## 1. Introduction

The ensemble Kalman filter (EnKF; Evensen 1994) is a Monte Carlo approximation to the traditional filter of Kalman (1960) that is suitable for high-dimensional problems such as numerical weather prediction (NWP). One of the strengths of ensemble Kalman filters is the ability to evolve in time estimates of forecast error covariance, using the flow-dependent information inherent in an ensemble of model runs.

Localization is a technique by which the impact of observations from distant regions upon an analysis is suppressed. There are two categories of localization techniques (discussed in detail in section 2b): those that operate on background error covariances 𝗕, which we call B localization; and those that operate on observation error covariances 𝗥, which we call R localization. Adaptive localization techniques, such as the hierarchical filter of Anderson (2007) and ensemble correlations raised to a power (ECO-RAP) of Bishop and Hodyss (2009a, 2009b), are beyond the scope of this work.

It is the error covariances between model variables, along with the observation error characteristics, that ultimately describe the impact pattern of an observation upon the analysis via the Kalman gain 𝗞. In practice, the accuracy of the background error covariance estimate is limited by the size of the ensemble, which must be kept small for computational feasibility (typically of order 20–100 for NWP). Empirically, at larger geographical distances background error covariance estimates tend to be dominated by noise rather than signal (Hamill et al. 2001); it is this “distance-dependent assumption” that motivates the technique of (nonadaptive) localization to eliminate correlations that are deemed to be spurious.

The background error covariance determined from an ensemble of *P* members has at most *P* − 1 degrees of freedom to express uncertainty. However, in local regions of large error growth the atmosphere has been shown to exhibit low dimensionality (Patil et al. 2001). When using localization, the ensemble needs to account for the instabilities in a local region. Additionally, if local analyses can choose different linear combinations of ensemble members in different regions, this allows the analysis to greatly reduce the previously noted dimensionality limitation (Hunt et al. 2007). Lorenc (2003) notes that the assimilation of a perfect observation removes a degree of freedom from the ensemble, but that localization with a Schur product allows for extra degrees of freedom in the analysis.

Localization can also lead to significant savings in computational resources. The analysis at each grid point only needs to consider local observations and the values at nearby model grid points that are linked to these observations by the observation operator. Analyses for local regions can thus be considered independently, allowing for more efficient parallelization of the code (Hunt et al. 2007; Szunyogh et al. 2008).

Successful NWP depends upon well-balanced initial conditions to avoid the generation of spurious inertial gravity waves such as those that ruined the 1922 Richardson forecast. By balanced, we mean an atmospheric state in the slow manifold that approximately follows physical balance equations appropriate to the scale and location, such as the geostrophic relationship. In practice, there are initialization techniques for improving the balance of an analysis, such as nonlinear normal mode initialization and digital filters (Lynch and Huang, 1992). However, once an analysis is filtered the resulting atmospheric state cannot be guaranteed to be optimal. Daley (1991, chapter 6) notes that there is no unique balanced state corresponding to a given unbalanced state; a filter may merely ignore the increment and move the solution back toward the balanced background state. Thus, an ideal data assimilation system should avoid or reduce the initialization by filtering and try to create a well-balanced analysis.

The impact of localization on the balance of an analysis is discussed in Cohn et al. (1998) who noted an unrealistically high ratio of divergence to vorticity as a consequence of local observation selection. Mitchell et al. (2002) show that the optimum localization distance (in terms of improving analysis error) grows with ensemble size, and that balance is improved with longer localization distances. Lorenc (2003) provides an example of how localization produces imbalance. Consider the assimilation of a single height observation located at the origin (*x* = 0) of Fig. 1. The solid lines in Fig. 1 represent a perfect scenario where the height *h* and meridional wind *υ* are in geostrophic balance in the context of the shallow-water equations (see section 2 for details). The black line is proportional to the error covariances between *h* at the location *x* and *h* at the origin, while the gray line is proportional to the error covariances between *υ* at *x* and *h* at the origin. In the assimilation of a single *h* observation, these lines are also proportional to the respective elements of the Kalman gain matrix 𝗞, and therefore the analysis increments. Localization is then applied to these error covariances by multiplying them by a Gaussian function with length scale 250 km based upon distance from the observation, so that the error covariances decay to zero for larger *x* (dashed lines). In the region of *x* = 250 km, the analysis increment of *υ* is reduced by localization. If geostrophic balance is to be maintained, then the magnitude of the height gradient with respect to *x* should also be smaller. However, the height gradient is actually increased by localization and therefore the wind becomes significantly ageostrophic in this region (dash–dot line). In general, EnKF covariance localization modifies the elements of either the 𝗕 matrix or the 𝗥 matrix, which in turn reduces the elements of 𝗞 as one moves farther from the observation. Thus, as in this example, the analysis increments asymptote to zero as the analysis converges to the background in the absence of observation information. During this transition the geostrophic balance of the analysis increment is disrupted.

Kepert (2009) demonstrates how assimilation of wind and height observations with localized covariances produce imbalanced analyses with excess divergence, and proposes assimilation in terms of streamfunction *ψ* and velocity potential *χ* rather than *u* and *υ* wind components. This technique results in a smaller (and more natural) ratio of divergence to rotation in the analysis, and hence balance is improved, but these improvements are less noticeable after initialization.

The purpose of this paper is to compare the B and R localizations and their impact on balance. Following a description of the EnKF and localization techniques (section 2), we first compare the localizations using a simple model (section 3), and then apply them to a global atmospheric model (section 4).

## 2. Methods

### a. Ensemble Kalman filter data assimilation

**x**

_{a}is improved through optimal combination of forecast

**x**

_{b}and observations

**y**

_{o}: The optimal weight matrix 𝗞, or Kalman gain, is given by where 𝗕 is the background error covariance matrix, 𝗥 is the observation error covariance matrix, and 𝗛 is the linearization of the observation operator

*h*

_{op}. In ensemble data assimilation methods, the background error covariance matrix is estimated using an ensemble of

*P*forecasts: where 𝗫

_{b}is the matrix of background ensemble perturbations from the ensemble mean with each row referring to a model variable, and each column to an ensemble member. The exact technique for updating the analysis ensemble members depends on the version of EnKF.

### b. Localization techniques

*f*

_{loc}of distance

*d*between grid points

*i*and

*j*(Houtekamer and Mitchell 2001). Gaspari and Cohn (1999) describe a Gaussian localization function: where

*L*is a localization distance used for scaling the width of the localization. Gaspari and Cohn (1999) also introduced a piecewise polynomial approximation of a Gaussian localization function with compact support (this means it becomes zero beyond some finite distance, in this case at about 3.65 times

*L*). Physically, this means that the background errors at model grid points that are far apart should have no statistical relationship.

*d*is the distance between observation

*i*and model grid point

*j*. Since

*d*varies depending upon which grid point the analysis is being performed at, the rows of 𝗞 [in (2)] must be computed independently because the (𝗛𝗕𝗛

^{T}+ 𝗥) term will be different at each grid point location. Physically, this means that far away observations can be considered to have infinite error, and thus do not impact the analysis.

_{p}is the

*P*×

*P*identity matrix. For this study, we employ Gaussian localization [(4) and (5)] with a cutoff distance of approximately 3.65 times

*L*beyond which there is no observation impact (the localization function is set to 0). The application of (5) to a diagonal 𝗥 using an observation cutoff radius of 3.65

*L*puts an upper bound on the conditioning number for 𝗥 at 10

^{3}for the case of uniform observation errors. Localization can also be applied by dividing the diagonal elements of 𝗥

^{−1}in (6) by

*f*

_{Rloc}. This reduces the size of the rightmost term of the bracketed expression in (6); as this smaller term is then added to the identity matrix, the inversion of the bracketed expression remains a stable calculation. Note that some studies (i.e., Houtekamer and Mitchell 2005) report localization values in terms of cutoff distance rather than

*L*.

For NWP applications, 𝗕 (*N* × *N*, where *N* is the dimension of *x*) is too large to be represented explicitly, therefore the 𝗕𝗛^{T} and 𝗛𝗕𝗛^{T} terms of (2) are calculated directly from the ensemble, as in Houtekamer and Mitchell (2001). For the serial ensemble square-root filters (EnSRF; Whitaker and Hamill 2002), localization by a distance-dependent function is performed upon 𝗕𝗛^{T}, where each element represents the covariance between a model grid point and observation. Because 𝗛𝗕𝗛^{T} is a scalar, it does not require localization. In the case of observations on grid points (which is the case used in this study), this form of localization (on 𝗕𝗛^{T}) is equivalent to B localization. When observations are located off grid points, or relate to more than one grid point, this technique exhibits hybrid properties of B localization and R localization. The problem of defining distance for vertically integrated measurements, such as satellite observations (Campbell et al. 2010), is equally challenging for 𝗕𝗛^{T} and R-localization techniques, as both require a distance between an observation and model grid point, and this issue is a motivation for adaptive localization (Anderson 2007; Bishop and Hodyss 2009b). This study focuses on horizontal localization with point observations; vertical localization in the LETKF is addressed in Miyoshi and Sato (2007).

## 3. Simple model experiments

The goal of this section is to demonstrate the impact of EnKF localization on balance using a simple model consisting of one-dimensional balanced waveforms. These initially balanced wave solutions (which are not integrated forward in time) serve as truth and background ensemble states for identical twin data assimilation experiments; any disruption to the balance of the resulting analysis is thus easily detectable and attributable to the properties of the EnKF technique.

### a. Simple model description

*x*direction for a rotating (constant Coriolis parameter

*f*), inviscid fluid: The geostrophic balance between the pressure gradient and Coriolis terms can thus be stated as Here

*υ*is the geostrophic wind. Assuming that the wave structure is uniform in the

_{g}*y*direction, harmonic form is applied to the perturbation variables to achieve a wave solution for

*h*, with

*h*

_{depth}being the mean depth of the fluid,

*h*

_{amp}the amplitude of the height perturbation,

*k*the wavenumber, and

*x*

_{ps}a wave phase shift: Assuming geostrophically balanced wind field, we arrive at the wave solution for

*υ*: For the simple model, consider a one-dimensional nonperiodic domain of 5000 km along the

*x*axis, with model grid points spaced regularly at 50-km intervals. The Coriolis parameter

*f*was selected to be 10

^{−4}s

^{−1}, a reasonable value for the midlatitudes.

### b. Experiment design

The truth state and five background ensemble members, plotted in Fig. 2, are defined for both height and *υ* component of the wind. Each ensemble member is generated by randomly selecting a height perturbation amplitude from a uniform distribution of [9, 11] m, a wavelength from [1950, 2050] km and phase shift from [−50, 50] km. The truth waveform (amplitude = 10 m, wavelength = 2100 km, offset = −100 km) is fixed in order to avoid having a mean background state too close to the ensemble mean. This would be undesirable, as an analysis that moves farther from the background toward an observation would be overly penalized, whereas one that remained close to the background would be falsely rewarded. The meridional wind waveform is then generated to be in geostrophic balance with the height waveform. These waves are represented discretely as height and meridional wind values at each of the 101 model grid points. Observations of both *h* and *υ* at regularly spaced grid points 250 km apart are chosen based upon the truth value at the corresponding model grid point plus a random observation error equal to 10% of the wave amplitude.

Ensemble mean analyses resulting from assimilation using no localization, B localization, and R localization using various localization distances *L* are compared. As the wind can be partitioned into geostrophic and ageostrophic components (*υ* = *υ _{g}* +

*υ*), the RMS value of

_{a}*υ*over all grid points is used as a summary metric of imbalance; accuracy is also assessed as the RMS difference from the truth. To obtain significant results not dependent upon the peculiarities of a specific random configuration of ensemble members and observation errors, each configuration is repeated 100 times in a Monte Carlo experiment. Note that the model is not advanced in time, so boundary conditions are not needed.

_{a}### c. Simple model results

Figure 3a shows the dependence of RMSE for each analysis as a function of localization distance *L*. LETKF rather than the generic EnKF formula is used for R localization; the differences in accuracy and balance metrics between LETKF and EnSRF R localization for this experiment (not shown) are on the order of 1%, so the comparison is fair. The R localization has an optimal scale of *L* = 500 km, whereas B localization is close to optimal at around *L* = 1000 km and larger for 5 ensemble members. A scenario using 40 ensemble members and no localization is also plotted as a best-case performance scenario to which the localized 5-ensemble member analyses aspire. Note that results for *υ*-wind error (not shown) are similar. An explanation for the disparity in optimal length scales is provided in the appendix.

Figure 3b shows the dependence of RMS imbalance (ageostrophic wind) for each analysis as a function of localization distance *L*. Analyses without localization show no ageostrophic wind, which is to be expected from the design of the experiment. For the localized cases, as the localization distance increases, the analysis becomes more balanced. The R localization is always more balanced than B localization for the same localization distance *L*, although the levels of imbalance are comparable when considering the optimal configuration of each method.

## 4. SPEEDY model experiments

### a. Measuring balance in a realistic model

In a realistic atmospheric model we can no longer assume that the background state is initially balanced, since an atmosphere with purely geostrophic flow would not allow for interesting weather such as intense baroclinic development and the vertical motion associated with heavy precipitation. Therefore, although much of the energy in the atmosphere is associated with the slow mode (Daley 1991), there is a natural level of imbalance in the atmosphere. The challenge is to differentiate between this background amount of imbalance, and additional spurious amounts introduced as an artifact of data assimilation.

There are several metrics for evaluating atmospheric imbalance. Section 3 (and Lorenc 2003) uses the magnitude of the ageostrophic wind. While this metric is straightforward to compute, it is not applicable at all latitudes; there are also more sophisticated balance equations, such as nonlinear balance (Raymond 1992), to consider. High-frequency oscillations can be diagnosed directly by examining the second derivative of the surface pressure field in time (Houtekamer and Mitchell 2005). Finally, the analysis can be compared to an initialized (filtered) version of itself using a Lynch and Huang (1992) Lanczos digital filter [as in Mitchell et al. (2002)] that removes high-frequency oscillations, and thus inertial-gravity waves, from the model time series (not included in this study). Similarly, Kepert (2009) used the magnitude of the nonlinear normal mode initialization (NNMI) increment as a measure of balance. The surface pressure and digital filter metrics require model output from several time steps at a relatively fine temporal resolution (smaller than 1 h).

### b. Experiment design

The Simplified Parameterizations, Primitive Equation Dynamics (SPEEDY) model (Molteni, 2003) is an atmospheric global circulation model of intermediate complexity designed for climate experiments. While containing many of the physics components found in larger models (including convection, condensation, cloud, radiation, and surface flux parameterizations), it is computationally inexpensive so it can be run on a single processor. There are seven vertical levels using the sigma coordinate system, with a horizontal spectral resolution of T30, which corresponds to a standard Gaussian grid of 96 by 48 points. The time scheme is leapfrog. There are five dynamical variables included in the output: zonal wind *u*, meridional wind *υ*, temperature *T*, specific humidity, and surface pressure *p _{s}*. Miyoshi (2005) modified the SPEEDY model for weather forecasting by creating output every 6 h, and implemented several data assimilation techniques on the SPEEDY model. Horizontal diffusion (of vorticity, divergence, temperature, and specific humidity) in the SPEEDY model is done with the fourth power of the Laplacian, and is applied on the sigma surfaces. Maximum damping time is 18 h for temperature and vorticity, and 9 h for divergence, with an additional 12 h applied at the top level (representing the stratosphere). There is also vertical diffusion that simulates shallow convection in regions with conditional instability, as well as water vapor and static energy vertical diffusion (Molteni 2003). Frequency damping with a Robert–Asselin filter (with filter parameter equal to 0.05) is included in the SPEEDY model to suppress the spurious computational mode. Amezcua et al. (2011) has examined the use of a Robert–Asselin–Williams (RAW) filter (which successfully dampens the computational mode without damping the physical solution; Williams, 2009) with the SPEEDY model, and found that there are very few changes to the model climatology that pass a field significance test, and the quality of the forecasts was slightly improved. This change in the high-frequency damping did not seem to affect the model balance. Note that the RAW filter is not employed in the experiments presented in this paper.

The ultimate goal of using the SPEEDY model is a realistic comparison of B localization and R localization in terms of balance and accuracy. Here, B localization is employed with the EnSRF algorithm (Whitaker and Hamill 2002), whereas R localization is used with LETKF (Hunt et al. 2007). In addition, a third configuration using the EnSRF with R localization is employed to investigate whether any differences between the first two configurations are primarily due to variation in localization technique rather than assimilation algorithm (serial vs simultaneous, etc.); see Holland and Wang (2010) for an independent comparison of EnSRF and LETKF. All systems use identical observations, which are generated as random perturbations from the nature run, or true state, in an identical twin experiment. The observation network used for this study approximately follows the rawinsonde locations (see Fig. 7), with all observations located on model grid points. Observations are located at each of the seven model levels. Observation error is 1 K for temperature, 1 m s^{−1} for *u* and *υ* wind magnitudes, 1 g kg^{−1} for specific humidity, and 1 mb for surface pressure. Multiplicative inflation of 2% is applied to the background ensemble spread. Vertical localization is by model level so that an observation corresponding to one of the model’s seven levels does not impact any other level; previous experience with the SPEEDY model has shown that vertical correlations for wind and temperature errors are minimal. The ensembles are composed of 20 members, with initial conditions taken from consecutive dates in January 1982.

The forecast-assimilation cycle is every 6 h over a period of 48 days from 1 February to 20 March 1982. The assessment of accuracy is made by comparing the ensemble mean analysis of wind magnitude to the truth at each 6-h period. Balance is assessed through the magnitude of the ageostrophic wind, as well as the second derivative of surface pressure. These metrics are applied during the month-long period of 20 February–20 March following 20 days of spinup. Wind metrics are obtained from model level 4 (∼500 hPa). Results are reported as an areal mean, either globally or over midlatitude bands (∼30°–60°) separately for the Northern Hemisphere (NH) and Southern Hemisphere (SH).

### c. SPEEDY model results

Figure 4 shows the accuracy of analyses (measured by mean absolute wind error at ∼500 hPa) for the EnSRF B localization and LETKF R localization relative to the true state as a function of localization distance parameter *L* [see the discussion surrounding (4) and (5)]. The performance of the system is highly dependent upon the choice of localization parameter. Too long a localization distance and the system is dominated by spurious observation increments that prevent it from converging to the truth, whereas too short a localization distance and observations introduce imbalanced increments, as well as fail to adequately impact their neighborhood of grid points. An optimal localization distance parameter *L* with respect to accuracy is 500 km for R localization, and 750 km for B localization. Error is higher and the optimal length scale is slightly longer for the SH compared to the NH (not shown), as the former has a relative paucity of observations. The performance for R localization in both LETKF and EnSRF is similar, particularly for *L* < 500 km where the results are essentially identical. The results for wind error at other vertical levels (not shown) reveal a similar dependence on localization, with slightly higher errors as altitude increases. Note that the areal mean ensemble spread (not shown) is also highly sensitive to *L*, with shorter *L* corresponding to greater ensemble spread. Observation information reduces the uncertainty of an analysis; for shorter localization distances this reduction in analysis spread takes place over smaller regions (nearest to the observations), and thus the areal mean ensemble spread remains high.

Figure 5 reveals the performance of the two systems with respect to balance, measured by the mean magnitude of the ageostrophic wind at ∼500 hPa as a function of the localization distance parameter *L*. There exists a larger natural state of geostrophic imbalance in the NH (∼3 m s^{−1}) compared to the SH (∼2 m s^{−1}) due to the presence of the Himalayan plateau protruding into the midlatitude belt as well as the fact that the experiment occurred in the NH winter with its stronger wind speeds. In all cases, the imbalance of the analyses is larger than that of the true state, indicating that data assimilation has introduced artificial imbalance. Although the magnitudes of the mean ageostrophic winds are higher for the NH, the difference in imbalance between the nature run and assimilation runs (assimilation-induced imbalance) is greater for the SH. Short localization distances (*L* < 300 km) are detrimental to balance, which agrees with the results of section 3 using a simple model. For very long localization distances (*L* = 2000 km), presumed spurious correlations can lead to larger values of both error and imbalance. Examination of performance time series reveals that values of imbalance tend to stabilize, along with the error, after 20 days of spinup, although there are day-to-day fluctuations on the order of 0.5 m s^{−1} that are reflected in both the nature run and assimilation analyses.

Figure 6 also depicts imbalance, but measured by the second derivative of surface pressure at each model time step. As in Fig. 5, short localization distances (*L* < 300 km) are very harmful to balance. Here, the NH is significantly more balanced than the SH, which agrees with the result for assimilation-induced imbalance in Fig. 5. Optimal values of *L* are slightly larger using this metric compared to Fig. 5; averaging the optimal *L* values for both metrics of imbalance results in an optimal *L* that agrees with the results for accuracy in Fig. 4. The occasional lack of smoothness in the relationship curves between imbalance and *L* in Figs. 5 –6 reveal that an evaluation time period of at least one month is required to overcome sampling error for these techniques.

Figure 7 reveals the spatial distribution of imbalance as a time mean over the period from 20 February to 20 March. For short localization distances, imbalance is large in the immediate vicinity of observations. For long localization distances, imbalance is smaller and spread over broader areas. This finding agrees with the Lorenc (2003) explanation using Fig. 1 in that imbalance can be introduced in the region where the impact of an observation moves toward zero. The circular patterns of imbalance surrounding the Southern Ocean islands in the case of *L* = 250 km demonstrate the detrimental impact of strong localization resulting from a sharp transition between a region with strong observation impact and a region with little observation impact. Imbalance is greatest along the Pacific coast of South America; the lack of observations in the South Pacific leads to large observation increments in the region. Inaccurate background fields, which require larger subsequent analysis increments resulting in greater potential for imbalance introduced by data assimilation, may explain the somewhat unexpected increase in imbalance for large *L* in Figs. 5 and 6.

## 5. Conclusions

This study has examined the impact of EnKF localization techniques upon the accuracy and balance of analyses. Localization is used to combat spurious correlations due to sampling error from finite ensemble size, to take advantage of low-dimensionality in local regions, and for efficient computation. Localization techniques can be classified into two methods: B localization, where the background error covariance is modified by a distant-dependent localization function, and R localization, where observation error variances are increased as distance from the analysis grid point increases. Variations of the B-localization technique are appropriate for EnSRF where the entire domain is updated with each observation, whereas R localization is used for LETKF as the background error covariances are specified in ensemble space and each model grid point is updated independently. In addition to accurately depicting the state of the system, atmospheric data assimilation should produce a balanced analysis so that information is not lost through spurious inertial-gravity wave propagation.

We first described experiments with simple, one-dimensional waveforms based upon the shallow-water equations. As the background ensemble is initially balanced, imbalance introduced by data assimilation is easy to measure as the magnitude of the ageostrophic wind. The two techniques have differing optimal localization distances *L* with respect to analysis accuracy; approximately 500 km for R localization, and 1000 km or larger for B localization. For the same localization length R localization is more balanced than B localization, but the balance of both techniques improves as *L* grows larger.

We then made a more realistic comparison between EnSRF B localization and LETKF R localization involving the global SPEEDY model in identical twin experiments. Here, the background state can no longer be assumed to be in balance. Two methods for evaluating imbalance are used: the magnitude of the ageostrophic wind and the second derivative of surface pressure. The two localization techniques are roughly comparable in performance with respect to localization and balance when the optimal length scale of *L* is selected: 500 km for R localization, and 750 km for B localization. This result is consistent with the discussion in the appendix, which demonstrates that B localization is more severe than R localization for the same *L*. We conclude that the differences in data assimilation algorithm (LETKF vs EnSRF) are smaller than differences in localization technique when identifying the optimal localization distance *L*.

Both types of localization introduce imbalance; as the solution reverts toward the background at long distances from observations, the damping of the height and wind increments results in a smaller wind increment, but a larger height gradient, which does not satisfy the geostrophic relationship. Localization can also introduce excess divergence to an analysis (Kepert 2009). The localization parameter *L* should be tuned depending on the particular scale and application of data assimilation, as well as the size of the ensemble. Tuning inflation values for each localization parameter *L* may result in improved performance. Future studies should consider balance in the context of the adaptive localization methods, as these techniques do not necessarily require a specification of *L*.

We are grateful to the UMD Weather Chaos Group, Jeff Anderson, Craig Bishop, Dale Barker, two anonymous reviewers, and attendees of the WWRP/THORPEX Workshop on 4D-VAR and Ensemble Kalman Filter Inter-Comparisons in Buenos Aires (October 2008) for their helpful comments and critiques of this project. This work was supported by NASA Grants NNX07AM97G and NNX08AD40G, DOE Grant DEFG0207ER64437, ONR Grant N000140910418, and NOAA Grant NA09OAR4310178.

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# APPENDIX

## Mathematical Analysis of B and R Localizations

*x*

_{1}and

*x*

_{2}at grid points 1 and 2, respectively. Consider a single observation of

*x*

_{1}, with 𝗛 = [1, 0]. Using (2), the Kalman gain matrix (without localization) can be specified as follows: where

*B*is the background covariance between

_{ij}*x*and

_{i}*x*, and

_{j}*R*

_{1}is the observation covariance.

*f*

_{Bloc}(4) to (A1). Using

*f*

_{Bloc}(

*d*) = 1, where

_{ij}*d*is the distance between grid points

_{ij}*i*and

*j*,

*K*

_{1}remains the same but

*K*

_{2}becomes Note that since we are assimilating a single observation located on a grid point, (A2) is identical for both B localization and the 𝗕𝗛

^{T}localization described at the end of section 2. Now we apply the R-localization function

*f*

_{Rloc}(5). Again,

*K*

_{1}remains the same as in (A1). Using the fact that

*K*

_{2}becomes

Comparing (A2) and (A3), the R localization (A3) has an extra localization term in the denominator. The localization function *f*_{Bloc} ranges from 1 to 0. Therefore, the amplitude of *K*_{2} (and hence the corresponding analysis increment) will be larger at grid point 2 for R localization than for B localization. This means that with B localization, the analysis reverts to the background (ignores observation information) more quickly with distance compared to R localization. In this respect, B localization can be considered more “severe” than R localization for the same localization distance parameter *L*; see discussion of (11) and (12) in Miyoshi and Yamane (2007).

*R*

_{1}and

*R*

_{2}represent the error variances of the two observations. Because the analysis process is the same for

*x*

_{1}and

*x*

_{2}by permuting the indices 1 and 2, we consider the impact of the localizations on

*x*

_{1}(i.e.,

*K*

_{1}) only. The application of the B-localization function leads to The application of the R-localization function with

*f*

_{Bloc}, we note that the 𝗕𝗛

^{T}terms are identical. However, the 𝗛𝗕𝗛

^{T}terms differ. Using this formulation, we arrive at an 𝗛𝗕𝗛

^{T}matrix for R localization in (A7) that is no longer symmetric, although the original formulation of R localization in terms of the R-localization function had symmetric covariance matrices (A6). Consequently, it is not straightforward to compute a priori the quantitative difference in localization strength between the techniques in the case of multiple observations. With localized serial EnSRF, the resulting analysis depends upon the order in which the observations are assimilated; this is not true for the simultaneous assimilation of LETKF. For this study we focus on R localization with the LETKF algorithm, performing EnSRF R localization in order to confirm that differences in the results are primarily due to difference in localization technique rather than algorithm. Note that EnSRF R localization requires a unique 𝗥 (and hence a separate computation) for every gridpoint–observation pair.