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## Abstract

Localization is a method for reducing the impact of sampling errors in ensemble Kalman filters. Here, the regression coefficient, or gain, relating ensemble increments for observed quantity *y* to increments for state variable *x* is multiplied by a real number *α* defined as a localization. Localization of the impact of observations on model state variables is required for good performance when applying ensemble data assimilation to large atmospheric and oceanic problems. Localization also improves performance in idealized low-order ensemble assimilation applications. An algorithm that computes localization from the output of an ensemble observing system simulation experiment (OSSE) is described. The algorithm produces localizations for sets of pairs of observations and state variables: for instance, all state variables that are between 300- and 400-km horizontal distance from an observation. The algorithm is applied in a low-order model to produce localizations from the output of an OSSE and the computed localizations are then used in a new OSSE. Results are compared to assimilations using tuned localizations that are approximately Gaussian functions of the distance between an observation and a state variable. In most cases, the empirically computed localizations produce the lowest root-mean-square errors in subsequent OSSEs. Localizations derived from OSSE output can provide guidance for localization in real assimilation experiments. Applying the algorithm in large geophysical applications may help to tune localization for improved ensemble filter performance.

## Abstract

Localization is a method for reducing the impact of sampling errors in ensemble Kalman filters. Here, the regression coefficient, or gain, relating ensemble increments for observed quantity *y* to increments for state variable *x* is multiplied by a real number *α* defined as a localization. Localization of the impact of observations on model state variables is required for good performance when applying ensemble data assimilation to large atmospheric and oceanic problems. Localization also improves performance in idealized low-order ensemble assimilation applications. An algorithm that computes localization from the output of an ensemble observing system simulation experiment (OSSE) is described. The algorithm produces localizations for sets of pairs of observations and state variables: for instance, all state variables that are between 300- and 400-km horizontal distance from an observation. The algorithm is applied in a low-order model to produce localizations from the output of an OSSE and the computed localizations are then used in a new OSSE. Results are compared to assimilations using tuned localizations that are approximately Gaussian functions of the distance between an observation and a state variable. In most cases, the empirically computed localizations produce the lowest root-mean-square errors in subsequent OSSEs. Localizations derived from OSSE output can provide guidance for localization in real assimilation experiments. Applying the algorithm in large geophysical applications may help to tune localization for improved ensemble filter performance.

## Abstract

Covariance localization is an essential component of ensemble-based data assimilation systems for large geophysical applications with limited ensemble sizes. For integral observations like the satellite radiances, where the concepts of location or vertical distance are not well defined, vertical localization in observation space is not as straightforward as in model space. The detailed differences between model space and observation space localizations are examined using a real radiance observation. Counterintuitive analysis increments can be obtained with model space localization; the magnitude of the increment can increase and the increment can change sign when the localization scale decreases. This occurs when there are negative background-error covariances and a predominately positive forward operator. Too narrow model space localization can neglect the negative background-error covariances and result in the counterintuitive analysis increments. An idealized 1D model with integral observations and known true error covariance is then used to compare errors resulting from model space and observation space localizations. Although previous studies have suggested that observation space localization is inferior to model space localization for satellite radiances, the results from the 1D model reveal that observation space localization can have advantages over model space localization when there are negative background-error covariances. Differences between model space and observation space localizations disappear as ensemble size, observation error variance, and localization scale increase. Thus, large ensemble sizes and vertical localization length scales may be needed to more effectively assimilate radiance observations.

## Abstract

Covariance localization is an essential component of ensemble-based data assimilation systems for large geophysical applications with limited ensemble sizes. For integral observations like the satellite radiances, where the concepts of location or vertical distance are not well defined, vertical localization in observation space is not as straightforward as in model space. The detailed differences between model space and observation space localizations are examined using a real radiance observation. Counterintuitive analysis increments can be obtained with model space localization; the magnitude of the increment can increase and the increment can change sign when the localization scale decreases. This occurs when there are negative background-error covariances and a predominately positive forward operator. Too narrow model space localization can neglect the negative background-error covariances and result in the counterintuitive analysis increments. An idealized 1D model with integral observations and known true error covariance is then used to compare errors resulting from model space and observation space localizations. Although previous studies have suggested that observation space localization is inferior to model space localization for satellite radiances, the results from the 1D model reveal that observation space localization can have advantages over model space localization when there are negative background-error covariances. Differences between model space and observation space localizations disappear as ensemble size, observation error variance, and localization scale increase. Thus, large ensemble sizes and vertical localization length scales may be needed to more effectively assimilate radiance observations.

## Abstract

The analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced during a model integration is often computed once and assumed to be constant for an assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus, the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems.

To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) used with the EnKF is presented. It constructs time-varying analysis increments by applying all observations in an assimilation window to state variables at different times during the assimilation window. It then gradually applies these time-varying analysis increments through the assimilation window. Results from a dry two-layer primitive equation model and the NCEP GFS show that EnKF with 4DIAU (EnKF-4DIAU) and 3DIAU (EnKF-3DIAU) reduce imbalances in the analysis compared to EnKF without initialization (EnKF-RAW). EnKF-4DIAU retains the time-varying information in the analysis increments better than EnKF-3DIAU, and produces better analysis and forecast than either EnKF-RAW or EnKF-3DIAU.

## Abstract

The analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced during a model integration is often computed once and assumed to be constant for an assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus, the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems.

To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) used with the EnKF is presented. It constructs time-varying analysis increments by applying all observations in an assimilation window to state variables at different times during the assimilation window. It then gradually applies these time-varying analysis increments through the assimilation window. Results from a dry two-layer primitive equation model and the NCEP GFS show that EnKF with 4DIAU (EnKF-4DIAU) and 3DIAU (EnKF-3DIAU) reduce imbalances in the analysis compared to EnKF without initialization (EnKF-RAW). EnKF-4DIAU retains the time-varying information in the analysis increments better than EnKF-3DIAU, and produces better analysis and forecast than either EnKF-RAW or EnKF-3DIAU.

## Abstract

Objective data assimilation methods such as variational and ensemble algorithms are attractive from a theoretical standpoint. Empirical nudging approaches are computationally efficient and can get around some amount of model error by using arbitrarily large nudging coefficients. In an attempt to take advantage of the strengths of both methods for analyses, combined nudging-ensemble approaches have been recently proposed. Here the two-scale Lorenz model is used to elucidate how the forecast error from nudging, ensemble, and nudging-ensemble schemes varies with model error. As expected, an ensemble filter and smoother are closest to optimal when model errors are small or absent. Model error is introduced by varying model forcing, coupling between scales, and spatial filtering. Nudging approaches perform relatively better with increased model error; use of poor ensemble covariance estimates when model error is large harms the nudging-ensemble performance. Consequently, nudging-ensemble methods always produce error levels between the objective ensemble filters and empirical nudging, and can never provide analyses or short-range forecasts with lower errors than both. As long as the nudged state and the ensemble-filter state are close enough, the ensemble statistics are useful for the nudging, and fully coupling the ensemble and nudging by centering the ensemble on the nudged state is not necessary. An ensemble smoother produces the overall smallest errors except for with very large model errors. Results are qualitatively independent of tuning parameters such as covariance inflation and localization.

## Abstract

Objective data assimilation methods such as variational and ensemble algorithms are attractive from a theoretical standpoint. Empirical nudging approaches are computationally efficient and can get around some amount of model error by using arbitrarily large nudging coefficients. In an attempt to take advantage of the strengths of both methods for analyses, combined nudging-ensemble approaches have been recently proposed. Here the two-scale Lorenz model is used to elucidate how the forecast error from nudging, ensemble, and nudging-ensemble schemes varies with model error. As expected, an ensemble filter and smoother are closest to optimal when model errors are small or absent. Model error is introduced by varying model forcing, coupling between scales, and spatial filtering. Nudging approaches perform relatively better with increased model error; use of poor ensemble covariance estimates when model error is large harms the nudging-ensemble performance. Consequently, nudging-ensemble methods always produce error levels between the objective ensemble filters and empirical nudging, and can never provide analyses or short-range forecasts with lower errors than both. As long as the nudged state and the ensemble-filter state are close enough, the ensemble statistics are useful for the nudging, and fully coupling the ensemble and nudging by centering the ensemble on the nudged state is not necessary. An ensemble smoother produces the overall smallest errors except for with very large model errors. Results are qualitatively independent of tuning parameters such as covariance inflation and localization.

## Abstract

To investigate the impacts of frequently assimilating only surface pressure (PS) observations, the Data Assimilation Research Testbed and the Community Atmosphere Model (DART/CAM) are used for observing system simulation experiments with the ensemble Kalman filter. An empirical localization function (ELF) is used to effectively spread the information from PS in the vertical. The ELF minimizes the root-mean-square difference between the truth and the posterior ensemble mean for state variables. The temporal frequency of the observations is increased from 6 to 3 h, and then 1 h. By observing only PS, the uncertainty throughout the entire depth of the troposphere can be constrained. The analysis error over the entire depth of the troposphere, especially the middle troposphere, is reduced with increased assimilation frequency. The ELF is similar to the vertical localization function used in the Twentieth-Century Reanalysis (20CR); thus, it demonstrates that the current vertical localization in the 20CR is close to the optimal localization function.

## Abstract

To investigate the impacts of frequently assimilating only surface pressure (PS) observations, the Data Assimilation Research Testbed and the Community Atmosphere Model (DART/CAM) are used for observing system simulation experiments with the ensemble Kalman filter. An empirical localization function (ELF) is used to effectively spread the information from PS in the vertical. The ELF minimizes the root-mean-square difference between the truth and the posterior ensemble mean for state variables. The temporal frequency of the observations is increased from 6 to 3 h, and then 1 h. By observing only PS, the uncertainty throughout the entire depth of the troposphere can be constrained. The analysis error over the entire depth of the troposphere, especially the middle troposphere, is reduced with increased assimilation frequency. The ELF is similar to the vertical localization function used in the Twentieth-Century Reanalysis (20CR); thus, it demonstrates that the current vertical localization in the 20CR is close to the optimal localization function.

## Abstract

Two techniques for estimating good localization functions for serial ensemble Kalman filters are compared in observing system simulation experiments (OSSEs) conducted with the dynamical core of an atmospheric general circulation model. The first technique, the global group filter (GGF), minimizes the root-mean-square (RMS) difference between the estimated regression coefficients using a hierarchical ensemble filter. The second, the empirical localization function (ELF), minimizes the RMS difference between the true values of the state variables and the posterior ensemble mean. Both techniques provide an estimate of the localization function for an observation’s impact on a state variable with few a priori assumptions about the localization function. The ELF localizations can have values larger than 1.0 at small distances, indicating that this technique addresses localization but also can correct the prior ensemble spread in the same way as a variance inflation when needed. OSSEs using ELF localizations generally have smaller root-mean-square error (RMSE) than the optimal Gaspari and Cohn (GC) localization function obtained by empirically tuning the GC width. The localization functions estimated by the GGF are broader than those from the ELF, and the OSSEs with the GGF localization generally have larger RMSE than the optimal GC localization function. The GGFs are too broad because of spurious correlation biases that occur in the OSSEs. These errors can be reduced by using a stochastic EnKF with perturbed observations instead of a deterministic EAKF.

## Abstract

Two techniques for estimating good localization functions for serial ensemble Kalman filters are compared in observing system simulation experiments (OSSEs) conducted with the dynamical core of an atmospheric general circulation model. The first technique, the global group filter (GGF), minimizes the root-mean-square (RMS) difference between the estimated regression coefficients using a hierarchical ensemble filter. The second, the empirical localization function (ELF), minimizes the RMS difference between the true values of the state variables and the posterior ensemble mean. Both techniques provide an estimate of the localization function for an observation’s impact on a state variable with few a priori assumptions about the localization function. The ELF localizations can have values larger than 1.0 at small distances, indicating that this technique addresses localization but also can correct the prior ensemble spread in the same way as a variance inflation when needed. OSSEs using ELF localizations generally have smaller root-mean-square error (RMSE) than the optimal Gaspari and Cohn (GC) localization function obtained by empirically tuning the GC width. The localization functions estimated by the GGF are broader than those from the ELF, and the OSSEs with the GGF localization generally have larger RMSE than the optimal GC localization function. The GGFs are too broad because of spurious correlation biases that occur in the OSSEs. These errors can be reduced by using a stochastic EnKF with perturbed observations instead of a deterministic EAKF.

## Abstract

The empirical localization algorithm described here uses the output from an observing system simulation experiment (OSSE) and constructs localization functions that minimize the root-mean-square (RMS) difference between the truth and the posterior ensemble mean for state variables. This algorithm can automatically provide an estimate of the localization function and does not require empirical tuning of the localization scale. It can compute an appropriate localization function for any potential observation type and kind of state variable. The empirical localization algorithm is investigated in the Community Atmosphere Model, version 5 (CAM5). The empirical localization function (ELF) is computed for the horizontal and vertical separately so that the vertical localization is explored explicitly. The horizontal and vertical ELFs are also computed for different geographic regions. The ELFs varying with region have advantages over the single global ELF in the horizontal and vertical, because different localization functions are more effective in different regions. The ELFs computed from an OSSE can be used as the localization in a subsequent OSSE. After three iterations, the ELFs appear to have converged. When used as localization in an OSSE, the converged ELFs produce a significantly smaller RMS error of temperature and zonal and meridional winds than the best Gaspari–Cohn (GC) localization for a dependent verification period using the observations from the original OSSE, and a similar RMS error to the best GC for an independent verification period. The converged ELFs have a significantly smaller RMS error of surface pressure than the best GC for both dependent and independent verification periods.

## Abstract

The empirical localization algorithm described here uses the output from an observing system simulation experiment (OSSE) and constructs localization functions that minimize the root-mean-square (RMS) difference between the truth and the posterior ensemble mean for state variables. This algorithm can automatically provide an estimate of the localization function and does not require empirical tuning of the localization scale. It can compute an appropriate localization function for any potential observation type and kind of state variable. The empirical localization algorithm is investigated in the Community Atmosphere Model, version 5 (CAM5). The empirical localization function (ELF) is computed for the horizontal and vertical separately so that the vertical localization is explored explicitly. The horizontal and vertical ELFs are also computed for different geographic regions. The ELFs varying with region have advantages over the single global ELF in the horizontal and vertical, because different localization functions are more effective in different regions. The ELFs computed from an OSSE can be used as the localization in a subsequent OSSE. After three iterations, the ELFs appear to have converged. When used as localization in an OSSE, the converged ELFs produce a significantly smaller RMS error of temperature and zonal and meridional winds than the best Gaspari–Cohn (GC) localization for a dependent verification period using the observations from the original OSSE, and a similar RMS error to the best GC for an independent verification period. The converged ELFs have a significantly smaller RMS error of surface pressure than the best GC for both dependent and independent verification periods.

## Abstract

Ensemble sensitivities have proven a useful alternative to adjoint sensitivities for large-scale dynamics, but their performance in multiscale flows has not been thoroughly examined. When computing sensitivities, the analysis covariance is usually approximated with the corresponding diagonal matrix, leading to a simple univariate regression problem rather than a more general multivariate regression problem. Sensitivity estimates are affected by sampling error arising from a finite ensemble and can lead to an overestimated response to an analysis perturbation. When forecasts depend on many details of an analysis, it is reasonable to expect that the diagonal approximation is too severe. Because spurious covariances are more likely when correlations are weak, computing the sensitivity with a multivariate regression that retains the full analysis covariance may increase the need for sampling error mitigation. The purpose of this work is to clarify the effects of the diagonal approximation, and investigate the need for mitigating spurious covariances arising from sampling error. A two-scale model with realistic spatial covariances is the basis for experimentation. For most problems, an efficient matrix inversion is possible by finding a minimum-norm solution, and employing appropriate matrix factorization. A published hierarchical approach for estimating regression factors for tapering (localizing) covariances is used to measure the effects of sampling error. Compared to univariate regressions in the diagonal approximation, skill in predicting a nonlinear response from the linear sensitivities is superior when localized multivariate sensitivities are used, particularly when fast scales are present, model error is present, and the observing network is sparse.

## Abstract

Ensemble sensitivities have proven a useful alternative to adjoint sensitivities for large-scale dynamics, but their performance in multiscale flows has not been thoroughly examined. When computing sensitivities, the analysis covariance is usually approximated with the corresponding diagonal matrix, leading to a simple univariate regression problem rather than a more general multivariate regression problem. Sensitivity estimates are affected by sampling error arising from a finite ensemble and can lead to an overestimated response to an analysis perturbation. When forecasts depend on many details of an analysis, it is reasonable to expect that the diagonal approximation is too severe. Because spurious covariances are more likely when correlations are weak, computing the sensitivity with a multivariate regression that retains the full analysis covariance may increase the need for sampling error mitigation. The purpose of this work is to clarify the effects of the diagonal approximation, and investigate the need for mitigating spurious covariances arising from sampling error. A two-scale model with realistic spatial covariances is the basis for experimentation. For most problems, an efficient matrix inversion is possible by finding a minimum-norm solution, and employing appropriate matrix factorization. A published hierarchical approach for estimating regression factors for tapering (localizing) covariances is used to measure the effects of sampling error. Compared to univariate regressions in the diagonal approximation, skill in predicting a nonlinear response from the linear sensitivities is superior when localized multivariate sensitivities are used, particularly when fast scales are present, model error is present, and the observing network is sparse.

## Abstract

To ameliorate suboptimality in ensemble data assimilation, methods have been introduced that involve expanding the ensemble size. Such expansions can incorporate model space covariance localization and/or estimates of climatological or model error covariances. Model space covariance localization in the vertical overcomes problematic aspects of ensemble-based satellite data assimilation. In the case of the ensemble transform Kalman filter (ETKF), the expanded ensemble size associated with vertical covariance localization would also enable the simultaneous update of entire vertical columns of model variables from hyperspectral and multispectral satellite sounders. However, if the original formulation of the ETKF were applied to an expanded ensemble, it would produce an analysis ensemble that was the same size as the expanded forecast ensemble. This article describes a variation on the ETKF called the gain ETKF (GETKF) that takes advantage of covariances from the expanded ensemble, while producing an analysis ensemble that has the required size of the unexpanded forecast ensemble. The approach also yields an inflation factor that depends on the localization length scale that causes the GETKF to perform differently to an ensemble square root filter (EnSRF) using the same expanded ensemble. Experimentation described herein shows that the GETKF outperforms a range of alternative ETKF-based solutions to the aforementioned problems. In cycling data assimilation experiments with a newly developed *storm-track* version of the Lorenz-96 model, the GETKF analysis root-mean-square error (RMSE) matches the EnSRF RMSE at shorter than optimal localization length scales but is superior in that it yields smaller RMSEs for longer localization length scales.

## Abstract

To ameliorate suboptimality in ensemble data assimilation, methods have been introduced that involve expanding the ensemble size. Such expansions can incorporate model space covariance localization and/or estimates of climatological or model error covariances. Model space covariance localization in the vertical overcomes problematic aspects of ensemble-based satellite data assimilation. In the case of the ensemble transform Kalman filter (ETKF), the expanded ensemble size associated with vertical covariance localization would also enable the simultaneous update of entire vertical columns of model variables from hyperspectral and multispectral satellite sounders. However, if the original formulation of the ETKF were applied to an expanded ensemble, it would produce an analysis ensemble that was the same size as the expanded forecast ensemble. This article describes a variation on the ETKF called the gain ETKF (GETKF) that takes advantage of covariances from the expanded ensemble, while producing an analysis ensemble that has the required size of the unexpanded forecast ensemble. The approach also yields an inflation factor that depends on the localization length scale that causes the GETKF to perform differently to an ensemble square root filter (EnSRF) using the same expanded ensemble. Experimentation described herein shows that the GETKF outperforms a range of alternative ETKF-based solutions to the aforementioned problems. In cycling data assimilation experiments with a newly developed *storm-track* version of the Lorenz-96 model, the GETKF analysis root-mean-square error (RMSE) matches the EnSRF RMSE at shorter than optimal localization length scales but is superior in that it yields smaller RMSEs for longer localization length scales.

## Abstract

For ensemble-based data assimilation, localization is used to limit the impact of observations on physically distant state variables to reduce spurious error correlations caused by limited ensemble size. Traditionally, the localization value applied is spatially homogeneous. Yet there are potentially larger errors and different covariance length scales in precipitation systems, and that may justify the use of different localization functions for precipitating and nonprecipitating regions. Here this is examined using empirical localization functions (ELFs). Using output from an ensemble observing system simulation experiment (OSSE), ELFs provide estimates of horizontal and vertical localization for different observation types in regions with and without precipitation. For temperature and *u*- and *υ*-wind observations, the ELFs for precipitating regions are shown to have smaller horizontal localization scales than for nonprecipitating regions. However, the ELFs for precipitating regions generally have larger vertical localization scales than for nonprecipitating regions. The ELFs are smoothed and then applied in three additional OSSEs. Spatially homogeneous ELFs are found to improve performance relative to a commonly used localization function with compact support. When different ELFs are applied in precipitating and nonprecipitating regions, performance is further improved, but varying ELFs by observation type was not found to be as important. Imbalance in initial states caused by use of different localization functions is diagnosed by the domain-averaged surface pressure tendency. Forecasts from analyses with ELFs have smaller surface pressure tendencies than the standard localization, indicating improved initial balance with ELFs.

## Abstract

For ensemble-based data assimilation, localization is used to limit the impact of observations on physically distant state variables to reduce spurious error correlations caused by limited ensemble size. Traditionally, the localization value applied is spatially homogeneous. Yet there are potentially larger errors and different covariance length scales in precipitation systems, and that may justify the use of different localization functions for precipitating and nonprecipitating regions. Here this is examined using empirical localization functions (ELFs). Using output from an ensemble observing system simulation experiment (OSSE), ELFs provide estimates of horizontal and vertical localization for different observation types in regions with and without precipitation. For temperature and *u*- and *υ*-wind observations, the ELFs for precipitating regions are shown to have smaller horizontal localization scales than for nonprecipitating regions. However, the ELFs for precipitating regions generally have larger vertical localization scales than for nonprecipitating regions. The ELFs are smoothed and then applied in three additional OSSEs. Spatially homogeneous ELFs are found to improve performance relative to a commonly used localization function with compact support. When different ELFs are applied in precipitating and nonprecipitating regions, performance is further improved, but varying ELFs by observation type was not found to be as important. Imbalance in initial states caused by use of different localization functions is diagnosed by the domain-averaged surface pressure tendency. Forecasts from analyses with ELFs have smaller surface pressure tendencies than the standard localization, indicating improved initial balance with ELFs.