1. Data assimilation with residual nudging

A finite, often small, ensemble size has some well-known effects that may substantially influence the behavior of an ensemble Kalman filter (EnKF). These effects include, for instance, rank deficient sample error covariance matrices, systematically underestimated error variances, and, in contrast, exceedingly large error cross covariances of the model state variables (Whitaker and Hamill 2002). In the literature, the latter two issues are often tackled through covariance localization (Hamill et al. 2001), while the first issue, underestimation of sample variances, is often handled by covariance inflation (Anderson and Anderson 1999), in which one artificially increases the sample variances, either multiplicatively (see, e.g., Anderson and Anderson 1999; Anderson 2007, 2009; Bocquet and Sakov 2012; Miyoshi 2011) or additively (see, e.g., Hamill and Whitaker 2011), or in a hybrid way by combining both multiplicative and additive inflation methods (see, e.g., Whitaker and Hamill 2012), or through other ways such as relaxation to the prior (Zhang et al. 2004), multischeme ensembles (Meng and Zhang 2007), modification of the eigenvalues of sample error covariance matrices (Altaf et al. 2013; Luo and Hoteit 2011; Ott et al. 2004; Triantafyllou et al. 2013), or back projection of the residuals to construct new ensemble members Song et al. (2010), to name but a few. In general, covariance inflation tends to increase the robustness of the EnKF against uncertainties in data assimilation (Luo and Hoteit 2011) and, often, also improves the filter performance in terms of estimation accuracy.

The focus of this article is on the study of the effect of covariance inflation from the point of view of residual nudging (Luo and Hoteit 2012). Here, the “residual” with respect to an m-dimensional system state x is a vector in the observation space, defined as x − y,¹ where : → is a linear observation operator and y is the corresponding p-dimensional observation vector. Throughout this paper, our discussion is confined to the filtering (or analysis) step of the EnKF, so that the time index in the EnKF is dropped. The linearity assumption in the observation operator is taken in order to simplify our discussion. The results to be presented later, though, might also provide insights into more complex situations.

Before introducing the concept of residual nudging, let us define some additional notation. We assume that the observation system is given by

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where v is the vector of observation error, with zero mean and a nonsingular covariance matrix . We further decompose as = ^{1/2 T/2}, where ^1/2 is a nonsingular square root of and ^T/2 denotes the transpose of ^1/2.

To measure the length of a vector z in the observation space, we adopt the following weighted Euclidean norm:

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One may convert the weighted Euclidean norm to the standard Euclidean norm by noticing that ‖z‖ = ‖^−1/2z‖₂, where ‖•‖₂ denotes the standard Euclidean norm. As a result, many topological properties with respect to the standard Euclidean norm, for example, the triangle inequality [see (3) below], still hold with respect to the weighted Euclidean norm.

The idea of data assimilation with residual nudging (DARN) is the following. Let x^tr be the true system state (truth), y^o = x^tr + v^o the recorded observation for a specific realization v^o of the observation error, and the state estimate (e.g., either the prior or posterior estimate) obtained from a data assimilation (DA) algorithm. Then, the residual . By the triangle inequality, the weighted Euclidean norm of the residual (residual norm hereafter) satisfies

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If the DA algorithm performs reasonably well, one may expect that the magnitude of may not be significantly larger than ‖v^o‖. As a result, one may obtain an upper bound of in terms of ‖v^o‖ (e.g., in the form of β‖v^o‖, where β is a nonnegative scalar coefficient). In practice, though, ‖v^o‖ is often unknown. As a remedy, we replace ‖v^o‖ by an upper bound of the expectation of the weighted Euclidean norm of the observation error v, where denotes the expectation operator. One such upper bound can be obtained by noticing that

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where the operator “trace” evaluates the trace of a matrix and _p is the p-dimensional identity matrix. From (4), we have the upper bound . Consequently, we want to find a state estimate whose residual norm satisfies

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for a previously chosen β. It is worthy of mentioning that in general it may be difficult to identity which β gives the best state estimation accuracy with respect to the truth x^tr. Therefore, in Luo and Hoteit (2012), we mainly used DARN as a safeguard strategy; that is, if a state estimate is found to have a too large residual norm, then we try to introduce some correction to the state estimate in order to reduce its residual norm, which in turn might also improve the estimation accuracy.

In Luo and Hoteit (2012), we introduced DARN to the analysis in the ensemble adjustment Kalman filter (EAKF; see Anderson 2001). In the EAKF with residual nudging (EAKF-RN), if the residual norm of is less than , then we accept as a reasonable estimate and no change is made. Otherwise, a correction is introduced to in a way such that the residual norm of the modified state estimate is exactly , and that among all possible state estimates whose residual norms are equal to , the simulated (or predicted) observation of the modified state estimate has the shortest distance to the one of the original state estimate . Numerical results in Luo and Hoteit (2012) show that the EAKF-RN exhibits (sometimes substantially) improved filter performance, in terms of estimation accuracy and/or stability against filter divergence, compared to the EAKF. Extension of DARN to other types of filters is also possible (see, e.g., Luo and Hoteit 2013).

2. Covariance inflation from the point of view of residual nudging

Here, we examine the effect of covariance inflation on the analysis residual norm. To this end, we first recall that the mean update formula in the EnKF (without perturbing the observation) is given by

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where and are the sample means of the background and analysis ensembles, respectively; is the Kalman gain; and is a certain symmetric, positive semidefinite matrix in accordance with the chosen inflation scheme. In general, may be related, but not necessarily proportional, to the sample error covariance matrix of the background ensemble. For instance, in the hybrid EnKF, can be a mixture of and a “background covariance” (Hamill and Snyder 2000) or may be partially time varying, as in Hoteit et al. (2002).

Our objective here is to examine under which conditions the residual norm of the analysis satisfies , where β_l and β_u (0 ≤ β_l ≤ β_u) represent, respectively, the lower and upper values of β that one wants to set for the analysis residual norm in DARN. Different from the previous works (Luo and Hoteit 2012, 2013), the lower bound is introduced here in order to make our discussion below slightly more general. In practice, it may also be used to prevent too small residual norms in certain circumstances in order to avoid, for instance, a state estimate that overfits the observation, a phenomenon that may be caused by “overinflation,” as will be shown later.

Inserting (6) into , one has

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where . Multiplying both sides of (7) by ^−1/2, one obtains

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To derive the bounded residual norm, we first consider under which conditions the upper bound is guaranteed to hold. Given that [cf. (19) later]

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a sufficient condition is thus

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Let

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and λ_max and λ_min be the maximum and minimum eigenvalues of , respectively. Recalling that the induced 2-norm of a symmetric positive semidefinite matrix is exactly the maximum eigenvalue of that matrix (Horn and Johnson 1990, section 5.6.6), we have

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Therefore, (10) leads to

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If is relatively small such that , then (13) automatically holds. However, if , and that λ_min is very small, then there is no guarantee that (13) will hold. A small λ_min may appear, for instance, when the ensemble size n is smaller than the dimension p of the observation space. In such circumstances, the matrix may be singular with λ_min = 0, and the singularity may not be avoided only through the multiplicative covariance inflation. If one cannot afford to increase the ensemble size n, then a few alternative strategies may be adopted to address (or at least mitigate) the problem of singularity. These include, for instance, (a) introducing covariance localization (Hamill et al. 2001) to in order to increase its rank (Hamill et al. 2009); (b) replacing the sample error covariance by a hybrid of and some full-rank matrix, similar to that in Hamill and Snyder (2000); and (c) reducing the dimension p of the observation in the update formula, for instance, by assimilating the observation in a serial way (see, e.g., Whitaker and Hamill 2002) or by assimilating the observation within the framework of a local EnKF (see, e.g., Bocquet 2011; Ott et al. 2004). Once the problem of singularity is solved so that the smallest eigenvalue of becomes positive, a (large enough) multiplicative inflation factor can be introduced to make sure that (13) holds.

Inequality (13) provides insights into what the constraints there may be in choosing the inflation factor. In what follows, we study the problem in a slightly more general setting. Concretely, we consider a family of mean update formulas in the form of

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where α, δ, and γ are some positive coefficients and is the gain matrix, which in general differs from the Kalman gain in (6) with the presence of these three extra coefficients. Without loss of generality, though, one may let α = 1 (e.g., by moving α inside the parentheses) so that the gain matrix is simplified to

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If δ = 1, then resembles the Kalman gain in the EnKF, with 1/γ being analogous to the multiplicative covariance inflation factor, as used in Anderson and Anderson (1999). In our discussion below, we first derive some inflation constraints in the general case with δ > 0, and then examine the more specific situation with δ = 1. It is expected that one can also obtain constraints for other types of inflations in a similar way, but the results themselves may be case dependent.

Using (14a) and (15) as the update formulas and with some algebra, the weighted residual is given by

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where , , and are defined as previously. Let

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then one has

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For our purposes, the following two matrix inequalities are useful. First, given a matrix and a vector z with suitable dimensions, one has

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where ‖‖₂, the induced 2-norm of , is the maximum of the absolute singular values of or, equivalently, ‖‖₂ is equal to the square root of the largest eigenvalue of ^T (Horn and Johnson 1990, chapter 5). Second, if in addition is nonsingular, then (see, e.g., Grcar 2010 and the references therein)

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The first inequality, (19), can be applied to obtain the sufficient conditions under which the inequality is achieved. Let the maximum and minimum eigenvalues of Φ be μ_max and μ_min, respectively. Then, by (17),

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We remark that both μ_max and μ_min can be negative (e.g., when δ < 1 and γ → 0); therefore, ‖Φ‖₂ = max(|μ_max|, |μ_min|). By (18) and (19), a sufficient condition for is . For notational convenience, we define and .

Depending on the signs and magnitudes of μ_max and μ_min, there are in general four possible scenarios: (a) μ_max ≥ 0 and μ_min ≥ 0, so that ‖Φ‖₂ = μ_max; (b) μ_max ≤ 0 and μ_min ≤ 0, so that ‖Φ‖₂ = −μ_min; (c) μ_max ≥ 0, μ_min ≤ 0, and μ_max + μ_min ≥ 0, so that ‖Φ‖₂ = μ_max; and (d) μ_max ≥ 0, μ_min ≤ 0, and μ_max + μ_min ≤ 0, so that ‖Φ‖₂ = −μ_min. Inserting (21) into the above conditions, one obtains some inequalities with respect to the variables δ and γ (subject to δ > 0 and γ > 0), which are omitted in this paper for brevity.

Similarly, the second inequality, (20), can be used to find the sufficient conditions for . By (18) and (20), one such sufficient condition can be . By (17), it can be shown that

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Let the maximum and minimum eigenvalues of Φ⁻¹ be ν_max and ν_min, respectively; then,

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Similar to the previous discussion, we require that , which also leads to four possible scenarios: (a) ν_max ≥ 0 and ν_min ≥ 0, so that ‖Φ⁻¹‖₂ = ν_max; (b) ν_max ≤ 0 and ν_min ≤ 0, so that ‖Φ⁻¹‖₂ = −ν_min; (c) ν_max ≥ 0, ν_min ≤ 0, and ν_max + ν_min ≥ 0, so that ‖Φ⁻¹‖₂ = ν_max; and (d) ν_max ≥ 0, ν_min ≤ 0, and ν_max + ν_min ≤ 0, so that ‖Φ⁻¹‖₂ = −ν_min. Again, inserting (23) into the above conditions, one obtains some inequalities with respect to the variables δ and γ.

Despite the complexity in the general situation, the analysis in the case of δ = 1 (corresponding to the update formula in the EnKF) is significantly simplified. Indeed, when δ = 1, the maximum and minimum eigenvalues in (21) and (23) are all positive. Therefore, the following conditions,

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are sufficient for the objective . Note that if ξ_u ≥ 1, that is, , then any γ > 0 would guarantee that [indeed by (16) and (19), the analysis residual norm is guaranteed to be no larger than since ‖Φ₂‖ ≤ 1 with δ = 1] and that inequality (24a) holds. On the other hand, if ξ_l ≥ 1 such that , then in most cases,² it is impossible for the EnKF to have no less than (hence ), for the same aforementioned reason. Therefore, the inequality (24b) becomes infeasible. With these said, in what follows we focus on the cases in which ξ_u, ξ_l ∈ [0, 1). With some algebra, it can be shown that γ should be bounded by

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Let κ = λ_max/λ_min be the condition number of the (normalized) matrix . From (25), we have , which leads to a constraint in choosing β_l and β_u, in terms of

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Inequality (25) suggests that the upper and lower bounds of γ are related to the minimum and maximum eigenvalues of , respectively. In particular, to avoid a too small residual norm (i.e., observation overfitting), γ should be lower bounded; hence, its inverse 1/γ, resembling the multiplicative inflation factor, should be upper bounded, as mentioned previously.

In practice, if the dimension p of the observation space is large, then it may be expensive to evaluate λ_max and λ_min. In certain circumstances, though, there may be cheaper ways to compute an interval for γ. For instance, if in the mean update formula is in the form of with c₁ and c₂ being some positive scalars and a constant, symmetric, and positive-definite matrix, then

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The additive Weyl inequality (Horn and Johnson 1991, chapter 3) suggests that the following bounds hold for λ_max and λ_min:

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where τ and ρ are the eigenvalues of and , respectively. In many situations, may be rank deficient; therefore, a singular value decomposition (SVD) analysis shows that τ_max is equal to the largest eigenvalue of , where is a square root of that can be directly constructed based on the background ensemble (Bishop et al. 2001; Luo and Moroz 2009; Wang et al. 2004). Note that is a matrix with its dimension determined by the ensemble size n and is, in fact, the same as the one used in the ensemble transform Kalman filter (ETKF; Bishop et al. 2001; Wang et al. 2004) in order to obtain the transform matrix. Therefore, τ_max can be taken as a by-product within the framework of ETKF. On the other hand, if both and are time invariant, then the eigenvalues ρ_max and ρ_min of ^{−1/2T−T/2} can be calculated offline once and for all. Taking these considerations into account, (25) can be modified as follows:

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Accordingly, (26) is changed to

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with being a modified “condition number.”

3. Numerical verification

Here, we focus on using the 40-dimensional Lorenz 96 (L96) model (Lorenz and Emanuel 1998) to verify the above analytic results, while more intensive filter (with residual nudging) performance investigations are reported in Luo and Hoteit (2012). The experiment settings are the following. A reference trajectory (truth) is generated by numerically integrating the L96 model (with the driving force term F = 8) forward through the fourth-order Runge–Kutta method, with the integration step being 0.05 and the total number of integration steps being 1500. The first 500 steps are discarded to avoid the transition effect, and the remaining 1000 steps are used for data assimilation. To obtain a long-term “background covariance” ^lt (“background mean” x, respectively), we also conduct a separate long model run with 100 000 integration steps, and take ^lt (x) as the temporal covariance (mean) of the generated model trajectory. The synthetic observations are generated by adding the Gaussian white noise N(0, 1) to each odd numbered element (x₁, x₃, …, x₃₉) of the state vector x = (x₁, x₂, …, x₄₀)^T every four integration steps. This corresponds to the ½ observation scenario used in Luo and Hoteit (2012). An initial ensemble with 20 ensemble members is generated by drawing samples from the Gaussian distribution N(x, ^lt), and the ETKF is adopted for data assimilation.

For distinction later, we call the ETKF without residual nudging the normal ETKF, and the ETKF with residual nudging the ETKF-RN. In the normal ETKF, (6) is used for the mean update, with equal to the sample error covariance of the background ensemble.³ Neither covariance inflation nor covariance localization is introduced to the normal ETKF, since for our purposes we wish to use this plain filter setting as the baseline for comparison. One may adopt various inflation and localization techniques to enhance the filter performance, but such an investigation is beyond the scope of this paper.

In the ETKF-RN, we adopt the hybrid scheme to address the issue of possible singularity in the matrix [cf. (11)]. Equation (14) is adopted for the mean update, with α = δ = 1 and γ constrained by (28) and (29). For convenience, we denote the lower and upper bounds of γ in (28) by γ_min and γ_max, respectively, and rewrite γ in terms of γ = γ_min + c(γ_max − γ_min) with c being a corresponding scalar coefficient that is involved in our discussion later. Note that in general the background residual norm changes with time, as do the values of ξ_u and ξ_l in (25). This implies that, in general, γ_min and γ_max (hence γ) also change with time; therefore, they need to be calculated at each data assimilation cycle.

An additional remark is that the normal ETKF and the ETKF-RN share the same square root update formula as in Wang et al. (2004), where it is the sample error covariance , rather than its hybrid with ^lt, that is used to generate the background square root. Such a choice is based on the following considerations. On the one hand, if one uses the hybrid covariance for square root update, then it would require a matrix factorization (e.g., singular value decomposition) in order to compute a square root of the hybrid covariance at each data assimilation cycle, which can be very expensive in large-scale applications. On the other hand, for the L96 model used here, numerical investigations show that using the hybrid covariance for the square root update does not necessarily improve the filter performance (results not shown).

The procedures in the ETKF-RN are summarized as follows. Because the matrix ^{−1/2T−T/2} is time invariant, its maximum and minimum eigenvalues, ρ_max and ρ_min [cf. (28)], respectively, are calculated and saved for later use. Then, with the background ensemble at each data assimilation cycle, calculate the sample mean , the corresponding background residual norm , and a square root of the sample error covariance following Bishop et al. (2001), Luo and Moroz (2009), and Wang et al. (2004). Update to its analysis counterpart by calculating a transform matrix , together with a “centering” matrix following Wang et al. (2004). During the square root update process, the maximum eigenvalue τ_max of is obtained as a by-product following our discussion in the previous section. With this information, one is ready to calculate the interval bounds γ_min and γ_max in (28) and, hence, obtain γ = γ_min + c(γ_max − γ_min) for a given value of c (c can be constant or variable during the whole data assimilation time window). This γ value is then inserted into (14) (with α = δ = 1 there) to obtain the analysis mean . With and , an analysis ensemble can be generated in the same way as in Bishop et al. (2001) and Wang et al. (2004). Propagating this ensemble forward in time, one starts a new data assimilation cycle, and so on. Comparing the above procedures to those in Luo and Hoteit (2012), the observation inversion used in Luo and Hoteit (2012) is avoided.

The experiment below aims to show that, at each data assimilation cycle, if a γ value lies in the interval = [γ_min, γ_max] given by (28), then the corresponding analysis residual norm is bounded by the interval , with β_l and β_u satisfying the constraint (29). In the experiment we fix β_u = 2, and let , where the small fraction 0.1 is introduced for convenience of visualization.⁴

Figure 1 shows the time series of the background (dashed–dotted) and analysis (thick solid) residual norms in different filter settings (for convenience of visualization, the residual norm values are plotted in the logarithmic scale). For reference we also plot the targeted lower and upper bounds (dashed and thin solid lines, respectively), and (p = 20), respectively. In the normal ETKF (Fig. 1a), in most of the time the analysis residual norms are larger than the targeted upper bound (no targeted lower bound is calculated and plotted in this case). With residual nudging, the analysis residual norms of the ETKF-RN migrate into the targeted interval, as long as the coefficient c lies in [0, 1] (Figs. 1b–d). Also see the caption of Fig. 1 to find out how the corresponding c values are chosen. When c is outside the interval [0, 1], the corresponding γ is not bounded by [γ_min, γ_max]; hence, there is no guarantee that the corresponding analysis residual norms are bounded by . Two such examples are presented in Figs. 1e and 1f, with c being 2.5 and −0.005, respectively (e.g., for c = −0.005 in Fig. 1f, breakthroughs of the lower bound are found around time step 220 and at a few other places). As “side” results, we also report in Table 1 on the time mean root-mean-square errors (RMSEs) [see Eq. (13) of Luo and Hoteit (2012)] that correspond to different filter settings in Fig. 1. In these tested cases, the filter performance of the ETKF-RN appears improved, in terms of the time mean RMSE, when compared to that of the normal ETKF.

Time series of the analysis residual norms in (a) the normal ETKF without residual nudging and (b)–(f) the ETKF-RN with the following different c values: (b) 0, (c) 1, (d) [0, 1], (e) 2.5, and (f) −0.005. For the normal ETKF there are no targeted lower and upper residual norm bounds. For reference, though, we still plot the targeted upper bound in (a). We also note that the c value in (d) is randomly drawn from the uniform distribution on the interval [0, 1] at each data assimilation cycle, while in the rest of the panels the c values are constant during the assimilation time window.

Citation: Monthly Weather Review 141, 10; 10.1175/MWR-D-13-00067.1

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Time series of the analysis residual norms in (a) the normal ETKF without residual nudging and (b)–(f) the ETKF-RN with the following different c values: (b) 0, (c) 1, (d) [0, 1], (e) 2.5, and (f) −0.005. For the normal ETKF there are no targeted lower and upper residual norm bounds. For reference, though, we still plot the targeted upper bound in (a). We also note that the c value in (d) is randomly drawn from the uniform distribution on the interval [0, 1] at each data assimilation cycle, while in the rest of the panels the c values are constant during the assimilation time window.

Citation: Monthly Weather Review 141, 10; 10.1175/MWR-D-13-00067.1

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Time series of the analysis residual norms in (a) the normal ETKF without residual nudging and (b)–(f) the ETKF-RN with the following different c values: (b) 0, (c) 1, (d) [0, 1], (e) 2.5, and (f) −0.005. For the normal ETKF there are no targeted lower and upper residual norm bounds. For reference, though, we still plot the targeted upper bound in (a). We also note that the c value in (d) is randomly drawn from the uniform distribution on the interval [0, 1] at each data assimilation cycle, while in the rest of the panels the c values are constant during the assimilation time window.

Citation: Monthly Weather Review 141, 10; 10.1175/MWR-D-13-00067.1

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Table 1.

Time mean RMSEs in the normal ETKF and the ETKF-RN with the same c values as in Fig. 1.

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4. Discussion and conclusions

We derived some sufficient inflation constraints in order for the analysis residual norm to be bounded in a certain interval. The analytic results showed that these constraints are related to the maximum and minimum eigenvalues of certain matrices [cf. (11)]. In certain circumstances, the constraint with respect to the minimum eigenvalue [e.g., (13)] may impose a nonsingularity requirement on relevant matrices. A few strategies in the literature that can be adopted to address or mitigate this issue are highlighted.

Some remaining issues are manifest in our deduction. These include, for instance, the nonlinearity in the observation operator and the choice of β_u and β_l. For the former problem, under a suitable smoothness assumption on the observation operator, one may also obtain inflation constraints similar to those in section 2. On the other hand, though, more investigations may be needed to make the results more practical in terms of computational complexity. For the latter problem, numerical results in Luo and Hoteit (2012) show that the β values influence the overall performance of the EnKF in terms of filter stability and accuracy. Intuitively, smaller (larger) β values tend to make residual nudging happen more (less) often. Therefore, if the normal EnKF performs well (poorly), then a larger (smaller) β value may be suitable. In this aspect, it is expected that an objective criterion is needed. This will be investigated in the future.