A new structure for error covariance matrices and their adaptive estimation in EnKF assimilation

Authors


Abstract

Correct estimation of the forecast and observational error covariance matrices is crucial for the accuracy of a data assimilation algorithm. In this article we propose a new structure for the forecast error covariance matrix to account for limited ensemble size and model error. An adaptive procedure combined with a second-order least squares method is applied to estimate the inflated forecast and adjusted observational error covariance matrices. The proposed estimation methods and new structure for the forecast error covariance matrix are tested on the well-known Lorenz-96 model, which is associated with spatially correlated observational systems. Our experiments show that the new structure for the forecast error covariance matrix and the adaptive estimation procedure lead to improvement of the assimilation results. Copyright © 2012 Royal Meteorological Society

1.  Introduction

Data assimilation is a procedure for producing an optimal combination of model outputs and observations. The combined result should be closer to the true state than either the model forecast or the observation are. However, the quality of data assimilation depends crucially on the estimation accuracy of the forecast and observational error covariance matrices. If these matrices are estimated appropriately, then the analysis states can be generated by minimizing an objective function, which is technically straightforward and can be accomplished using existing engineering solutions (Reichle, 2008).

The ensemble Kalman filter (EnKF) is a popular sequential data assimilation approach, which has been widely studied and applied since its introduction by Evensen (1994a, 1994b). In EnKF, the forecast error covariance matrix is estimated as the sampling covariance matrix of the ensemble forecast states, which is usually underestimated due to the limited ensemble size and model error. This may eventually lead to the divergence of the EnKF assimilation scheme (e.g. Anderson and Anderson, 1999; Constantinescu et al., 2007).

One of the forecast error covariance matrix inflation techniques is additive inflation, in which a noise is added to the ensemble forecast states that samples the probability distribution of model error (Hamill and Whitaker, 2005). Another widely used forecast error covariance matrix inflation technique is multiplicative inflation, that is, to multiply the matrix by an appropriate factor.

In early studies of multiplicative inflation, researchers determined the inflation factor by repeated experimentation and chose a value according to their prior knowledge. Hence such experimental tuning is rather empirical and subjective. Wang and Bishop (2003) proposed an on-line estimation method for the inflation factor of the forecast error covariance matrix in a model with a linear observational operator. Building on that work, Li et al.(2009) further developed the algorithm. All these methods are based on the first moment estimation of the squared observation-minus-forecast residual, which was first introduced by Dee (1995). Anderson (2007, 2009) used a Bayesian approach to covariance matrix inflation for the spatially independent observational errors, and Miyoshi (2011) further simplified Anderson's inflation approach by making a number of additional simplifying assumptions.

In practice, the observational error covariance matrix may also need to be adjusted (Liang et al., 2011). Zheng (2009) and Liang et al.(2011) proposed an approach to simultaneously optimize the inflation factor of the forecast error covariance matrix and the adjustment factor of the observational error covariance matrix. Their approach is based on the optimization of the likelihood function of the observation-minus-forecast residual, an idea proposed by Dee and colleagues (Dee and Da Silva, 1999; Dee et al., 1999). However, the likelihood function of the observation-minus-forecast residual is nonlinear and involves the computationally expensive determinant and inverse of the residual's covariance matrix. In this article, the second-order least squares (SLS; Wang and Leblanc, 2008) statistic of the squared observation-minus-forecast residual is introduced as the objective function instead. The main advantage of the SLS objective function is that it is a quadratic function of the factors, and therefore the closed forms of the estimators of the inflation factors can be obtained. Compared with the method proposed by Liang et al.(2011), the computational cost is greatly reduced.

Another innovation of this article is to propose a new structure for the forecast error covariance matrix that is different from the sampling covariance matrix of the ensemble forecast states used in the conventional EnKF. In the ideal situation, an ensemble forecast state is assumed to be a random vector with the true state as its population mean. Hence it is more appropriate to define the ensemble forecast error by the ensemble forecast states minus true state rather than by the perturbed forecast states minus their ensemble mean (Evensen, 2003). This is because in a model with large error and limited ensemble size, the ensemble mean of the forecast states can be very different from the true state. Therefore, the sampling covariance matrix of the ensemble forecast states can be very different from the true forecast error covariance matrix. As a result, the estimated analysis state can be substantially inaccurate. However, in reality the true state is unknown, and the analysis state is a better estimate of the true state than the forecast state. Therefore, in this article we propose to use the information feedback from the analysis state to update the forecast error covariance matrix. In fact, our proposed forecast error covariance matrix is a combination of multiplicative and additive inflation. Bai and Li (2011) also used the feedback from the analysis state to improve assimilation but in a different way.

This article consists of four sections. Section 2 proposes an EnKF scheme with a new structure for the forecast error covariance matrix and its adaptive estimation procedure based on the second-order least squares method. Section 3 presents the assimilation results on the Lorenz model with a correlated observational system. Conclusions and discussion are provided in section 4.

2.  Methodology

2.1.  EnKF with SLS inflation

Using the notations of Ide et al.(1997), a nonlinear discrete-time forecast and linear observational system is written as

equation image(1)

and

equation image(2)

where i is the time index; equation image{equation image is the n-dimensional true state vector at time step equation image is the n-dimensional analysis state vector which is an estimate of equation image is a nonlinear forecast operator such as a weather forecast model; equation image is an observational vector with dimension pi;Hi is an observational matrix of dimension pi × n that maps model states to the observational space, ηi and εi are the forecast error vector and the observational error vector respectively, which are assumed to be statistically independent of each other, time-uncorrelated, and have mean zero and covariance matrices Pi and Ri respectively. The goal of the EnKF assimilation is to find a series of analysis states equation image that are sufficiently close to the corresponding true states equation image, using the information provided by Mi and equation image.

It is well-known that any EnKF assimilation scheme should include a forecast error inflation scheme. Otherwise, the EnKF may diverge (Anderson and Anderson, 1999). In this article, a procedure for estimating the multiplicative inflation factor of Pi and adjustment factor of Ri is proposed based on the SLS principle (Wang and Leblanc, 2008). The basic filter algorithm in this article uses perturbed observations (Burgers et al., 1998) without localization (Houtekamer and Mitchell, 2001). The estimation steps of this algorithm equipped with SLS inflation are as follows.

  • (1)Calculate the perturbed forecast states
    equation image(3)
    where equation image is the perturbed analysis state derived from the previous time step (1 ≤ jm, with m the number of ensemble members).
  • (2)Estimate the inflated forecast and adjusted observational error covariance matrices.Define the forecast state equation image to be the ensemble mean of equation image and suppose that the initial forecast error covariance matrix is
    equation image(4)
    and the initial observational error covariance matrix is Ri. Then, the adjusted forms of forecast and observational error covariance matrices are equation image and μiRi respectively.There are several approaches for estimating the inflation factor λi and adjustment factor μi. For example, Wang and Bishop (2003), Li et al.(2009) and Miyoshi (2011) use the first order least square of the squared observation-minus-forecast residual equation image to estimate λi; Liang et al.(2011) maximize the likelihood of di to estimate λi and μi. In this article, we propose to use the SLS approach for estimating λi and μi. That is, λi and μi are estimated by minimizing the objective function
    equation image(5)
    This leads to
    equation image(6)
    and
    equation image(7)
    (See Appendix A for detailed derivation.) Similar to Wang and Bishop (2003) and Li et al.(2009), this procedure does not use the Bayesian approach (Anderson 2007, 2009; Miyoshi 2011).
  • (3)Compute the perturbed analysis states
    equation image(8)
    where εi,j′ is a normal random variable with mean zero and covariance matrix equation image (Burgers et al., 1998). Here equation image can be effectively calculated using the Sherman–Morrison–Woodbury formula (Golub and Van Loan, 1996; Tippett et al., 2003; Liang et al., 2011). Further, the analysis state equation image is estimated as the ensemble mean of equation image. Finally, set i = i + 1 and return to step (1) for the assimilation at next time step.

2.2.  EnKF with SLS inflation and new structure for forecast error covariance matrix

By Eqs (1) and (3), the ensemble forecast error is equation image is an estimate of equation image without knowing observations. The ensemble forecast error is initially estimated as equation image, which is used to construct the forecast error covariance matrix in section 2.1. However, due to limited ensemble size and model error, equation image can be biased. Therefore, equation image can be a biased estimate of equation image.

In this article, we propose to use observations for improving the estimation of the ensemble forecast error. The idea is as follows: after analysis state equation image is derived, it is generally a better estimate of equation image than the forecast state equation image. So equation image in Eq. (4) is substituted by equation image for updating the forecast error covariance matrix. This procedure is repeated iteratively until the corresponding objective function (Eq. (5)) converges. For the computational details, step (2) in section 2.1 is modified to the following adaptive procedure.

(2a) Use step (2) in section 2.1 to inflate the initial forecast error covariance matrix to equation image and adjust initial observational error covariance matrix to equation image. Then use step (3) in section 2.1 to estimate the initial analysis state equation image and set k = 1.

(2b) Update the forecast error covariance matrix as

equation image(9)

Then, adjust the forecast and observational error covariance matrices to equation image and equation image, where

equation image(10)

and

equation image(11)

are estimated by minimizing the objective function

equation image(12)

If equation image, where δ is a predetermined threshold to control the convergence of Eq. (12), then estimate the k-th updated analysis state as

equation image(13)

set k = k + 1 and return to Eq. (9); otherwise, accept equation image and equation image as the estimated forecast and observational error covariance matrices at the ith time step, and go to step (3) in section 2.1.

A flowchart of our proposed assimilation scheme is shown in Figure 1. Moreover, our proposed forecast error covariance matrix (Eq. (9)) can be expressed as

equation image(14)

which is a multiplicatively inflated sampling error covariance matrix plus an additive inflation matrix (see Appendix B for the proof).

Figure 1.

Flowchart of our proposed assimilation scheme.

2.3.  Notes

2.3. 1.  Correctly specified observational error covariance matrix

If the observational error covariance matrix Ri is correctly known, then its adjustment is no longer required. In this case, the inflation factor equation image can be estimated by minimizing the following objective function

equation image(15)

This leads to a simpler estimate

equation image(16)

2.3. 2.  Smoothing observational adjustment factor

If the observational error covariance matrix Ri is assumed to be time invariant, then the adjustment factor μi should also be time invariant. In this case we propose to estimate μi as

equation image(17)

where K is the number of previous assimilation time steps that involve smoothing adjustment factor of Ri,μi is estimated using Eq. (11) and equation image are previous estimates using the smoothing procedure. A larger K does not increase the computational burden, but if K is too small, the smoothing effect may be weakened. In this article, K is selected as 10. As we demonstrate later, the estimated equation image using Eq. (11) has relatively large error and formula (17) gives a much better estimate.

2.3. 3.  Validation statistics

In any toy model, the ‘true’ state equation image is known by experimental design. In this case, we can use the root-mean-square error (RMSE) of the analysis state to evaluate the accuracy of the assimilation results. The RMSE at the ith step is defined as

equation image(18)

where ‖ · ‖ represents the Euclidean norm and n is the dimension of the state vector. A smaller RMSE indicates a better performance of the assimilation scheme.

3.  Experiment on the Lorenz-96 model

In this section we apply our proposed data assimilation schemes to a nonlinear dynamical system with properties relevant to realistic forecast problems: the Lorenz-96 model (Lorenz, 1996) with model error and a linear observational system. We evaluate the performances of the assimilation schemes in section 2 through the following experiments.

3.1.  Description of dynamic and observational systems

The Lorenz-96 model (Lorenz, 1996) is a strongly nonlinear dynamical system with quadratic nonlinearity, governed by the equation

equation image(19)

where k = 1,2,···,K (K = 40, hence there are 40 variables). For Eq. (19) to be well defined for all values of k, we define X−1 = XK−1,X0 = XK,XK+1 = X1. The dynamics of Eq. (19) are ‘atmosphere-like’ in that the three terms on the right-hand side consist of a nonlinear advection-like term, a damping term and an external forcing term respectively. These terms can be thought of as some atmospheric quantity (e.g. zonal wind speed) distributed on a latitude circle.

In our assimilation schemes, we set F = 8, so that the leading Lyapunov exponent implies an error-doubling time of about eight time steps, and the fractal dimension of the attractor is 27.1 (Lorenz and Emanuel, 1998). The initial condition is chosen to be Xk = F when k≠20 and X20 = 1.001F. We solve Eq. (19) using a fourth-order Runge-Kutta time integration scheme (Butcher, 2003) with a time step of 0.05 non-dimensional unit to derive the true state. This is roughly equivalent to 6 h in real time, assuming that the characteristic time-scale of the dissipation in the atmosphere is 5 days (Lorenz, 1996).

In this study, we assume the synthetic observations are generated at every model grid point by adding random noises that are multivariate normally distributed with mean zero and covariance matrix Ri to the true states. The leading diagonal elements of Ri are equation image and the off-diagonal elements at site pair (j,k) are

equation image(20)

By considering spatially correlated observational errors, the scheme may potentially be applied for assimilating remote sensing observations and radiances data.

We added model errors in the Lorenz-96 model because it is inevitable in real dynamic systems. Thus, we chose different values of F in our assimilation schemes, while retaining F = 8 when generating the ‘true’ state. We simulate observations every four time steps for 100000 steps to ensure robust results (Sakov and Oke, 2008; Oke et al., 2009). The ensemble size is selected as 30. The predetermined threshold δ to control the convergence of Eq. (12) is set to be 1, because the values of objective functions are in the order of 105. In most cases of the following experiment, the objective functions converge after 3–4 iterations, and the estimated analysis states also converge.

3.2.  Comparison of assimilation schemes

In section 2.1 we outlined the EnKF assimilation scheme with SLS error covariance matrix inflation. In section 2.2, we summarized the EnKF assimilation scheme with the SLS error covariance matrix inflation and the new structure for the forecast error covariance matrix. In this section, we assess the impacts of these estimation methods on EnKF data assimilation schemes using the Lorenz-96 model.

The Lorenz-96 model is a forced dissipative model with a parameter F that controls the strength of the forcing (Eq. (19)). The model behaves quite differently with different values of F and produces chaotic systems with integer values of F larger than 3. As such, we used a set of values of F to simulate a wide range of model errors. In all cases, the true states were generated by a model with F = 8. These observations were then assimilated into models with F = 4, 5, , 12.

3.2.1.  Correctly specified observational error covariance matrix

Suppose the observational error covariance matrix Ri is correctly specified, we first take inflation adjustment on equation image in each assimilation cycle and estimate the inflation factor λi by the method described in section 2.1. Then we conduct the adaptive assimilation scheme with the new structure for the forecast error covariance matrix proposed in section 2.2.

Figure 2 shows the time-mean analysis RMSE of the two assimilation schemes averaged over 100000 time steps, as a function of F. Overall, the analysis RMSE of the two assimilation schemes gradually grows when the model error increases. When F is around the true value 8, the two assimilation schemes have almost indistinguishable values of the analysis RMSE. However, when F becomes increasingly distant from 8, the analysis RMSE of the assimilation scheme with the new structure for the forecast error covariance matrix becomes progressively smaller than that of the assimilation scheme with the forecast error covariance matrix inflation only.

Figure 2.

Time-mean values of the analysis RMSE as a function of forcing F when observational errors are spatially correlated and their covariance matrix is correctly specified, by using three EnKF schemes. (1) SLS only (solid line, described in section 2.1); (2) SLS and new structure (dashed line, described in section 2.2); and (3) SLS and true ensemble forecast error (dotted line, described in section 2.2).

For the Lorenz-96 model with large error (F = 12), the time-mean analysis RMSEs of the two assimilation schemes are listed in Table 1, as well as the time-mean values of the objective functions. The EnKF assimilation scheme is also included for comparison. These results show clearly that our two schemes have significantly smaller RMSEs than the EnKF assimilation scheme. Moreover, the assimilation scheme with the new structure of the forecast error covariance matrix performs much better than the assimilation scheme with forecast error covariance matrix inflation only.

Table 1. The time-mean analysis RMSE and the time-mean objective function values in four EnKF schemes for the Lorenz-96 model when observational errors are spatially correlated and their covariance matrix is correctly specified: (1) EnKF (non-inflation); (2) the SLS scheme in section 2.1 (SLS); (3) the SLS scheme in section 2.2 (SLS and new structure); (4) the SLS scheme in the discussion (SLS and true ensemble forecast error). The forcing term F = 12.
EnKF schemesTime-mean RMSETime-mean L
Non-inflation5.652298754
SLS1.89148468
SLS and new structure1.2238125
SLS and true ensemble forecast error0.4819652

3.2.2.  Incorrectly specified observational error covariance matrix

In this section, we suppose that the observational error covariance matrix is correct only up to a constant factor. We estimate this factor using different estimation methods and evaluate the corresponding assimilation results.

We set the observational error covariance matrix Ri as four times the true matrix and introduce another factor μi to adjust Ri. We conduct the assimilation scheme in four cases: (i) inflate forecast and adjust observational error covariance matrices only (section 2.1); (ii) inflate forecast and adjust observational error covariance matrices and use the smoothing technique for μi (section 2.3.2) where the number K (the smoothing interval) is 10; (iii) inflate forecast and adjust observational error covariance matrices and use the new structure for the forecast error covariance matrix (section 2.2); (iv) inflate forecast and adjust observational error covariance matrices with smoothing μi and use the new structure for the forecast error covariance matrix. The model parameter F again takes values 4, 5, , 12 when assimilating observations, but F = 8 is used when generating the true states in all cases.

Figure 3 shows the time-mean analysis RMSE of the four cases averaged over 100000 time steps, as a function of F. Overall, the analysis RMSE of the four cases gradually grows when the model error increases. However, the analysis RMSE in the cases using the new structure for the forecast error covariance matrix (cases 3 and 4) are smaller than those in the cases using the error covariance matrix inflation technique only (cases 1 and 2). Moreover, using the smoothing technique for estimating the adjustment factor μi (cases 2 and 4) can improve the assimilation result to some extent.

Figure 3.

Time-mean values of the analysis RMSE as a function of forcing F when observational errors are spatially correlated and their covariance matrix is incorrectly specified, by using five EnKF schemes. (1) SLS only (thick solid line); (2) SLS with smoothing μi (thick dashed line); (3) SLS and new structure (thin solid line); (4) SLS with smoothing μi and new structure (thin dashed line); and (5) SLS and true ensemble forecast error (dotted line).

For the Lorenz-96 model with model parameter F = 12, the time-mean analysis RMSE of the four cases are listed in Table 2, along with the time-mean values of the objective functions. These results show clearly that when the observational error covariance matrix is specified incorrectly, the assimilation result is much better if the new structure for the forecast error covariance matrix is used (cases 3 and 4). Moreover, the assimilation result can be further improved by using the smoothing technique to estimate μi (cases 2 and 4).

Table 2. The time-mean analysis RMSE and the time-mean objective function values in five EnKF schemes for Lorenz-96 model when observational errors are spatially correlated and their covariance matrix is incorrectly specified: (1) SLS; (2) SLS with smoothing μi; (3) SLS and new structure; (4) SLS with smoothing μi and new structure; (5) SLS and true ensemble forecast error. The forcing term F = 12.
EnKF schemesEnsemble size 30Ensemble size 20
 Time-mean RMSETime-mean L Time-mean RMSETime-mean L
SLS2.4314265413.511492685
SLS with smoothing μi2.251276432.86207964
SLS and new structure1.35413261.4595685
SLS with smoothing μi and new structure1.22379531.4058466
SLS and true ensemble forecast error0.58215850.6021355

The estimated equation image over 100000 time steps in the two cases of using the new structure of the forecast error covariance matrix (cases 3 and 4) are plotted in Figure 4. It can be seen that the time-mean value of estimated equation image is 0.75 in case 3, but is 0.36 after using the smoothing technique (case 4). The latter is closer to the reciprocal of the constant that we multiplied to the observational error covariance matrix (0.25).

Figure 4.

The times series of estimated equation image when observational error covariance matrix is incorrectly specified (solid line) and using smoothing technique (dashed line). The dotted line is the correct scale (0.25) of the observational error covariance matrix.

To investigate the effect of ensemble size on the assimilation result, Figure 3 is reproduced with the ensemble size 20. The results are shown in Figure 5 as well as in Table 2. Generally speaking, Figures 5 is similar to Figure 3, but with larger analysis error. This indicates that the smaller ensemble size corresponds to the larger forecast error and the analysis error. We also repeated our analysis with the ensemble size 10. However in this case, both the inflation and new structure are not effective. This could be due to the ensemble size 10 being too small to generate robust covariance estimation.

Figure 5.

Similar to Figure 3, but ensemble size is 20.

4.  Discussion and conclusions

It is well-known that accurately estimating the error covariance matrix is one of the most important steps in data assimilation. In the EnKF assimilation scheme, the forecast error covariance matrix is estimated as the sampling covariance matrix of the ensemble forecast states. Due to the limited ensemble size and model error, however, the forecast error covariance matrix is usually underestimated, which may lead to the divergence of the filter. So we multiplied the estimated forecast error covariance matrix by an inflation factor λi and proposed the SLS estimation for this factor.

In fact, the true forecast error should be represented as the ensemble forecast states minus the true state. However, since in real problems the true state is not available, we use the ensemble mean of the forecast states instead. Consequently the forecast error covariance matrix is represented as the sampling covariance matrix of the ensemble forecast states. If the model error is large, however, the ensemble mean of the forecast states may be far from the true state. In this case, the estimated forecast error covariance matrix will remain far from the truth, no matter which inflation technique is used.

To verify this point, a number of EnKF assimilation schemes with necessary error covariance matrix inflation are applied to the Lorenz-96 model, but with the forecast state equation image in the forecast error covariance matrix (Eq. (4)) substituted by the true state equation image. The corresponding RMSEs are shown in Figures 2, 3 and 5 and Tables 1 and 2. All the figures and tables show that the analysis RMSE is significantly reduced.

However, since the true state equation image is unknown, we use the analysis state equation image to replace the forecast state equation image, because equation image is closer to equation image than equation image. To achieve this goal, a new structure for the forecast error covariance matrix and an adaptive procedure for estimating the new structure are proposed here to iteratively improve the estimation. As shown in section 3, the RMSEs of the corresponding analysis states are indeed smaller than those of the EnKF assimilation scheme with the error covariance matrix inflation only. For example, in the experiment in section 3.1, when the error covariance matrix inflation technique is applied, the RMSE is 1.89, which is much smaller than that for the original EnKF. When we further used the new structure of the forecast error covariance matrix in addition to the inflation, the RMSE is reduced to 1.22 (see Table 1).

In practice, the observational error covariance matrix is not always known correctly, and hence needs to be adjusted too. Another factor μi is introduced to adjust the observational error covariance matrix, which can be estimated simultaneously with λi by minimizing the second-order least squares function of the squared observation-minus-forecast residual. If the observational error covariance matrices are time invariant, a smoothing technique (Eq. (17)) can be used to improve the estimation. Our experiment in section 3.2 shows that the smoothing technique plays an important role in retrieving the correct scale of the observational error covariance matrix.

The second-order least squares function of the squared observation-minus-forecast residual seems to be a good objective function to quantify the goodness of fit of the error covariance matrix. The SLS can be used to estimate the factors for adjusting both the forecast and observational error covariance matrices, while the first order method can estimate only the inflation factor of the forecast error covariance matrix. The SLS can also provide a stopping criterion for the iteration in the adaptive estimation procedure when the new structure of the forecast error covariance matrix is used. This is important for preventing the proposed forecast error covariance matrix to depart from the truth in the iteration. In most cases in this study, the minimization algorithms converge after 3–4 iterations and the objective function decreases sharply. On the other hand, the improved forecast error covariance matrix indeed leads to the improvement of analysis state. In fact, as shown in Tables 1 and 2, a small objective function value always corresponds to a small RMSE of the analysis state.

We also investigated the difference between our proposed scheme and the assimilation scheme proposed by Liang et al.(2011). Generally speaking, the RMSE of the analysis state derived using the MLE inflation scheme proposed by Liang et al.(2011) is slightly smaller than that derived using the SLS inflation scheme only, but is larger than that derived using the SLS inflation with the new structure for forecast error covariance matrix. For example, for the Lorenz-96 model with forcing F = 12, the RMSE is 1.69 for MLE inflation (table 3 in Liang et al., 2011), 1.89 for SLS inflation only and 1.22 for SLS inflation and new structure (Table 1). Whether this is a general rule or not is unclear, and is subject to further investigation. However, in the MLE inflation scheme, the objective function is nonlinear and especially involves the determinant of the observation-minus-forecast residual's covariance matrix, which is computationally expensive. The objective function in our proposed scheme is quadratic, so its minimizer is analytic and can be calculated easily.

Similar to other EnKF assimilation schemes with single parameter inflation, this study also assumes the inflation factor to be constant in space. Apparently this is not the case in many practical applications, especially when the observations are distributed unevenly. Persistently applying the same inflation values that are reasonably large to address problems in densely observed areas to all state variables can systematically overinflate the ensemble variances in sparsely observed areas (Hamill and Whitaker, 2005; Anderson, 2009). Even if the adaptive procedure for estimating the error covariance matrix is applied, the problem may still exist to some extent. In the two case studies conducted in this article, the observational systems are distributed relatively evenly.

In our future study we will investigate how to modify the adaptive procedure to suit the system with unevenly distributed observations. We also plan to apply our methodology to error covariance localization and to validate the proposed methodologies using more sophisticated dynamic and observational systems.

Acknowledgements

This work was supported by National High-tech R&D Program of China (Grant No. 2009AA122104), National Program on Key Basic Research Project of China (Grant Nos 2010CB951604), the National Natural Science foundation of China General Program (Grant Nos 40975062), and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors gratefully acknowledge the anonymous reviewers for their constructive and relevant comments, which helped greatly in improving the quality of this manuscript. The authors are also grateful to the editors for their hard work and suggestions on this manuscript.

Appendix A

The forecast error covariance matrix equation image is inflated to equation image. The estimation of the inflation factor λ is based on the observation-minus-forecast residual

equation image(A1)

The covariance matrix of the random vector di can be expressed as a second-order regression equation (Wang and Leblanc, 2008):

equation image(A2)

where E is the expectation operator and Ξ is the error matrix. The left-hand side of (A2) can be decomposed as

equation image(A3)

Since the forecast and observational errors are statistically independent, we have

equation image(A4)

and

equation image(A5)

From Eq. (2), equation image is the observational error at the ith time step, and hence

equation image(A6)

Further, since the forecast state equation image is treated as a random vector with the true state equation image as its population mean,

equation image(A7)

Substituting Eqs (A3)–(A7) into Eq (A2), we have

equation image(A8)

It follows that the second-order moment statistic of error Ξ can be expressed as

equation image(A9)

Therefore, λ can be estimated by minimizing objective function Li(λ). Since Li(λ) is a quadratic function of λ with positive quadratic coefficients, the inflation factor can be easily expressed as

equation image(A10)

Similarly, if the amplitude of the observational error covariance matrix is not correct, we can adjust Ri to μiRi as well (Li et al., 2009; Liang et al., 2011). Then the objective function becomes

equation image(A11)

As a bivariate function of λ and μ, the first partial derivative with respect to the two parameters respectively are

equation image(A12)

and

equation image(A13)

Setting Eqs (A12) and (A13) to zero and solving them lead to

equation image(A14)

and

equation image(A15)

Appendix B

In fact,

equation image(B1)

Since equation image is the ensemble mean forecast, we have

equation image(B2)

and similarly

equation image(B3)

That is, the last two terms of Eq. (B1) vanish. Therefore our proposed forecast error covariance matrix can be expressed as

equation image(B4)

Ancillary