f = ( 1 / N ) i = 1 N x i f is the prior ensemble mean, x i f = x i f x ¯ f is the ith ensemble perturbation corresponding to the ith ensemble forecast member x i f , and X = ( x 1 f , x 2 f , , x N f ) / N 1 whose multiplication of its own tranpose is the sample background error covariance matrix P f = XX T, is the square root of P f .

By assimilating observations y with observation error covariance matrix R , the posterior ensemble mean x ¯ a is given by
x ¯ a = x ¯ f + ρ o { X ( HX ) T [ ( HX ) ( HX ) T + R ] 1 } [ y H ( x f ) ¯ ] = x ¯ f + ρ o [ P f H T ( H P f H T + R ) 1 ] [ y H ( x f ) ¯ ] ,
where H is the observation forward operator, H is the Jacobian matrix that is the partial derivative of the observation operator with respect to the model state, and ρ ο is localization matrix in observation space with dimension k × p (k and p are numbers of state variables and observations), and denotes the Schur (elementwise) product. The ith (i = 1,…, N) ensemble perturbation is updated by
x i a = x i f ρ o { X ( HX ) T [ ( ( HX ) ( HX ) T + R ) 1 ] T ×  [ ( HX ) ( HX ) T + R + R ] 1 } H x i f = x i f ρ o { P f H T [ ( H P f H T + R ) 1 ] T ×  [ ( H P f H T + R ) + R ] 1 } H x i f .
The traditional Kalman gain matrix:
K = X ( HX ) T [ ( HX ) ( HX ) T + R ] 1 = P f H T ( H P f H T + R ) 1 ,
and the reduced Kalman gain matrix:
K ˜ = X ( HX ) T { [ ( HX ) ( HX ) T + R ] 1 } T ×  [ ( HX ) ( HX ) T + R + R ] 1 = P f H T [ ( H P f H T + R ) 1 ] T ×  [ ( H P f H T + R ) + R ] 1 ,
are matrices with dimension k × p.

b. Hybrid covariance data assimilation (HCDA)

The three-dimensional variational method (3DVAR; Lorenc 1986) solves for the best estimate of state vector x by minimizing the cost function:
J ( x ) = ( x x f ) T B 1 ( x x f ) + [ y H ( x ) ] T R 1 [ y H ( x ) ] ,
where x f is a deterministic forecast, and B is the static background error covariance matrix. Hamill and Snyder (2000) proposed the hybrid covariance method that linearly combines the static background error covariance B and the localized sample background error covariance P f , so that the cost function (5) can be modified to
J ( x ) = ( x x f ) T [ α B + ( 1 α ) ρ m P f ] 1 ( x x f ) + [ y H ( x ) ] T R 1 [ y H ( x ) ] ,
where α is the hybrid weight that determines the relative weighting for B and P f , and ρ m is localization matrix in model space with dimension k × k. Wang et al. (2007) showed the equivalence of a similar hybrid covariance approach to the control-variable method of Lorenc (2003).

c. Hybrid gain data assimilation (HGDA)

Instead of combing background error covariance matrices B and P f , Penny (2014) proposed the hybrid gain method that combines the gain matrices of the 3DVAR and EnKF methods. The gain matrix of 3DVAR minimizes the cost function J in (5) and can be written as
K B = B H T ( HB H T + R ) 1 .
As (6) in Penny (2014), the general hybrid gain matrix is constructed by linearly combining the gain matrices of 3DVAR and EnKF:
K ^ = β 1 ρ o K + β 2 K B + β 3 K B H ( ρ o K ) ,
where β 1, β 2, and β 3 are the weighting coefficients. To compare with the hybrid covariance method HCDA, the weighting coefficients are chosen as β 1 = 1 − α, β 2 = α, and β 3 = 0.
Using the hybrid gain matrix, the ensemble-mean update in (1) can be written as
x ¯ a = x ¯ f + K ^ [ y H ( x f ) ¯ ] = x ¯ f + [ ( 1 α ) ρ o K + α K B ] [ y H ( x f ) ¯ ] .
By splitting x ¯ f into two parts with weights of (1 − α) and α, the hybrid gain matrix can further lead to hybrid mean for the ensemble mean update, as
x ¯ a = ( 1 α ) { x ¯ f + ρ o K [ y H ( x f ) ¯ ] } + α { x ¯ f + K B [ y H ( x f ) ¯ ] } .

Despite the weighting coefficient, the first term on the right-hand-side of (10) is the ensemble-mean update of EnKF, and the other term is the 3DVAR solution when the prior ensemble mean is used as the background. The weighting coefficient α balances the corrections made by 3DVAR and EnKF, which plays a similar role as the weighting coefficient of the hybrid covariance method as in (6).

d. Integrated hybrid ensemble Kalman filter with augmented perturbations (IHCEnKF)

HCDA uses the variational solver to find the best estimate of x that minimizes the cost function (6) with hybrid covariance of P f and B . To operate a similar hybrid covariance approach in an ensemble framework rather than a variational framework, an integrated hybrid ensemble Kalman filter with augmented perturbations (IHCEnKF) is proposed here. The schematic illustration of IHCEnKF is shown by Fig. 1a.

Fig. 1.
Fig. 1.

Schematic illustrations of (a) IHCEnKF and (b) IHGEnKF. For IHCEnKF, cycling and climatological perturbations need also be multiplied by N + N c 1 , because N + N c 1 is cancelled when computing the background error covariance.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

A set of climatological ensemble perturbations X c = [ x c , 1 f , x c , 2 f , , x c , N c f ] / N c 1 gives a climatological background error covariance matrix P c = X c X c T ; N c is the ensemble size of climatological perturbations, which needs to be larger than that of cycling ensemble perturbations N. The climatological background error covariance matrix P c is used to approximate the static background error covariance matrix B . Similar to the modulation approach proposed by Bishop and Hodyss (2009), Bocquet (2016), Lei et al. (2018), and Farchi and Bocquet (2019), the climatological ensemble perturbations X c with a weighting coefficient α are attached to the cycling ensemble perturbations X with a weighting coefficient 1 α . The augmented ensemble perturbations are now
X ^ = [ 1 α N 1 ( x 1 f , x 2 f , , x N f ) , α N c 1 ( x c , 1 f , x c , 2 f , , x c , N c f ) ] ,
which is the square root of the hybrid background error covariance matrix P ^ f = X ^ X ^ T . Given the augmented ensemble perturbations (11), the hybrid background error covariance matrix can be separated to two parts, P ^ f = ( 1 α ) P f + α P c , which is the same as the hybrid background error covariance matrix in (6) of HCDA.
Using the augmented ensemble perturbations, the update equation for the ensemble mean is
x ¯ a = x ¯ f + ρ o { X ^ ( H X ^ ) T [ ( H X ^ ) ( H X ^ ) T + R ] 1 } [ y H ( x f ) ¯ ] = x ¯ f + ρ o [ P ^ f H T ( H P ^ f H T + R ) 1 ] [ y H ( x f ) ¯ ] ,
and the update equation for the augmented ensemble perturbations is
x a = x f ρ o { X ^ ( H X ^ ) T [ ( ( H X ^ ) ( H X ^ ) T + R ) 1 ] T ×  [ ( H X ^ ) ( H X ^ ) T + R + R ] 1 } H x f = x f ρ o { P ^ f H T [ ( H P ^ f H T + R ) 1 ] T ×  [ ( H P ^ f H T + R ) + R ] 1 } H x f .

By adding the first N updated perturbations (i.e., the cycling ensemble perturbations) with a multiplication of N 1 / 1 α to the updated ensemble mean, N posterior ensemble members that are consistent with the N prior ensemble members are obtained. These N posterior ensemble members are then used to advance the model to the next assimilation cycle (Fig. 1a).

Compared to HCDA, the IHCEnKF uses a set of climatological ensemble perturbations to construct a background error covariance, which with localization applied approximates the static background error covariance B . Given the same x ¯ f in (12) as x f in (6) and a sufficiently large size of climatological ensemble perturbations in (11) that approximates the static background error covariance B , the posterior ensemble mean x ¯ a given by (12) should be very similar to the solution x a that minimizes (6), despite the impact of localization (Lei and Whitaker 2015). Compared to HCDA in which the static background error covariance has no influence on ensemble perturbations, IHCEnKF is able to update the ensemble perturbations with the hybrid background error covariance. The influences caused by updating the ensemble perturbations with the hybrid background error covariance in the IHCEnKF can be accumulated through cycling data assimilation and subsequently lead to different ensemble mean update from the HCDA.

Compared to the EnSRF, the IHCEnKF approach combines the sample background error covariances with the approximated static background error covariances through the augmented ensemble perturbations. But IHCEnKF requires no modification of the EnSRF code, although the computational cost of data assimilation increases since the ensemble size increases from N to N + N c .

e. Integrated hybrid ensemble Kalman filter with additive perturbations (IHGEnKF)

HGDA combines the gain matrices of the 3DVAR and EnKF methods, but a variational solver for the 3DVAR is still needed. To operate a similar hybrid gain approach in a purely ensemble framework, an integrated hybrid ensemble Kalman filter with additive climatological perturbations (IHGEnKF) is proposed. Similar to IHCEnKF, IHGEnKF also uses a set of climatological ensemble perturbations X c to approximate the static background error covariance B , but IHGEnKF keeps the climatological ensemble perturbations X c and cycling ensemble perturbations X separately. The schematic illustration of IHGEnKF is shown by Fig. 1b.

Given X c and the corresponding climatological background error covariance matrix P c , a Kalman gain matrix that approximates K B in (7) can be formed similarly to (3) as
K c = X c ( H X c ) T [ ( H X c ) ( H X c ) T + R ] 1 = P c H T ( H P c H T + R ) 1 ,
and a reduced Kalman gain can be formed similarly to (4) as
K ˜ c = X c ( H X c ) T { [ ( H X c ) ( H X c ) T + R ] 1 } T ×  [ ( H X c ) ( H X c ) T + R + R ] 1 = P c H T [ ( H P c H T + R ) 1 ] T ×  [ ( H P c H T + R ) + R ] 1 .
Thus the ensemble mean can be updated by a hybrid gain matrix that is a linear combination of K and K c , given by K ^ = ( 1 α ) ρ o K + α ρ o K c . Then the update equation for the ensemble mean is
x ¯ a = ( 1 α ) { x ¯ f + ρ o K [ y H ( x f ) ¯ ] } + α { x ¯ f + ρ o K c [ y H ( x f ) ¯ ] } ,
which is similar to (10) of HGDA. If the cycling ensemble perturbations are updated by the reduced Kalman gain K ˜ while the climatological ensemble perturbations are updated by the reduced Kalman gain K ˜ c , the IHGEnKF is the same as the HGDA, except that the climatological background error covariance matrix P c is used to approximate the static background error covariance matrix B . However, IHGEnKF can update the ensemble perturbations with a hybrid reduced gain matrix, K ˜ ^ = ( 1 α ) ρ o K ˜ + α ρ o K ˜ c . The update equation for the ensemble perturbations is
x a = ( 1 α ) ( x f ρ o KH ˜ x f ) + α ( x f ρ o K ˜ c H x f ) .

Although the posterior ensemble mean given by (16) is similar to that of (10), the influences of the hybrid reduced gain matrix on the cycling ensemble perturbations can be accumulated through cycling data assimilation and have impacts on the ensemble mean update.

3. Model and experimental design

a. The L05 model

The model II described in L05 contains a single large scale (slow) variable X. The one-scale model II is governed by
d X n d t = [ X , X ] K , n X n + F ,
where the subscript n indexes the grid point, K is a constant, and F is the forcing parameter. Given a total number of grid points of N = 960, the grid spacing is 0.375°. The constant K should be much smaller than N, which is chosen as 32. Parameter F is a constant forcing, which varies for different experiments. The advection term [X, X] K,n is formulated by
[ X , X ] K , n = W n 2 K W n K + j = J J ' W n K + j X n + K + j / K ,
where W n = Σ i = J ' J X n i / K with J = K/2 when K is even and W n = Σ i = J J X n i / K with J = (K − 1)/2 when K is odd. The special sum Σ′ is the same as the ordinary sum except that the first and last terms are divided by 2.
The one-scale model II can be extended to the two-scale model III (L05) by including a fast variable Y. The two-scale model III is written as
d Z n d t = [ X , X ] K , n + b 2 [ Y , Y ] 1 , n + c [ Y , X ] 1 , n X n b Y n + F ,
in which Z is the integration variable. Coefficient b = 10 determines the relative frequency and amplitude of Y compared to X. Coupling coefficient c = 3 gives the strength of coupling between X and Y. The construction of X and Y through Z is given by
X n = i = I I ' ( α β | i | ) Z n + i and
Y n = Z n X n .

The smoothing scale I is chosen as 12. The constants α and β are chosen such that X n will equal Z n whenever Z varies quadratically over the interval nI through n + I. The two-scale model III in (20) reduces to the one-scale model II in (18) when the fast variable Y is eliminated.

b. Experimental design

The data assimilation methods described in section 2 (Table 1) are evaluated in both one-scale model II and two-scale model III, to consider the influence from the fast variable. Imperfect-model experiments are conducted, in which model error is included by varying the parameter F. The nature run has F of 15, and assimilation experiments have F of 16 and 18 that represents small and large model errors, respectively. Consistent results are also obtained for perfect-model experiments, although differences among data assimilation methods are smaller than those with model errors (figures are not shown).

Initial conditions of the nature run and ensembles are randomly drawn from a large set of independent states. Synthetic observations are created by adding random perturbations drawn from a normal distribution N(0, R) to the true values from the nature run. The observation error R is the diagonal element of the diagonal observation error covariance matrix R . Synthetic observations contain correlations adopted from the dynamical system but have no added correlated observation errors. The default observation error variance R is set to 1.0. The default observing network is every 4 grid points (240 observing locations). Synthetic observations are created every 50 time steps (~6 h).

The default ensemble size of cycling assimilation N is 40. The default ensemble size of climatological perturbations N c is 800. To combat the sampling error resulted from a limited ensemble size, covariance inflation and localization are applied. The constant multiplicative inflation (Anderson and Anderson 1999) is applied to enlarge the ensemble spread and prevent filter divergence. The Gaspari and Cohn (GC; Gaspari and Cohn 1999) function is used to localize the impact of observations and mitigate the spurious error correlations between observations and state variables. The GC localization function is a fifth-order piecewise rational function, and its width is determined by a single real parameter.

A group of data assimilation experiments for each data assimilation method described in section 2 are conducted for 360 days, with default settings and varying assimilation parameters (inflation, localization, and hybrid weight α). The first 10 days are discarded to avoid transients, and the last 350 days are used to obtain the optimal assimilation parameters. The manually tuned optimal assimilation parameters are shown in Table 2. Sensitivity experiments that vary one assimilation parameter but keep the other assimilation parameters as the default (ensemble sizes of cycling members and climatological perturbations, observation error variance and observation density) and optimal choices (inflation, localization and hybrid weight) are also conducted for 360 days. Assimilation cycles from the last 350 days are used for evaluation.

Table 2.

The manually tuned optimal assimilation parameters for each data assimilation method of model II (upper row for each experiment) and model III (lower row for each experiment) with parameter F of 16 (left value in each column) and 18 (right boldface value in each column).


Table 2.

4. Results

a. Comparisons among various data assimilation approaches

The integrated hybrid ensemble Kalman filters (IHCEnKF and IHGEnKF) can update both the ensemble mean and ensemble perturbations by the hybrid background error covariance. To demonstrate the advantages of updating the ensemble perturbations with the hybrid background error covariance, an intermediate solution of IHGEnKF (IHGEnKF-Mean) that updates the ensemble mean by the hybrid gain matrix K ^ and updates the ensemble perturbations by the reduced gain matrix K ˜ , is performed. It is not straightforward to compute the intermediate solution for IHCEnKF, since IHCEnKF has climatological perturbations augmented to cycling perturbations and the augmented perturbations are used to update both the ensemble mean and perturbations. Table 3 shows the mean root-mean-square errors (RMSEs) of model II with F = 16 for different data assimilation methods. IHGEnKF-Mean has slightly smaller RMSE to HGDA, which indicates that the static background error covariance can be approximated by the climatological ensemble perturbations in an EnKF. IHGEnKF-Mean is not equivalent to HGDA, since IHGEnKF-Mean uses the hybrid gain matrix K ^ to update ensemble mean for serially assimilated observations while HGDA has hybrid analyses after all observations are assimilated. IHGEnKF further reduces the RMSE comparing to IHGEnKF-Mean and HGDA. Thus updating the ensemble perturbations by the hybrid background error covariance can be beneficial for hybrid data assimilation.

Table 3.

The mean RMSEs averaged from 350 days for experiments HGDA, IHGEnKF-Mean, and IHGEnKF of model II with F = 16.


Table 3.

Figure 2 displays the times series of RMSEs from various data assimilation approaches for models II and III with different magnitudes of model errors. When only slow variables exist, hybrid data assimilation methods (HCDA and HGDA) have smaller RMSEs than EnSRF. This demonstrates the advantages of updating ensemble mean with the hybrid background error covariance compared to the pure sample background error covariance (Hamill and Snyder 2000; Etherton and Bishop 2004; Wang et al. 2009). The integrated hybrid ensemble Kalman filters, IHCEnKF and IHGEnKF, obtain similar RMSEs, and both produce smaller RMSEs than HCDA and HGDA. Thus the advantages of updating the ensemble perturbations with the hybrid background error covariance through either augmented climatological perturbations or additive climatological perturbations have been proved. Similar results are obtained when fast variables are included, which indicates that the advantages of using the hybrid background error covariance to update the ensemble perturbations are retained even with fast variables.

Fig. 2.
Fig. 2.

Time series of RMSEs with different data assimilation approaches for (a) model II with F = 16, (b) model II with F = 18, (c) model III with F = 16, and (d) model III F = 18. The bars on the right side of each panel denote the mean RMSEs.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

b. Sensitivities to data assimilation parameters

Sensitivity experiments with varying data assimilation parameters are displayed for model III, since the realistic atmosphere contains both slow and fast variables. Consistent results are obtained for model II (figures are not shown). Ensemble size, observation error variance and observation density follow the default configuration, and the inflation, localization and hybrid weight are optimally tuned based on the default configuration. Sensitivities of the integrated hybrid EnKFs to the size of climatological ensemble perturbations are first shown in Fig. 3. RMSEs of IHCEnKF and IHGEnKF quickly decrease when the size of climatological ensemble perturbations increases from 10 to 200, for both small and large magnitudes of model errors. IHCEnKF and IHGEnKF have RMSEs slightly reduced when the size of climatological ensemble perturbations increases larger than 200, while the RMSE reduction is more obvious with large magnitude of model error than that with small magnitude of model error. Figure 4 shows the variances and mean correlations estimated from the climatological ensemble perturbations. Compared to the variances of static background error covariances, the estimated variances from 80 climatological ensemble perturbations have larger variations over the model grid points, while the domain averaged variance is slightly smaller than those with larger sizes of climatological ensemble perturbations and the static background error covariance. The estimated variances with 200 or more climatological ensemble perturbations are similar to those of the static background error covariance. The mean correlation length scales estimated from climatological ensemble perturbations with sizes from 80 to 2000 are similar to those of the static background error covariance. Thus with an order of hundreds of climatological ensemble perturbations, the climatological background error covariance used by the integrated hybrid EnKFs can sufficiently approximate the static background error covariance.

Fig. 3.
Fig. 3.

The mean RMSEs averaged over 350 days for the integrated hybrid ensemble Kalman filters using model III for (a) F = 16 and (b) F = 18, with varying ensemble sizes of climatological perturbations.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1


Fig. 4.
Fig. 4.

Estimated (a) variances and (b) mean correlations with different sizes of climatological ensemble perturbations. The black solid lines denote the variances and mean correlations of the static background error covariance. The bars on the right side of (a) denote the domain-averaged variances.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

Figure 5a shows the mean RMSEs with different ensemble sizes for model III with F = 16. For all data assimilation methods (EnSRF, HCDA, HGDA, IHCEnKF and IHGEnKF), the RMSEs decrease with increasing ensemble sizes. But when ensemble size increases from 80 to 200, the RMSEs are not further reduced. This is possibly due to the inflation and localization parameters tuned optimally for 40 ensemble members, since less inflation and broader localization are expected when ensemble size increases. HCDA and HGDA produce smaller RMSEs than EnSRF, especially with small ensemble sizes. IHCEnKF and IHGEnKF obtain smaller RMSEs than EnSRF, HCDA and HGDA with different ensemble sizes. The advantages of IHCEnKF and IHGEnKF over the other data assimilation methods are persistent, from small to large ensemble sizes. By optimally tuning assimilation parameters (inflation, localization and hybrid weight) for each ensemble size, the RMSEs of each assimilation method slightly decrease with ensemble size increasing from 80 to 200, while the comparisons among assimilation methods are hold (figures not shown). Consistent results are also obtained with large magnitude of model error (F = 18, Fig. 5b). The differences between the hybrid data assimilation methods and EnSRF, and the differences between the integrated hybrid EnKFs and hybrid data assimilation methods, are increased when the model error is enlarged. Therefore, given different magnitudes of model error, the advantages of the integrated hybrid EnKFs over the hybrid data assimilation methods are retained with varying ensemble sizes.


Fig. 5.
Fig. 5.

The mean RMSEs averaged over 350 days for different data assimilation methods using model III for (a) F = 16 and (b) F = 18, with varying ensemble sizes.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

Figure 6 shows the sensitivities of different data assimilation approaches to multiplicative inflation. The default ensemble sizes of cycling members and climatological perturbations, observation error variance and observation density, and the optimal hybrid weight based on the default configuration are used. With different magnitudes of model error, IHCEnKF and IHGEnKF have smaller RMSEs than the other data assimilation methods when inflation is small, and all data assimilation approached obtain similar RMSEs when inflation is large. The data assimilation approaches are less sensitive to inflation when the model error is smaller, thus inflation used by ensemble-based assimilation methods can account for system error (Houtekamer and Mitchell 2005). Whitaker et al. (2008) showed that additive inflation can better capture the model error than multiplicative inflation. Therefore, sensitivities of data assimilation approaches on additive inflation are conducted here, while the other configurations are the same as the sensitivity experiments of multiplicative inflation. Consistent with the results with additive inflation, the RMSEs of IHCEnKF and IHGEnKF are smaller than the other data assimilation methods for small inflation values, and the RMSEs of all data assimilation methods become similar for large inflation values (Fig. 7). For each data assimilation method, the RMSEs with additive inflation are slightly smaller than those with multiplicative inflation, especially for the RMSE of EnKF with large model error. Since a combination of multiplicative and additive inflation could further improve the performance of EnKF (Whitaker and Hamill 2012), the error differences between the integrated hybrid EnKFs and the EnKF might decrease when model error is large and both multiplicative and additive inflation are applied. For both multiplicative inflation and additive inflation, the integrated hybrid EnKFs are superior to the hybrid data assimilation methods with varying values of inflation.


Fig. 6.
Fig. 6.

As in Fig. 5, but for varying multiplicative inflation values.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1


Fig. 7.
Fig. 7.

As in Fig. 5, but for varying additive inflation values.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

Figure 8 displays the sensitivities of different data assimilation approaches on localization, with the optimal multiplicative inflation based on the default configuration. When a small magnitude of model error is included, HCDA and HGDA have smaller RMSEs than EnSRF, and IHCEnKF and IHGEnKF further reduce the RMSEs than HCDA and HGDA, with varying localization values. Similar results are obtained when a large magnitude of model error is introduced. Compared to EnSRF, HCDA and HGDA are less sensitive to localization (Fig. 8), and they generally have larger optimal localization length scales (Table 2). This is because that static background error covariance B incorporated by HCDA and HGDA often having larger scales than the sample background error covariance P f and meanwhile a fraction of sample background error covariance P f from the cycling ensemble requires less localization than the full P f . IHCEnKF and IHGEnKF are more sensitive to localization than EnSRF (Fig. 8), and the optimal localization values of IHCEnKF and IHGEnKF are generally smaller than those of EnSRF (Table 2). This is because besides the cycling ensemble perturbations, the climatological ensemble perturbations incorporated by IHCEnKF and IHGEnKF are also impacted by sampling errors. With varying values of localization, the integrated hybrid EnKFs are superior to the hybrid data assimilation methods.


Fig. 8.
Fig. 8.

As in Fig. 5, but for varying localization length scales.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

Sensitivities of the data assimilation methods to the hybrid weight are shown in Fig. 9. HCDA and IHCEnKF prefer small amounts of the hybrid weight. IHCEnKF has smaller RMSEs than HCDA when the hybrid weight is less than 0.1. When the hybrid weight is larger than 0.1, IHCEnKF and HCDA produce similar RMSEs with small magnitude of model error, and IHCEnKF has slightly larger RMSEs than HCDA with large magnitude of model error. With different magnitudes of model error, HGDA and IHGEnKF are less sensitive to the hybrid weight than HCDA and IHCEnKF. IHGEnKF produces smaller RMSEs than HGDA when the hybrid weight is larger than 0 and smaller than 0.2 (0.3) for a small (large) magnitude of model error. IHGEnKF has larger RMSEs than HGDA with large values of the hybrid weight. Thus in general, IHCEnKF (IHGEnKF) has similar sensitivity of the hybrid weight to HCDA (HGDA), and the integrated hybrid EnKFs have smaller RMSEs than the hybrid data assimilation methods when the optimal hybrid weight is chosen.


Fig. 9.
Fig. 9.

As in Fig. 5, but for varying hybrid weights.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

Sensitivities of the data assimilation methods to the observation error variance (R) are also examined. As shown by Fig. 10, when the magnitude of model error is small, HGDA has slightly smaller RMSEs than EnSRF, and HCDA has smaller RMSEs than HGDA, with R smaller than 1.0. EnSRF, HCDA and HGDA obtain similar RMSEs with R larger than 1.0. The integrated hybrid EnKFs have smaller RMSEs than the other data assimilation methods when R is smaller than 2.0. When R is larger than 2.0, IHCEnKF has similar RMSE to EnSRF and the hybrid data assimilation methods, while IHGEnKF has larger RMSEs than the other data assimilation methods. When the magnitude of model error is large, EnSRF has larger RMSEs than the hybrid data assimilation methods and integrated hybrid EnKFs, with different values of R. When R is smaller than 2.0, IHCEnKF and IHGEnKF have small RMSEs than the hybrid data assimilation methods. When R is larger than 2.0, IHGEnKF (IHCEnKF) has smaller (larger) RMSEs than the hybrid data assimilation methods. Thus the integrated hybrid EnKFs are superior to the hybrid data assimilation methods when R is smaller than 2.0.


Fig. 10.
Fig. 10.

As in Fig. 5, but for varying observation error variances.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

Figure 11 shows the sensitivities of the data assimilation methods to the observation density. When the magnitude of model error is small, IHGEnKF has larger RMSEs than the other data assimilation methods with sparse observations (observations are fewer than 120). When observation density increases, the integrated hybrid EnKFs produce smaller RMSEs than the hybrid data assimilation methods, while EnSRF has the largest RMSEs. When the magnitude of model error is large, similar results are obtained, except that IHGEnKF has smaller RMSE than EnSRF with observation grids fewer than 120. The advantages of the integrated hybrid EnKFs over the hybrid data assimilation methods are more evident with a larger magnitude of model error. Therefore, given different magnitudes of model error, the integrated hybrid EnKFs produce smaller RMSEs than the hybrid data assimilation methods and EnSRF with denser observing networks.

Fig. 11.
Fig. 11.

As in Fig. 5, but for varying numbers of observing grids.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

As shown by Figs. 2 and 511, the differences between the hybrid data assimilation methods and EnSRF, and the differences between the integrated hybrid EnKFs and hybrid data assimilation methods, are increased when the model error is enlarged. With larger model errors, the benefits to utilize the hybrid background error covariance for ensemble mean update are more prominent, because the hybrid background error covariance can provide potential error growing modes to ensemble mean. This is a similar role to additive inflation that takes model error into account (Whitaker et al. 2008). With larger model errors, the benefits to utilize the hybrid background error covariance are further enlarged when the hybrid background error covariance is also used to update ensemble perturbations. Thus the hybrid background error covariance can also provide potential error growing modes to ensemble perturbations.

c. Computational cost

The climatological ensemble perturbations used by the integrated hybrid EnKFs can be drawn from previous cycling ensemble experiments or control simulations. Thus no additional computational cost is needed for generating the sample set of climatological ensemble perturbations, although additional storage for the sample set of climatological ensemble perturbations is required. Given N cycling ensemble members, data assimilation experiments have the same computational cost for advancing N members forward. But data assimilation experiments have different computational costs for the assimilation step. Figure 12 shows the computational costs of assimilation for different data assimilation methods, with default configuration of 40 cycling ensemble members, observing every four grid points and observation error variance as 1.0, and the optimal inflation, localization and hybrid weight based on the default configuration. EnSRF, HCDA and HGDA have similar computational costs, although HCDA and HGDA have slightly larger computational costs than EnSRF. IHCEnKF and IHGEnKF have similar computational costs, and both have similar computational costs to EnSRF when the size of climatological ensemble perturbations is small. The computational costs of IHCEnKF and IHGEnKF significantly increase with the increased size of climatological ensemble perturbations. This is as expected, because compared to EnSRF that updates N ensemble members, IHCEnKF has the ensemble size of assimilation directly increased to N + N c , and IHGEnKF has additional computations for K c and K ˜ c and also updating N c climatological ensemble perturbations.


Fig. 12.
Fig. 12.

Computational cost for different data assimilation methods with varying sizes of climatological ensemble perturbations.

Citation: Monthly Weather Review 149, 12; 10.1175/MWR-D-21-0002.1

Although computational costs of assimilation for IHCEnKF and IHGEnKF are much increased compared to EnSRF and hybrid methods, IHCEnKF and IHGEnKF use the same code structure as EnSRF, which can be well scaled with parallel computing. Thus it is practical to apply IHCEnKF and IHGEnKF with hundreds of climatological ensemble perturbations for realistic geophysical applications.

5. Discussion and conclusions

Two integrated hybrid EnKFs that are able to update both the ensemble mean and ensemble perturbations by the hybrid background error covariance in the framework of EnKF are proposed here. Both the integrated hybrid EnKFs approximate the static background error covariance by the background error covariance estimated from climatological ensemble perturbations. The integrated hybrid EnKF with augmented perturbations (IHCEnKF) appends the climatological ensemble perturbations to the cycling ensemble per turbations, which gives the hybrid background error covariance similar to the hybrid data assimilation with hybrid background error covariance (HCDA). The integrated hybrid EnKF with additive perturbations (IHGEnKF) computes the Kalman gain by the cycling ensemble perturbations and climatological ensemble perturbations, respectively, which is similar to the hybrid data assimilation with hybrid gains (HGDA). Compared to the hybrid data assimilation methods (HCDA and HGDA) that have ensemble perturbations updated by an EnKF with sample background error covariances, the integrated hybrid EnKFs (IHCEnKF and IHGEnKF) can update the ensemble perturbations using the hybrid background error covariance. The integrated hybrid EnKFs also avoid the inconsistencies that result from performing separate hybrid assimilation and EnKF systems.

The integrated hybrid EnKFs are tested in the L05 model II and III with different magnitudes of model error. The IHGEnKF with the ensemble mean updated by the hybrid background error covariance and ensemble perturbations up dated by the sample background error covariance (IHGEnKF-Mean) produces similar errors to HGDA, which indicates that the background error covariance estimated from climatological perturbations can approximate the static background error covariance. IHGEnKF further reduces the error than IHGEnKF-Mean and HGDA, thus it is beneficial to update the ensemble perturbations by the hybrid background error covariance. IHCEnKF and IHGEnKF outperform HCDA and HGDA for model II and model III with small and large magnitudes of model error. The advantages of updating the ensemble perturbations with the hybrid background error covariance through either augmented climatological perturbations or additive climatological perturbations have been demonstrated.

Sensitivity experiments with varying data assimilation parameters for the integrated hybrid EnKFs are examined. The integrated hybrid EnKFs can sufficiently approximate the static background error covariance with an order of hundreds of climatological perturbations. The advantages of the integrated hybrid EnKFs over the hybrid data assimilation are kept with varying ensemble sizes, inflation values and localization length scales. The integrated hybrid EnKFs are superior to the hybrid data assimilation methods when observation error variance is smaller than 2.0 and number of observing grids is larger than 120. IHCEnKF (IHGEnKF) has similar sensitivity of the hybrid weight to HCDA (HGDA), and the integrated hybrid EnKFs have smaller errors than the hybrid data assimilation methods when the optimal hybrid weight is applied.

Using the L05 model II and III, an order of hundreds of climatological ensemble perturbations is needed to sufficiently approximate the static background error covariance in the integrated hybrid EnKFs. Thus, at least hundreds of climatological ensemble perturbations are required for realistic applications. Additional computational costs and storage required for the integrated hybrid EnKFs also need to be taken into accounted for realistic applications. The sensitivity to climatological ensemble size needs further investigation with realistic models. Moreover, the same localization length scale is used here for both the cycling ensemble and climatological ensemble. However, a broader localization could be favorable for the climatological ensemble than the cycling ensemble, since the ensemble size of climatological ensemble is much larger than that of cycling ensemble. For IHGEnKF, it is straightforward to implement different localization length scales for cycling and climatological ensemble perturbations. For IHCEnKF, different localization length scales for cycling and climatological ensemble perturbations can be achieved by the modulation approach (Bishop et al. 2017; Lei et al. 2018). By implementing different localization length scales for the cycling and climatological ensembles, the integrated hybrid EnKFs could be further improved, which will be presented in a future study.

Acknowledgments

This work is supported by the National Key Research and Development Program of China under Grant 2017YFC1501603, the National Outstanding Youth Science Fund Project of National Natural Science Foundation of China under Grant 41922036, the Frontiers Science Center for Critical Earth Material Cycling Fund JBGS2102, and the Fundamental Research Funds for the Central Universities 0209-14380097. Thanks to three anonymous reviewers who help to significantly improve an earlier version.

REFERENCES

  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903, https://doi.org/10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 27412758, https://doi.org/10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Auligné, T., B. Ménétrier, A. C. Lorenc, and M. Buehner, 2016: Ensemble–variational integrated localized data assimilation. Mon. Wea. Rev., 144, 36773696, https://doi.org/10.1175/MWR-D-15-0252.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., and D. Hodyss, 2009: Ensemble covariances adaptively localized with ECO-RAP. Part 2: A strategy for the atmosphere. Tellus, 61A, 97111, https://doi.org/10.1111/j.1600-0870.2008.00372.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., J. S. Whitaker, and L. Lei, 2017: Gain form of the ensemble transform Kalman filter and its relevance to satellite data assimilation with model space ensemble covariance localization. Mon. Wea. Rev., 145, 45754592, https://doi.org/10.1175/MWR-D-17-0102.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bocquet, M., 2016: Localization and the iterative ensemble Kalman smoother. Quart. J. Roy. Meteor. Soc., 142, 10751089, https://doi.org/10.1002/qj.2711.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bonavita, M., L. Isaksen, and E. Hólm, 2012: On the use of EDA background error variances in the ECMWF 4D-VAR. Quart. J. Roy. Meteor. Soc., 138, 15401559, https://doi.org/10.1002/qj.1899.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bonavita, M., E. Hólm, L. Isaksen, and M. Fisher, 2016: The evolution of the ECMWF hybrid data assimilation system. Quart. J. Roy. Meteor. Soc., 142, 287303, https://doi.org/10.1002/qj.2652.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buehner, M., J. Morneau, and C. Charette, 2013: Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction. Nonlinear Processes Geophys., 20, 669682, https://doi.org/10.5194/npg-20-669-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Burgers, G., P. J. van Leeuwen, and G. Evensen, 1998: Analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126, 17191724, https://doi.org/10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Campbell, W. F., C. H. Bishop, and D. Hodyss, 2010: Vertical covariance localization for satellite radiances in ensemble Kalman filters. Mon. Wea. Rev., 138, 282290, https://doi.org/10.1175/2009MWR3017.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Clayton, A. M., A. C. Lorenc, and D. M. Barker, 2013: Operational implementation of a hybrid ensemble/4D-VAR global data assimilation system at the Met Office. Quart. J. Roy. Meteor. Soc., 139, 14451461, https://doi.org/10.1002/qj.2054.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Counillon, F., P. Sakov, and L. Bertino, 2009: Application of a hybrid EnKF-OI to ocean forecasting. Ocean Sci., 5, 389401, https://doi.org/10.5194/os-5-389-2009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Etherton, B. J., and C. H. Bishop, 2004: Resilience of hybrid ensemble/3DVAR analysis schemes to model error and ensemble covariance error. Mon. Wea. Rev., 132, 10651080, https://doi.org/10.1175/1520-0493(2004)132<1065:ROHDAS>2.0.CO;2.

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

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Farchi, A., and M. Bocquet, 2019: On the efficiency of covariance localisation of the ensemble Kalman filter using augmented ensembles. Front. Appl. Math. Stat., 5, 3, https://doi.org/10.3389/fams.2019.00003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gaspari, G., and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723757, https://doi.org/10.1002/qj.49712555417.

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

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter–3D variational analysis scheme. Mon. Wea. Rev., 128, 29052919, https://doi.org/10.1175/1520-0493(2000)128<2905:AHEKFV>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 2005: Ensemble Kalman filtering. Quart. J. Roy. Meteor. Soc., 131, 32693289, https://doi.org/10.1256/qj.05.135.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112126, https://doi.org/10.1016/j.physd.2006.11.008.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kalnay, E., 2002: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 341 pp.

  • Kleist, D. T., and K. Ide, 2015: An OSSE-based evaluation of hybrid variational–ensemble data assimilation for the NCEP GFS. Part II: 4DEnVar and hybrid variants. Mon. Wea. Rev., 143, 452470, https://doi.org/10.1175/MWR-D-13-00350.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kuhl, D. D., T. E. Rosmond, C. H. Bishop, J. McClay, and N. L. Baker, 2013: Comparison of hybrid ensemble/4DVAR and 4DVAR within the NAVDAS-AR data assimilation framework. Mon. Wea. Rev., 141, 27402758, https://doi.org/10.1175/MWR-D-12-00182.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lei, L., and J. S. Whitaker, 2015: Model space localization is not always better than observation space localization for assimilation of satellite radiances. Mon. Wea. Rev., 143, 39483955, https://doi.org/10.1175/MWR-D-14-00413.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lei, L., J. S. Whitaker, and C. H. Bishop, 2018: Improving assimilation of radiance observations by implementing model space localization in an ensemble Kalman filter. J. Adv. Model. Earth Syst., 10, 32213232, https://doi.org/10.1029/2018MS001468.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liu, C., and Q. Xiao, 2013: An ensemble-based four-dimensional variational data assimilation scheme. Part III: Antarctic applications with Advanced Research WRF using real data. Mon. Wea. Rev., 141, 27212739, https://doi.org/10.1175/MWR-D-12-00130.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liu, C., Q. Xiao, and B. Wang, 2008: An ensemble-based four-dimensional variational data assimilation scheme. Part I: Technical formulation and preliminary test. Mon. Wea. Rev., 136, 33633373, https://doi.org/10.1175/2008MWR2312.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liu, C., Q. Xiao, and B. Wang, 2009: An ensemble-based four-dimensional variational data assimilation scheme. Part II: Observing system simulation experiments with Advanced Research WRF. Mon. Wea. Rev., 137, 16871704, https://doi.org/10.1175/2008MWR2699.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112, 11771194, https://doi.org/10.1002/qj.49711247414.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., 2003: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-VAR. Quart. J. Roy. Meteor. Soc., 129, 31833203, https://doi.org/10.1256/qj.02.132.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1996: Predictability: A problem partly solved. Proc. Seminar on Predictability, Vol. 1, Reading, United Kingdom, ECMWF, https://www.ecmwf.int/node/10829.

  • Lorenz, E. N., 2005: Designing chaotic models. J. Atmos. Sci., 62, 15741587, https://doi.org/10.1175/JAS3430.1.

  • Penny, S. G., 2014: The hybrid local ensemble transform Kalman filter. Mon. Wea. Rev., 142, 21392149, https://doi.org/10.1175/MWR-D-13-00131.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Penny, S. G., D. W. Behringer, J. A. Carton, and E. Kalnay, 2015: A hybrid global ocean data assimilation system at NCEP. Mon. Wea. Rev., 143, 46604677, https://doi.org/10.1175/MWR-D-14-00376.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Penny, S. G., E. Bach, K. Bhargava, C.-C. Chang, C. Da, L. Sun, and T. Yoshida, 2019: Strongly coupled data assimilation in multiscale media: Experiments using a quasi-geostrophic coupled model. J. Adv. Model. Earth Syst., 11, 18031829, https://doi.org/10.1029/2019MS001652.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sakov, P., J. M. Haussaire, and M. Bocquet, 2018: An iterative ensemble Kalman filter in presence of additive model error. Quart. J. Roy. Meteor. Soc., 144, 12971309, https://doi.org/10.1002/qj.3213.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., C. Snyder, and T. M. Hamill, 2007: On the theoretical equivalence of differently proposed ensemble–3D-VAR hybrid analysis schemes. Mon. Wea. Rev., 135, 222227, https://doi.org/10.1175/MWR3282.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., D. M. Barker, C. Snyder, and T. M. Hamill, 2008: A hybrid ETKF–3DVAR data assimilation scheme for the WRF model. Part I: Observing system simulation experiments. Mon. Wea. Rev., 136, 51165131, https://doi.org/10.1175/2008MWR2444.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., T. M. Hamill, J. S. Whitaker, and C. H. Bishop, 2009: A comparison of the hybrid and EnSRF analysis schemes in the presence of model errors due to unresolved scales. Mon. Wea. Rev., 137, 32193232, https://doi.org/10.1175/2009MWR2923.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924, https://doi.org/10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., and T. M. Hamill, 2012: Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev., 140, 30783089, https://doi.org/10.1175/MWR-D-11-00276.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., T. M. Hamill, X. Wei, Y. Song, and Z. Toth, 2008: Ensemble data assimilation with the NCEP global forecast system. Mon. Wea. Rev., 136, 463482, https://doi.org/10.1175/2007MWR2018.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, M., and F. Zhang, 2012: E4DVAR: Coupling an ensemble Kalman filter with four-dimensional variational data assimilation in a limited-area weather prediction model. Mon. Wea. Rev., 140, 587600, https://doi.org/10.1175/MWR-D-11-00023.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save
  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903, https://doi.org/10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 27412758, https://doi.org/10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Auligné, T., B. Ménétrier, A. C. Lorenc, and M. Buehner, 2016: Ensemble–variational integrated localized data assimilation. Mon. Wea. Rev., 144, 36773696, https://doi.org/10.1175/MWR-D-15-0252.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., and D. Hodyss, 2009: Ensemble covariances adaptively localized with ECO-RAP. Part 2: A strategy for the atmosphere. Tellus, 61A, 97111, https://doi.org/10.1111/j.1600-0870.2008.00372.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bishop, C. H., J. S. Whitaker, and L. Lei, 2017: Gain form of the ensemble transform Kalman filter and its relevance to satellite data assimilation with model space ensemble covariance localization. Mon. Wea. Rev., 145, 45754592, https://doi.org/10.1175/MWR-D-17-0102.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bocquet, M., 2016: Localization and the iterative ensemble Kalman smoother. Quart. J. Roy. Meteor. Soc., 142, 10751089, https://doi.org/10.1002/qj.2711.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bonavita, M., L. Isaksen, and E. Hólm, 2012: On the use of EDA background error variances in the ECMWF 4D-VAR. Quart. J. Roy. Meteor. Soc., 138, 15401559, https://doi.org/10.1002/qj.1899.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bonavita, M., E. Hólm, L. Isaksen, and M. Fisher, 2016: The evolution of the ECMWF hybrid data assimilation system. Quart. J. Roy. Meteor. Soc., 142, 287303, https://doi.org/10.1002/qj.2652.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buehner, M., J. Morneau, and C. Charette, 2013: Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction. Nonlinear Processes Geophys., 20, 669682, https://doi.org/10.5194/npg-20-669-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Burgers, G., P. J. van Leeuwen, and G. Evensen, 1998: Analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126, 17191724, https://doi.org/10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Campbell, W. F., C. H. Bishop, and D. Hodyss, 2010: Vertical covariance localization for satellite radiances in ensemble Kalman filters. Mon. Wea. Rev., 138, 282290, https://doi.org/10.1175/2009MWR3017.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Clayton, A. M., A. C. Lorenc, and D. M. Barker, 2013: Operational implementation of a hybrid ensemble/4D-VAR global data assimilation system at the Met Office. Quart. J. Roy. Meteor. Soc., 139, 14451461, https://doi.org/10.1002/qj.2054.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Counillon, F., P. Sakov, and L. Bertino, 2009: Application of a hybrid EnKF-OI to ocean forecasting. Ocean Sci., 5, 389401, https://doi.org/10.5194/os-5-389-2009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Etherton, B. J., and C. H. Bishop, 2004: Resilience of hybrid ensemble/3DVAR analysis schemes to model error and ensemble covariance error. Mon. Wea. Rev., 132, 10651080, https://doi.org/10.1175/1520-0493(2004)132<1065:ROHDAS>2.0.CO;2.

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

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Farchi, A., and M. Bocquet, 2019: On the efficiency of covariance localisation of the ensemble Kalman filter using augmented ensembles. Front. Appl. Math. Stat., 5, 3, https://doi.org/10.3389/fams.2019.00003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gaspari, G., and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723757, https://doi.org/10.1002/qj.49712555417.

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

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter–3D variational analysis scheme. Mon. Wea. Rev., 128, 29052919, https://doi.org/10.1175/1520-0493(2000)128<2905:AHEKFV>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houtekamer, P. L., and H. L. Mitchell, 2005: Ensemble Kalman filtering. Quart. J. Roy. Meteor. Soc., 131, 32693289, https://doi.org/10.1256/qj.05.135.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112126, https://doi.org/10.1016/j.physd.2006.11.008.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kalnay, E., 2002: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 341 pp.

  • Kleist, D. T., and K. Ide, 2015: An OSSE-based evaluation of hybrid variational–ensemble data assimilation for the NCEP GFS. Part II: 4DEnVar and hybrid variants. Mon. Wea. Rev., 143, 452470, https://doi.org/10.1175/MWR-D-13-00350.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kuhl, D. D., T. E. Rosmond, C. H. Bishop, J. McClay, and N. L. Baker, 2013: Comparison of hybrid ensemble/4DVAR and 4DVAR within the NAVDAS-AR data assimilation framework. Mon. Wea. Rev., 141, 27402758, https://doi.org/10.1175/MWR-D-12-00182.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lei, L., and J. S. Whitaker, 2015: Model space localization is not always better than observation space localization for assimilation of satellite radiances. Mon. Wea. Rev., 143, 39483955, https://doi.org/10.1175/MWR-D-14-00413.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lei, L., J. S. Whitaker, and C. H. Bishop, 2018: Improving assimilation of radiance observations by implementing model space localization in an ensemble Kalman filter. J. Adv. Model. Earth Syst., 10, 32213232, https://doi.org/10.1029/2018MS001468.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liu, C., and Q. Xiao, 2013: An ensemble-based four-dimensional variational data assimilation scheme. Part III: Antarctic applications with Advanced Research WRF using real data. Mon. Wea. Rev., 141, 27212739, https://doi.org/10.1175/MWR-D-12-00130.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liu, C., Q. Xiao, and B. Wang, 2008: An ensemble-based four-dimensional variational data assimilation scheme. Part I: Technical formulation and preliminary test. Mon. Wea. Rev., 136, 33633373, https://doi.org/10.1175/2008MWR2312.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liu, C., Q. Xiao, and B. Wang, 2009: An ensemble-based four-dimensional variational data assimilation scheme. Part II: Observing system simulation experiments with Advanced Research WRF. Mon. Wea. Rev., 137, 16871704, https://doi.org/10.1175/2008MWR2699.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112, 11771194, https://doi.org/10.1002/qj.49711247414.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenc, A. C., 2003: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-VAR. Quart. J. Roy. Meteor. Soc., 129, 31833203, https://doi.org/10.1256/qj.02.132.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1996: Predictability: A problem partly solved. Proc. Seminar on Predictability, Vol. 1, Reading, United Kingdom, ECMWF, https://www.ecmwf.int/node/10829.

  • Lorenz, E. N., 2005: Designing chaotic models. J. Atmos. Sci., 62, 15741587, https://doi.org/10.1175/JAS3430.1.

  • Penny, S. G., 2014: The hybrid local ensemble transform Kalman filter. Mon. Wea. Rev., 142, 21392149, https://doi.org/10.1175/MWR-D-13-00131.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Penny, S. G., D. W. Behringer, J. A. Carton, and E. Kalnay, 2015: A hybrid global ocean data assimilation system at NCEP. Mon. Wea. Rev., 143, 46604677, https://doi.org/10.1175/MWR-D-14-00376.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Penny, S. G., E. Bach, K. Bhargava, C.-C. Chang, C. Da, L. Sun, and T. Yoshida, 2019: Strongly coupled data assimilation in multiscale media: Experiments using a quasi-geostrophic coupled model. J. Adv. Model. Earth Syst., 11, 18031829, https://doi.org/10.1029/2019MS001652.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sakov, P., J. M. Haussaire, and M. Bocquet, 2018: An iterative ensemble Kalman filter in presence of additive model error. Quart. J. Roy. Meteor. Soc., 144, 12971309, https://doi.org/10.1002/qj.3213.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., C. Snyder, and T. M. Hamill, 2007: On the theoretical equivalence of differently proposed ensemble–3D-VAR hybrid analysis schemes. Mon. Wea. Rev., 135, 222227, https://doi.org/10.1175/MWR3282.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., D. M. Barker, C. Snyder, and T. M. Hamill, 2008: A hybrid ETKF–3DVAR data assimilation scheme for the WRF model. Part I: Observing system simulation experiments. Mon. Wea. Rev., 136, 51165131, https://doi.org/10.1175/2008MWR2444.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X., T. M. Hamill, J. S. Whitaker, and C. H. Bishop, 2009: A comparison of the hybrid and EnSRF analysis schemes in the presence of model errors due to unresolved scales. Mon. Wea. Rev., 137, 32193232, https://doi.org/10.1175/2009MWR2923.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924, https://doi.org/10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., and T. M. Hamill, 2012: Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev., 140, 30783089, https://doi.org/10.1175/MWR-D-11-00276.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., T. M. Hamill, X. Wei, Y. Song, and Z. Toth, 2008: Ensemble data assimilation with the NCEP global forecast system. Mon. Wea. Rev., 136, 463482, https://doi.org/10.1175/2007MWR2018.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, M., and F. Zhang, 2012: E4DVAR: Coupling an ensemble Kalman filter with four-dimensional variational data assimilation in a limited-area weather prediction model. Mon. Wea. Rev., 140, 587600, https://doi.org/10.1175/MWR-D-11-00023.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Schematic illustrations of (a) IHCEnKF and (b) IHGEnKF. For IHCEnKF, cycling and climatological perturbations need also be multiplied by N + N c 1 , because N + N c 1 is cancelled when computing the background error covariance.

  • Fig. 2.

    Time series of RMSEs with different data assimilation approaches for (a) model II with F = 16, (b) model II with F = 18, (c) model III with F = 16, and (d) model III F = 18. The bars on the right side of each panel denote the mean RMSEs.

  • Fig. 3.

    The mean RMSEs averaged over 350 days for the integrated hybrid ensemble Kalman filters using model III for (a) F = 16 and (b) F = 18, with varying ensemble sizes of climatological perturbations.

  • Fig. 4.

    Estimated (a) variances and (b) mean correlations with different sizes of climatological ensemble perturbations. The black solid lines denote the variances and mean correlations of the static background error covariance. The bars on the right side of (a) denote the domain-averaged variances.

  • Fig. 5.

    The mean RMSEs averaged over 350 days for different data assimilation methods using model III for (a) F = 16 and (b) F = 18, with varying ensemble sizes.

  • Fig. 6.

    As in Fig. 5, but for varying multiplicative inflation values.

  • Fig. 7.

    As in Fig. 5, but for varying additive inflation values.

  • Fig. 8.

    As in Fig. 5, but for varying localization length scales.

  • Fig. 9.

    As in Fig. 5, but for varying hybrid weights.

  • Fig. 10.

    As in Fig. 5, but for varying observation error variances.

  • Fig. 11.

    As in Fig. 5, but for varying numbers of observing grids.

  • Fig. 12.

    Computational cost for different data assimilation methods with varying sizes of climatological ensemble perturbations.

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