In principle, localization can be implemented by using filtered estimates of the background covariances rather than the raw sample covariances to define the matrix, Pb, used in the computation of K by Eq. (). The filtered (localized) version of covariance matrix, P̃b, is obtained by computing the Schur (element-wise) product: P̃b=P^b∘C, where C is a correlation matrix, which has the same dimensions as the sample covariance matrix, P^b. In practice, however, the localization is often done by taking advantage of the fact that localization affects the analysis through PbHT and HPbHT, or, ultimately, through K. In particular – because a distance, d, can be defined for each entry, Kij, of K by the distance between the ith analyzed variable and the jth observation – the simplest localization strategy is to set all entries, Kij, that are associated with a distance longer than a prescribed localization radius, R (d>R), to zero, while leaving the remaining entries unchanged e.g.,.

Another approach is to localize PbHT and HPbHT by a tapering function e.g.,. The usual justification for this approach is that the localized matrix products provide good approximations of the products computed by using localized estimates of Pb. Note that PbHT is the matrix of background covariances between the state variables at the model grid points and at the observation locations, while HPbHT is the matrix of background covariances between the state variables at the observation locations. Thus, a distance can be associated with each entry of the two matrix products, which makes the distance-dependent localization of the two products possible. The approach becomes problematic, however, when h(⋅) is not a local function, which is the typical case for remotely sensed observations .

We consider the situation where localization is applied directly to the background error covariance matrix, P^b. Recall that the localized covariance matrix is expressed as in Eq. (). In particular, C is a positive-definite matrix with strictly positive eigenvalues, while the sample covariance matrix, P^b, may have zero eigenvalues (as it is only non-negative definite). The localization in Eq. () helps to eliminate those zero eigenvalues of P^b and alleviates the related large relative estimation errors. The positive definiteness of C ensures that localization does not introduce new zero eigenvalues in the process of eliminating the zero eigenvalues of P^b. The proper definition of the localization function that ensures that C is positive definite has been thoroughly investigated for the univariate case (N=1) in the literature (e.g. ).

We now consider a model with multiple state variables (N>1). For instance, we take a simple model based on the hydrostatic primitive equations. This model solves the equations for the two horizontal components of wind, the surface pressure, the virtual temperature and a couple of atmospheric constituents. The state of the model is represented by the state vector, x=(x1, x2, …, xN), where xi, i=1, 2, …, N, represents the spatially discretized state of the ith state variable in the model.

The sample background covariance matrix, P^b, can be partitioned as P^b=P^11bP^12b⋯P^1NbP^21bP^22b⋯P^2Nb⋮⋮⋱⋮P^N1bP^N2b⋯P^NNb. The entries of the submatrices, P^iib, i=1, …, N, are called the marginal covariances for the ith state variable. In practical terms if the ith state variable is the virtual temperature, for instance, each diagonal entry of P^iib represents the sample variance for the virtual temperature at a given model grid point, while each off-diagonal entry of P^iib represents the sample covariances between the virtual temperatures at a pair of grid points. Likewise, the entries of P^ijb, i≠j, are called the sample cross-covariances between the grid point values of the ith and the jth state variables at pairs of locations, where the two locations for an entry can be the same grid point.

We thus consider matrix-valued localization functions, ρ(d)={ρij(d)}i,j=1,…,N, which are continuous functions of d. The component ρij(d) of ρ(d) is the localization function used for the calculation of the covariances included in the submatrix Pijb of Pb. Each entry of the localization matrix C is computed by considering the value of the appropriate component of ρ(d) for a particular pair of state variables and the separation distance, d, associated with the related entry of P^b.

In order to get a proper matrix-valued localization function, ρ, a seemingly obvious approach to extend the results of would be to compute the entries of C based on a univariate correlation function for a multivariate variable. That is, for the pair of state variables i and j, we localize the corresponding sample background covariance matrix, P^ijb, by multiplying a localization matrix from the same correlation function for all i and j. Formally, this would be possible because the distance d is uniquely defined for each entry of P^b the same way in the multivariate case as in the univariate case. This approach, however, cannot guarantee the positive definiteness of the resulting matrix, C. As a simple illustrative example, consider the situation where the discretized state vector has only two components that are defined by two different scalar state variables at the same location (e.g., the temperature and the pressure). In this case, if n is the number of locations, the localization matrix for the two state variables together can be written as C=C0C0C0C0 independently of the particular choice of the localization function. Here C0 is an n × n localization matrix from a univariate localization function. From Eq. (), it is clear that n eigenvalues of C are zero and the rank of C is n, while its dimensions are 2n × 2n.

As in Eq. (), although C is rank-deficient and thus so is the localized covariance matrix P̃b (and thus P̃b may be rank-deficient as well), we may still be able to calculate the inverse of HP̃bHT+R, as R is a diagonal matrix. The smallest eigenvalue of HP̃bHT+R is the smallest (positive) value of R, and thus the matrix, HP̃bHT+R, is still invertible and has positive eigenvalues. However, unless the diagonal elements of R are large (which implies large observation error variance), the matrix HP̃bHT+R is seriously ill-conditioned and the computation of its inverse may be numerically unstable. Therefore, the numerical stability of the computation of the inverse of the matrix heavily relies on the observation error variance, which is an undesirable property.

We therefore propose two approaches to construct positive-definite (full rank) matrix-valued localization functions, ρ(d). The first proposed method takes advantage of the knowledge of a proper univariate localization function, ρ̃. Instead of using the same correlation function to localize multiple state variables, for a certain distance lag, we let ρ=ρ̃⋅B, where B is an N×N symmetric, positive-definite matrix whose diagonal entries are 1. It can be easily verified that ρ is a matrix-valued positive-definite function, which makes it a valid multivariate localization function. For instance, in the hypothetical case where the two components of the state vector are two different state variables at the same location, making the choice B=1ββ1 for β, with |β|<1, leads to C=C0βC0βC0C0 rather than what is given in Eq. (). Since the eigenvalues of the matrix B are 1±β>0, it can be easily verified that the matrix in Eq. () is positive definite. For the case with more than two state variables (N≥3), the matrix B can be parametrized as B=LLT, where L=ℓ1,10⋯0ℓ2,1ℓ2,2⋯0⋮⋮⋱0ℓN,1ℓN,2⋯ℓN,N is a lower triangular matrix with the constraints that ∑j=1iℓi,j2=1 and ℓi,i>0 for all i=1, …, N. The constraints are used to have the diagonal entries of B be 1. Other than these constraints, the elements of L can vary freely in order to guarantee the positive definiteness of B.

An attractive feature of this approach is that we can take advantage of any known univariate localization function to produce a multivariate localization function. However, the multivariate localization function from this approach is separable in the sense that the multivariate component (i.e., B) and the localization function (i.e., ρ̃) are factored. Another limitation of the approach is that the localization radius and decay rate are the same for each pair of state variables, leaving no flexibility to account for the potential differences in the correlation lengths and decay rate for the different state vector components.

The second proposed method takes advantage of the availability of multivariate compactly supported functions from the spatial statistics literature. To the best of our knowledge, only a few papers have been published on this subject; one of them is . The function class they considered was essentially a multivariate extension of the Askey function , f(d; ν, c)=(1-dc)+ν, with c, ν>0. Here, x+=max(x, 0) for x∈R. For instance, a bivariate Askey function, which is a special case of the results of , is given by (i, j=1, 2) ρij(d;ν,c)=βij1-dc+ν+μij, where c>0, μ12=μ21≤12(μ11+μ22), ν≥[12s]+2, βii=1 (i=1, 2), β12=β21, and |β12|≤Γ1+μ12Γ1+ν+μ12Γ1+ν+μ11Γ1+ν+μ22Γ1+μ11Γ1+μ22. Here, Γ(⋅) is the gamma function e.g., and s is the dimension of the Euclidean space where the state variable is defined. If the state is defined at a particular instant on a grid formed by latitude, longitude, and height, then s=3. Here, [x] is the largest integer that is equal to or smaller than x. The Askey function in Eq. () has the support c because it sets covariances beyond a distance c to zero. It can be seen from Eq. () that, if the scalars, μij, are chosen to be the same for all values of i and j, the condition on β12 for ρ to be valid is |β12|≤1. (Note that the case of equality here, with the same μij's, reduces to the rank-deficient case where the multivariate localization matrix has zero eigenvalues, similarly to the case of β=1 in Eq. .) For this choice, the second method is essentially the same as the first method with the Askey function set to ρ̃. The localization function given by Eq. () is more flexible than the functions of the first method with the Askey function set to ρ̃ because μij can be chosen to be different for each pair of indexes, i and j. The localization length, however, is still the same for the different pairs of the state variables. The multivariate Askey function is formed by ρij(d;ν,c)=cν+1Bμij+1,ν+11-|d|cν+μij+1,|d|<c and 0 otherwise, where ν≥(s+1)/2, μij=(μi+μj)/2, and μi>0 for all i=1, …, N. Here, B is the beta function .

To illustrate the differences between the shape of the Gaspari–Cohn and the Askey functions, we show the Gaspari–Cohn function for c=25 and the univariate Askey function for c=50 and ν=1, …, 4 (Fig. ). This figure shows that, for a given support, the Askey functions are narrower.

There are many different formulations of the EnKF update equations, which produce not only an updated estimate of the mean but also the ensemble of analysis perturbations that are added to the mean to obtain an ensemble of analyses. This ensemble of analyses serves as the ensemble of initial conditions for the model integration that produce the background ensemble. In our experiments, we use the method of perturbed observations. It obtains the analysis mean and the ensemble of analysis perturbations by the equations x‾a=x‾b+Ky-Hx‾b,x′ka=x′kb+Ky′ko-Hx′kb, where xk′, k=1, 2, …, M, are the ensemble perturbations and y′ko, k=1, 2, …, M, are random draws from the probability distribution of observation errors. As the notation suggests, we consider a linear observation function in our experiments. This choice is made for the sake of simplicity and limits the generality of our findings much less than the use of an idealized model of atmospheric dynamics.

For the case of multiple state variables, the ensemble members are considered to be in a single ensemble, that is, not being grouped into distinct subensembles.

discussed the bivariate Lorenz 95 model, which mimics the nonlinear dynamics of two linearly coupled atmospheric state variables, X and Y, on a latitude circle. This model provides a simple and conceptually satisfying representation of basic atmospheric processes but is not suitable for some atmospheric processes. The model 3 in made it more realistic and suitable with sacrifice of simplicity, by producing a rapidly varying small-scale activity superposed on the smooth large-scale waves. We use the Lorenz 95 model for simplicity in our following experiments.

In the bivariate Lorenz 95 model, the variable X is a “slow” variable represented by K discrete values, Xk, and Y is a “fast” variable represented by J × K discrete values. The governing equations are dXkdt=-Xk-1Xk-2-Xk+1-Xk-(ha/b)∑j=1JYj,k+F,dYj,kdt=-abYj+1,kYj+2,k-Yj-1,k-aYj,k+(ha/b)Xk, where Yj-J,k=Yj,k-1 and Yj+J,k=Yj,k+1 for k=1, …, K and j=1, …, J. The “boundary condition” is periodic; that is, Xk-K=Xk+K=Xk, and Yj,k-K=Yj,k+K=Yj,k. In our experiments, K=36 and J=10. The parameter h controls the strength of the coupling between X and Y, a is the ratio of the characteristic timescales of the slow motion of X to the fast motion of Y, b is the ratio of the characteristic amplitudes of X to Y, and F is a forcing term. We choose the parameters to be a=10, b=10, F=10, and h=2. These values of the model parameters are equal to those originally suggested by , except for the value of the coupling coefficient h, which is twice as large in our case. We made this change in h to increase the covariances between the errors in the estimates of X and Y, which makes the model more sensitive to the choices of the localization parameters. We use a fourth-order Runge–Kutta time integration scheme with a time step of 0.005 nondimensional units, as did. We define the physical distances between Xk1 and Xk2, between Yj1,k1 and Yj2,k2, and between Xk1 and Yj1,k2 by |10(k1-k2)|, |10(k1-k2)+j1-j2|, and |10(k1-k2)-j1|, respectively. Figure  shows a typical state of the model for the selected parameters. The figure shows that X tends to drive the evolution of Y: the hypothetical process represented by Y is more active (its variability is higher) with higher values of X.

Since the estimates of the cross-covariances play a particularly important role at locations where one of the variables is unobserved, we expect an improved treatment of the cross-covariances to lead to analysis improvements at locations where only one of the state variables is observed. This motivates us to consider an observation scenario in which X and Y are partially observed. The variable X is observed at 20 % of all locations, and Y is observed at 90 % of the locations where X is not observed. These observation locations for variables X and Y are randomly chosen. Spatial locations of the partially observed X and Y are illustrated in Fig. . The results from this experiment are compared to those from a control experiment, in which both X and Y are fully observed.

We first generate a time series of “true” model states by a 2000-time-step integration of the model. We initialize an ensemble by adding the standard Gaussian noise to the true state; then we discard the first 3000 time steps. We then generate simulated observations by adding random observation noise of mean 0 and variance 0.02 to the the appropriate components of the true state of X at each time step. We use the same procedure to generate simulated observations of Y, except that the variance of the observation noise is 0.005. Observations are assimilated at every time step by first using a 20-member ensemble with a constant covariance inflation factor of 1.015. The error in the analysis at a given verification time is measured by the root-mean-square distance between the analysis mean and the true state. We refer to the resulting measure as the root-mean-square error (RMSE). The probability distribution of the RMSE for the last 1000 time steps of 50 different realizations of each experiment is shown by a box plot. The box plot is an effective way of displaying a summary of the distribution of numbers. The lower and upper bounds of the box respectively give the 25th and 75th percentiles. The thick line going across the interior of the box gives the median. The whisker depends on the interquartile range (IQR), which is precisely equal to the vertical length of the box. The whiskers extend to the extreme values, which are no more than 1.5 IQR from the box. Any values that fall outside of the end points of whiskers are considered outliers, and they are displayed as circles.

In the box plot figures in the next section, we compare the RMSE for four different localization schemes. We use the following notation to distinguish between them in the figures:

S1 – the bivariate sample background covariance is used without localization;

Figure  shows the distribution of RMSE for variable X for different configurations of the localization scheme in the case where the state is only partially observed. This figure compares the Askey function and Gaspari–Cohn function which have the same support (localization radius), so setting all the covariances to zero beyond the same distance. We recall that, because X is much more sparsely observed than Y, we expect to see some sensitivity of the analyses of X to the treatment of the cross-covariance terms. The figure confirms this expectation. A comparison of the results for configurations S1 and S2 suggests that ignoring the cross-covariances is a better strategy than using them without localization. This conclusion does not hold once a univariate localization is applied to the marginal covariances, as using configuration S3 produces worse results than applying no localization at all (S1).

Figure  also shows that the distribution of the state estimation error is less sensitive to the choice of localization strategy for the larger values of support. Of all localization schemes, S4 with β=0.1 performs best regardless of the localization radius: the distribution of the state estimation error is narrow with a mean value that is lower than those for the other configurations of the localization scheme. For this choice of localization scheme and β, the Askey function produces smaller errors than the Gaspari–Cohn function, particularly for smaller localization radii.

Figure  is the same as Fig.  but for variable Y rather than for variable X. A striking feature of the results shown in this figure is that the Askey function clearly performs better than the Gaspari–Cohn function. Another obvious conclusion is that using a smaller localization radius (a lower value of support) is clearly advantageous for the estimation of Y. This result is not surprising, considering that Y is densely observed and its spatial variability is much higher than that of X. In contrast to the results for variable X, configuration S3 produces much more accurate estimates of variable Y than do configurations S1 and S2. In addition, configuration S4 performs only slightly better, and only for the lowest value of support, than does configuration S3. The latter observations indicate that the marginal covariances play a more important role than do the cross-covariances in the estimation of the densely observed Y. The proper filtering of the marginal covariances can thus greatly increase the accuracy of the estimates of Y. In other words, the densely observed Y is primarily estimated based on observations of Y. Hence, the low signal-to-noise ratio for the sample estimate of the marginal covariances for Y greatly limits the value of the observations of Y at longer distances.

Figure  is the same as Fig.  but for the case of a fully observed state. By comparing the two figures, we see that the analysis is far less sensitive to the localization radius in the fully observed case than in the partially observed case. As can be expected, the state estimates are also more accurate in the fully observed case. In the fully observed case, localization strategy S3 performs much better than do strategies S1 and S2 and similarly to S4. This result indicates that, in the fully observed case, X is primarily analyzed based on observations of X, making the analysis of X more sensitive to the localization of the marginal covariances than to the localization of the cross-covariances. Similar to the partially observed case, the Askey function tends to perform better than the Gaspari–Cohn function, but the differences between the accuracy of the state estimates for the two filter functions are negligible, except for the shortest localization radius.

Figure  shows the distribution of the errors for variable Y in the fully observed case. The best results are obtained by using a short localization radius with the Askey function, even though the variability of the error is relatively large in that case. The fact that localization strategies S3 and S4 perform similarly well shows that the estimates of the cross-covariances do not play an important role in this case; that is, X is primarily estimated based on observations of X, and Y is dominantly estimated based on observations of Y.

We also investigated the performance of EnKF with a 500-member ensemble. The results for the 500-member ensemble are shown in Figs.  to . We use an inflation factor of 1.005 for 500 ensembles, because the optimal value of the inflation factor is typically smaller for a larger ensemble. The rank of the 500-member ensemble covariance matrix is significantly larger than that of the 20-member ensemble covariance matrix, as expected.

Figures  to  show that, overall, S4 still performs better than the other localization schemes regardless of the choice of localization radius, as in the case of the 20-member ensemble. In particular, when observations are partially observed, S4 with β=0.01 provides the smallest RMSE. The cross-correlation between X and Y, calculated using 500-member ensembles without assimilating any observation, varies from -0.4 to 0.4, which indicates that the cross-correlation between the two variables are not negligible. Therefore, improved treatment of cross-covariance tends to lead to an improved accuracy in the state estimation.

The results with the 500-member ensemble also show that the distribution of the state estimation error is in general less sensitive to the choice of the localization function or the localization radius, compared to the 20-member-ensemble case. Figure , however, shows that, for the estimation of sparsely observed X, the localization scheme S3 with smaller localization radius performs worse than that with larger localization radius. For variable Y in the partially observed case (Fig. ) and both variables X and Y in the fully observed case (Figs.  and ), the best results are obtained with S3 and S4 regardless of the localization radius. They also show that the state estimation error is not sensitive but stable to the choice of localization radius.

Figures  and  show that the localization schemes S3 and S4 perform in a similar way, and obviously perform better than the other two localization schemes. This might imply that the cross-covariances do not have much influence on the state estimation in the fully observed case, once the covariances within each state variable are localized.