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## Abstract

A theory for estimating the probability distribution of the state of a model given a set of observations exists. This nonlinear filtering theory unifies the data assimilation and ensemble generation problem that have been key foci of prediction and predictability research for numerical weather and ocean prediction applications. A new algorithm, referred to as an ensemble adjustment Kalman filter, and the more traditional implementation of the ensemble Kalman filter in which “perturbed observations” are used, are derived as Monte Carlo approximations to the nonlinear filter. Both ensemble Kalman filter methods produce assimilations with small ensemble mean errors while providing reasonable measures of uncertainty in the assimilated variables. The ensemble methods can assimilate observations with a nonlinear relation to model state variables and can also use observations to estimate the value of imprecisely known model parameters. These ensemble filter methods are shown to have significant advantages over four-dimensional variational assimilation in low-order models and scale easily to much larger applications. Heuristic modifications to the filtering algorithms allow them to be applied efficiently to very large models by sequentially processing observations and computing the impact of each observation on each state variable in an independent calculation. The ensemble adjustment Kalman filter is applied to a nondivergent barotropic model on the sphere to demonstrate the capabilities of the filters in models with state spaces that are much larger than the ensemble size.

When observations are assimilated in the traditional ensemble Kalman filter, the resulting updated ensemble has a mean that is consistent with the value given by filtering theory, but only the expected value of the covariance of the updated ensemble is consistent with the theory. The ensemble adjustment Kalman filter computes a linear operator that is applied to the prior ensemble estimate of the state, resulting in an updated ensemble whose mean and also covariance are consistent with the theory. In the cases compared here, the ensemble adjustment Kalman filter performs significantly better than the traditional ensemble Kalman filter, apparently because noise introduced into the assimilated ensemble through perturbed observations in the traditional filter limits its relative performance. This superior performance may not occur for all problems and is expected to be most notable for small ensembles. Still, the results suggest that careful study of the capabilities of different varieties of ensemble Kalman filters is appropriate when exploring new applications.

## Abstract

A theory for estimating the probability distribution of the state of a model given a set of observations exists. This nonlinear filtering theory unifies the data assimilation and ensemble generation problem that have been key foci of prediction and predictability research for numerical weather and ocean prediction applications. A new algorithm, referred to as an ensemble adjustment Kalman filter, and the more traditional implementation of the ensemble Kalman filter in which “perturbed observations” are used, are derived as Monte Carlo approximations to the nonlinear filter. Both ensemble Kalman filter methods produce assimilations with small ensemble mean errors while providing reasonable measures of uncertainty in the assimilated variables. The ensemble methods can assimilate observations with a nonlinear relation to model state variables and can also use observations to estimate the value of imprecisely known model parameters. These ensemble filter methods are shown to have significant advantages over four-dimensional variational assimilation in low-order models and scale easily to much larger applications. Heuristic modifications to the filtering algorithms allow them to be applied efficiently to very large models by sequentially processing observations and computing the impact of each observation on each state variable in an independent calculation. The ensemble adjustment Kalman filter is applied to a nondivergent barotropic model on the sphere to demonstrate the capabilities of the filters in models with state spaces that are much larger than the ensemble size.

When observations are assimilated in the traditional ensemble Kalman filter, the resulting updated ensemble has a mean that is consistent with the value given by filtering theory, but only the expected value of the covariance of the updated ensemble is consistent with the theory. The ensemble adjustment Kalman filter computes a linear operator that is applied to the prior ensemble estimate of the state, resulting in an updated ensemble whose mean and also covariance are consistent with the theory. In the cases compared here, the ensemble adjustment Kalman filter performs significantly better than the traditional ensemble Kalman filter, apparently because noise introduced into the assimilated ensemble through perturbed observations in the traditional filter limits its relative performance. This superior performance may not occur for all problems and is expected to be most notable for small ensembles. Still, the results suggest that careful study of the capabilities of different varieties of ensemble Kalman filters is appropriate when exploring new applications.

## Abstract

The unstable normal modes of the barotropic vorticity equation, linearized around an observed zonally varying atmospheric flow, have been related to patterns of observed low-frequency variability. The sensitivity of this problem to changes in the model truncation and diffusion and to details of the basic state flow are examined. Normal modes that are highly sensitive to these changes are found to be of minimal relevance to the low-frequency variability of the atmosphere.

A new numerical method capable of efficiently finding a number of the most unstable modes of large eigenvalue problems is used to examine the effects of model truncation on the instability problem. Most previous studies are found to have utilized models of insufficiently high resolution. A small subset of unstable modes is found to be robust to changes in truncation. Sensitivity to changes in diffusion in a low-resolution model can partially reproduce the truncation results.

Sensitivity to the basic state is examined using a matrix method and by examining the normal modes of perturbed basic states. Again, a small subset of unstable normal modes is found to be robust. These modes appear to agree better with observed patterns of low-frequency variability than do less robust unstable modes.

## Abstract

The unstable normal modes of the barotropic vorticity equation, linearized around an observed zonally varying atmospheric flow, have been related to patterns of observed low-frequency variability. The sensitivity of this problem to changes in the model truncation and diffusion and to details of the basic state flow are examined. Normal modes that are highly sensitive to these changes are found to be of minimal relevance to the low-frequency variability of the atmosphere.

A new numerical method capable of efficiently finding a number of the most unstable modes of large eigenvalue problems is used to examine the effects of model truncation on the instability problem. Most previous studies are found to have utilized models of insufficiently high resolution. A small subset of unstable modes is found to be robust to changes in truncation. Sensitivity to changes in diffusion in a low-resolution model can partially reproduce the truncation results.

Sensitivity to the basic state is examined using a matrix method and by examining the normal modes of perturbed basic states. Again, a small subset of unstable normal modes is found to be robust. These modes appear to agree better with observed patterns of low-frequency variability than do less robust unstable modes.

## Abstract

An extremely simple chaotic model, the three-variable Lorenz convective model, is used in a perfect model setting to study the selection of initial conditions for ensemble forecasts. Observations with a known distribution of error are sampled from the “climate” of the simple model. Initial condition distributions that use only information about the observation and the observational error distribution (i.e., traditional Monte Carlo methods) are shown to differ from the correct initial condition distributions, which make use of additional information about the local structure of the model's attractor. Three relatively inexpensive algorithms for finding the local attractor structure in a simple model are examined; these make use of singular vectors. normal modes, and perturbed integrations. All of these are related to heuristic algorithms that have been applied to select ensemble members in operational forecast models. The method of perturbed integrations, which is somewhat similar to the “breeding” method used at the National Meteorological Center, is shown to be the most effective in this context. Validating the extension of such methods to realistic models is expected to be extremely difficult; however, it seems reasonable that utilizing all available information about the attractor structure of real forecast models when selecting ensemble initial conditions could improve the success of operational ensemble forecasts.

## Abstract

An extremely simple chaotic model, the three-variable Lorenz convective model, is used in a perfect model setting to study the selection of initial conditions for ensemble forecasts. Observations with a known distribution of error are sampled from the “climate” of the simple model. Initial condition distributions that use only information about the observation and the observational error distribution (i.e., traditional Monte Carlo methods) are shown to differ from the correct initial condition distributions, which make use of additional information about the local structure of the model's attractor. Three relatively inexpensive algorithms for finding the local attractor structure in a simple model are examined; these make use of singular vectors. normal modes, and perturbed integrations. All of these are related to heuristic algorithms that have been applied to select ensemble members in operational forecast models. The method of perturbed integrations, which is somewhat similar to the “breeding” method used at the National Meteorological Center, is shown to be the most effective in this context. Validating the extension of such methods to realistic models is expected to be extremely difficult; however, it seems reasonable that utilizing all available information about the attractor structure of real forecast models when selecting ensemble initial conditions could improve the success of operational ensemble forecasts.

## Abstract

Nearly stationary states (NSSs) of the barotropic vorticity equation (BVE) on the sphere that are closely related to observed atmospheric blocking patterns have recently been derived. Examining the way such NSSs affect integrations of the BVE is of interest. Unfortunately, the BVE rapidly evolves away from the neighborhood of blocking NSSs due to instability and never again generates sufficient amplitude to return to the vicinity of the blocking NSSs. However, forced versions of the BVE with both a high amplitude blocking NSS and more zonal low amplitude NSSs can be constructed. For certain parameter ranges, extended integrations of these forced BVFs exhibit two “regimes,” one strongly blocked and the other relatively zonal. Somewhat realistic simulators of low and high frequency variability and individual blocking event life cycles are also produced by these forced barotropic models. It is argued here that these regimes are related to “attractor-like” behavior of the NSSs of the forced BVE. Strong barotropic short waves apparently provide the push needed to cause a transition to or from the blocked regime. In the purely barotropic model used here, there is a rather delicate balance required between the forcing strength for different spatial scales in order to produce regimelike behavior. However, the mechanism proposed appears to be a viable candidate for explaining the observed behavior of blocking events in the atmosphere.

## Abstract

Nearly stationary states (NSSs) of the barotropic vorticity equation (BVE) on the sphere that are closely related to observed atmospheric blocking patterns have recently been derived. Examining the way such NSSs affect integrations of the BVE is of interest. Unfortunately, the BVE rapidly evolves away from the neighborhood of blocking NSSs due to instability and never again generates sufficient amplitude to return to the vicinity of the blocking NSSs. However, forced versions of the BVE with both a high amplitude blocking NSS and more zonal low amplitude NSSs can be constructed. For certain parameter ranges, extended integrations of these forced BVFs exhibit two “regimes,” one strongly blocked and the other relatively zonal. Somewhat realistic simulators of low and high frequency variability and individual blocking event life cycles are also produced by these forced barotropic models. It is argued here that these regimes are related to “attractor-like” behavior of the NSSs of the forced BVE. Strong barotropic short waves apparently provide the push needed to cause a transition to or from the blocked regime. In the purely barotropic model used here, there is a rather delicate balance required between the forcing strength for different spatial scales in order to produce regimelike behavior. However, the mechanism proposed appears to be a viable candidate for explaining the observed behavior of blocking events in the atmosphere.

## Abstract

A robust algorithm, capable of finding nearly stationary solutions of the unforced barotropic vorticity equation near to observed atmospheric streamfunctions, is presented. When applied to observed persistent anomaly patterns, the nearly stationary states (NSSs) produced by the algorithm usually have a distinctive appearance. NSSs produced for observed blocks tend to have even stronger blocks, and NSSs for intense jet anomaly patterns have intense jets. When applied to observed patterns that are not associated with persistent anomalies, the algorithm produces low-amplitude relatively zonal NSSs. The blocking and intense jet anomaly NSSs bear a striking resemblance to previously derived analytic stationary solutions of the vorticity equation. In particular, NSS blocking states are similar to certain types of modons.The algorithm is applied to a number of modified observed flows to better document what features of an observed pattern determine the nature of the resulting NSS. The short-wave components of an observed pattern need not be present in order for the algorithm to find interesting zonally varying NSSs. However, short waves play an essential part in the resulting NSSs by balancing the long-wave time tendencies. All the NSSs discovered are unstable to the introduction of small perturbations in the barotropic vorticity equation. Despite this instability, the NSSs still persist for many days when integrated in time. The existence of these persistent NSSs may play a significant role in the appearance and subsequent longevity of persistent anomaly patterns in the atmosphere.

## Abstract

A robust algorithm, capable of finding nearly stationary solutions of the unforced barotropic vorticity equation near to observed atmospheric streamfunctions, is presented. When applied to observed persistent anomaly patterns, the nearly stationary states (NSSs) produced by the algorithm usually have a distinctive appearance. NSSs produced for observed blocks tend to have even stronger blocks, and NSSs for intense jet anomaly patterns have intense jets. When applied to observed patterns that are not associated with persistent anomalies, the algorithm produces low-amplitude relatively zonal NSSs. The blocking and intense jet anomaly NSSs bear a striking resemblance to previously derived analytic stationary solutions of the vorticity equation. In particular, NSS blocking states are similar to certain types of modons.The algorithm is applied to a number of modified observed flows to better document what features of an observed pattern determine the nature of the resulting NSS. The short-wave components of an observed pattern need not be present in order for the algorithm to find interesting zonally varying NSSs. However, short waves play an essential part in the resulting NSSs by balancing the long-wave time tendencies. All the NSSs discovered are unstable to the introduction of small perturbations in the barotropic vorticity equation. Despite this instability, the NSSs still persist for many days when integrated in time. The existence of these persistent NSSs may play a significant role in the appearance and subsequent longevity of persistent anomaly patterns in the atmosphere.

## Abstract

A number of operational atmospheric prediction centers now produce ensemble forecasts of the atmosphere. Because of the high-dimensional phase spaces associated with operational forecast models, many centers use constraints derived from the dynamics of the forecast model to define a greatly reduced subspace from which ensemble initial conditions are chosen. For instance, the European Centre for Medium-Range Weather Forecasts uses singular vectors of the forecast model and the National Centers for Environmental Prediction use the “breeding cycle” to determine a limited set of directions in phase space that are sampled by the ensemble forecast.

The use of dynamical constraints on the selection of initial conditions for ensemble forecasts is examined in a perfect model study using a pair of three-variable dynamical systems and a prescribed observational error distribution. For these systems, one can establish that the direct use of dynamical constraints has no impact on the error of the ensemble mean forecast and a negative impact on forecasts of higher-moment quantities such as forecast spread. Simple examples are presented to show that this is not a result of the use of low-order dynamical systems but is instead related to the fundamental nature of the dynamics of these particular low-order systems themselves. Unless operational prediction models have fundamentally different dynamics, this study suggests that the use of dynamically constrained ensembles may not be justified. Further studies with more realistic prediction models are needed to evaluate this possibility.

## Abstract

A number of operational atmospheric prediction centers now produce ensemble forecasts of the atmosphere. Because of the high-dimensional phase spaces associated with operational forecast models, many centers use constraints derived from the dynamics of the forecast model to define a greatly reduced subspace from which ensemble initial conditions are chosen. For instance, the European Centre for Medium-Range Weather Forecasts uses singular vectors of the forecast model and the National Centers for Environmental Prediction use the “breeding cycle” to determine a limited set of directions in phase space that are sampled by the ensemble forecast.

The use of dynamical constraints on the selection of initial conditions for ensemble forecasts is examined in a perfect model study using a pair of three-variable dynamical systems and a prescribed observational error distribution. For these systems, one can establish that the direct use of dynamical constraints has no impact on the error of the ensemble mean forecast and a negative impact on forecasts of higher-moment quantities such as forecast spread. Simple examples are presented to show that this is not a result of the use of low-order dynamical systems but is instead related to the fundamental nature of the dynamics of these particular low-order systems themselves. Unless operational prediction models have fundamentally different dynamics, this study suggests that the use of dynamically constrained ensembles may not be justified. Further studies with more realistic prediction models are needed to evaluate this possibility.

## Abstract

A deterministic square root ensemble Kalman filter and a stochastic perturbed observation ensemble Kalman filter are used for data assimilation in both linear and nonlinear single variable dynamical systems. For the linear system, the deterministic filter is simply a method for computing the Kalman filter and is optimal while the stochastic filter has suboptimal performance due to sampling error. For the nonlinear system, the deterministic filter has increasing error as ensemble size increases because all ensemble members but one become tightly clustered. In this case, the stochastic filter performs better for sufficiently large ensembles. A new method for computing ensemble increments in observation space is proposed that does not suffer from the pathological behavior of the deterministic filter while avoiding much of the sampling error of the stochastic filter. This filter uses the order statistics of the prior observation space ensemble to create an approximate continuous prior probability distribution in a fashion analogous to the use of rank histograms for ensemble forecast evaluation. This rank histogram filter can represent non-Gaussian observation space priors and posteriors and is shown to be competitive with existing filters for problems as large as global numerical weather prediction. The ability to represent non-Gaussian distributions is useful for a variety of applications such as convective-scale assimilation and assimilation of bounded quantities such as relative humidity.

## Abstract

A deterministic square root ensemble Kalman filter and a stochastic perturbed observation ensemble Kalman filter are used for data assimilation in both linear and nonlinear single variable dynamical systems. For the linear system, the deterministic filter is simply a method for computing the Kalman filter and is optimal while the stochastic filter has suboptimal performance due to sampling error. For the nonlinear system, the deterministic filter has increasing error as ensemble size increases because all ensemble members but one become tightly clustered. In this case, the stochastic filter performs better for sufficiently large ensembles. A new method for computing ensemble increments in observation space is proposed that does not suffer from the pathological behavior of the deterministic filter while avoiding much of the sampling error of the stochastic filter. This filter uses the order statistics of the prior observation space ensemble to create an approximate continuous prior probability distribution in a fashion analogous to the use of rank histograms for ensemble forecast evaluation. This rank histogram filter can represent non-Gaussian observation space priors and posteriors and is shown to be competitive with existing filters for problems as large as global numerical weather prediction. The ability to represent non-Gaussian distributions is useful for a variety of applications such as convective-scale assimilation and assimilation of bounded quantities such as relative humidity.

## Abstract

The binned probability ensemble (BPE) technique is presented as a method for producing forecasts of the probability distribution of a variable using an ensemble of numerical model integrations. The ensemble forecasts are used to partition the real line into a number of bins, each of which has an equal probability of containing the “true” forecast. The method is tested for both a simple low-order dynamical system and a general circulation model (GCM) forced with observed sea surface temperatures (an ensemble of Atmospheric Model Intercomparison Project integrations). The BPE method can also be used to calculate the probability that probabilistic ensemble forecasts are consistent with the verifying observations. The method is not sensitive to the fact that the characteristics of the forecast probability distribution may change drastically for different initial condition (or boundary condition) probability distributions. For example, the method is capable of evaluating whether the variance of a set of ensemble forecasts is consistent with the verifying observed variance. Applying the method to the ensemble of boundary-forced GCM integrations demonstrates that the GCM produces probabilistic forecasts with too little variability for upper-level dynamical fields. Operational weather prediction centers including the U.K. Meteorological Office, the European Centre for Medium-Range Forecasts, and the National Centers for Environmental Prediction have been applying this method, referred to by them as Talagrand diagrams, to the verification of operational ensemble predictions. The BPE method only evaluates the consistency of ensemble predictions and observations and should be used in conjunction with additional verification tools to provide a complete assessment of a set of probabilistic forecasts.

## Abstract

The binned probability ensemble (BPE) technique is presented as a method for producing forecasts of the probability distribution of a variable using an ensemble of numerical model integrations. The ensemble forecasts are used to partition the real line into a number of bins, each of which has an equal probability of containing the “true” forecast. The method is tested for both a simple low-order dynamical system and a general circulation model (GCM) forced with observed sea surface temperatures (an ensemble of Atmospheric Model Intercomparison Project integrations). The BPE method can also be used to calculate the probability that probabilistic ensemble forecasts are consistent with the verifying observations. The method is not sensitive to the fact that the characteristics of the forecast probability distribution may change drastically for different initial condition (or boundary condition) probability distributions. For example, the method is capable of evaluating whether the variance of a set of ensemble forecasts is consistent with the verifying observed variance. Applying the method to the ensemble of boundary-forced GCM integrations demonstrates that the GCM produces probabilistic forecasts with too little variability for upper-level dynamical fields. Operational weather prediction centers including the U.K. Meteorological Office, the European Centre for Medium-Range Forecasts, and the National Centers for Environmental Prediction have been applying this method, referred to by them as Talagrand diagrams, to the verification of operational ensemble predictions. The BPE method only evaluates the consistency of ensemble predictions and observations and should be used in conjunction with additional verification tools to provide a complete assessment of a set of probabilistic forecasts.

## Abstract

Ensemble Kalman filters use the sample covariance of an observation and a model state variable to update a prior estimate of the state variable. The sample covariance can be suboptimal as a result of small ensemble size, model error, model nonlinearity, and other factors. The most common algorithms for dealing with these deficiencies are inflation and covariance localization. A statistical model of errors in ensemble Kalman filter sample covariances is described and leads to an algorithm that reduces ensemble filter root-mean-square error for some applications. This sampling error correction algorithm uses prior information about the distribution of the correlation between an observation and a state variable. Offline Monte Carlo simulation is used to build a lookup table that contains a correction factor between 0 and 1 depending on the ensemble size and the ensemble sample correlation. Correction factors are applied like a traditional localization for each pair of observations and state variables during an ensemble assimilation. The algorithm is applied to two low-order models and reduces the sensitivity of the ensemble assimilation error to the strength of traditional localization. When tested in perfect model experiments in a larger model, the dynamical core of a general circulation model, the sampling error correction algorithm produces analyses that are closer to the truth and also reduces sensitivity to traditional localization strength.

## Abstract

Ensemble Kalman filters use the sample covariance of an observation and a model state variable to update a prior estimate of the state variable. The sample covariance can be suboptimal as a result of small ensemble size, model error, model nonlinearity, and other factors. The most common algorithms for dealing with these deficiencies are inflation and covariance localization. A statistical model of errors in ensemble Kalman filter sample covariances is described and leads to an algorithm that reduces ensemble filter root-mean-square error for some applications. This sampling error correction algorithm uses prior information about the distribution of the correlation between an observation and a state variable. Offline Monte Carlo simulation is used to build a lookup table that contains a correction factor between 0 and 1 depending on the ensemble size and the ensemble sample correlation. Correction factors are applied like a traditional localization for each pair of observations and state variables during an ensemble assimilation. The algorithm is applied to two low-order models and reduces the sensitivity of the ensemble assimilation error to the strength of traditional localization. When tested in perfect model experiments in a larger model, the dynamical core of a general circulation model, the sampling error correction algorithm produces analyses that are closer to the truth and also reduces sensitivity to traditional localization strength.

## Abstract

An objective criterion for identifying blocking events is applied to a ten-year climate run of the National Meteorological Center's Medium-Range Forecast Model (MRF) and to observations. The climatology of blocking in the ten-year run is found to be somewhat realistic in the Northern Hemisphere, although when averaged over all longitudes and seasons a general lack of blocking is found. Previous studies have suggested that numerical models are incapable of producing realistic numbers of blocks, however, the ten-year model run is able to produce realistic numbers of blocks for selected geographic regions and seasons. In these regions, blocks are found to persist longer than observed blocking events. The ten-year run of the model is also able to reproduce the average longitudinal extent and motion of the observed blocks. These results suggest that the MRF is able to generate and persist realistic blocks, but only at longitudes and seasons for which the underlying model climate is conducive. In the Southern Hemisphere, the ten-year run blocking climatology is considerably less realistic. The appearance of “transient” blocking events in the model distinguishes it from the Southern Hemisphere observations and from the Northern Hemisphere.

A set of 60-day forecasts by the MRF is used to evaluate the evolution of the model blocking climatology with lead time (blocking climate drift) for a 90-day period in autumn of 1990. Although the ten-year run and observed blocking climates are quite similar at most longitudes at this time of year, it is found that blocking almost entirely disappears from the model forecasts at lead times of approximately 10 days before reappearing at leads greater than 15 days. It is argued that this lack of a direct transition between observed and model blocking climates is the result of a drift in the underlying climate (for example, the positions of the jet streams) in the MRF forecasts. If so, the climate drift of the MRF must he further reduced in order to produce more accurate medium-range forecasts of blocking events.

## Abstract

An objective criterion for identifying blocking events is applied to a ten-year climate run of the National Meteorological Center's Medium-Range Forecast Model (MRF) and to observations. The climatology of blocking in the ten-year run is found to be somewhat realistic in the Northern Hemisphere, although when averaged over all longitudes and seasons a general lack of blocking is found. Previous studies have suggested that numerical models are incapable of producing realistic numbers of blocks, however, the ten-year model run is able to produce realistic numbers of blocks for selected geographic regions and seasons. In these regions, blocks are found to persist longer than observed blocking events. The ten-year run of the model is also able to reproduce the average longitudinal extent and motion of the observed blocks. These results suggest that the MRF is able to generate and persist realistic blocks, but only at longitudes and seasons for which the underlying model climate is conducive. In the Southern Hemisphere, the ten-year run blocking climatology is considerably less realistic. The appearance of “transient” blocking events in the model distinguishes it from the Southern Hemisphere observations and from the Northern Hemisphere.

A set of 60-day forecasts by the MRF is used to evaluate the evolution of the model blocking climatology with lead time (blocking climate drift) for a 90-day period in autumn of 1990. Although the ten-year run and observed blocking climates are quite similar at most longitudes at this time of year, it is found that blocking almost entirely disappears from the model forecasts at lead times of approximately 10 days before reappearing at leads greater than 15 days. It is argued that this lack of a direct transition between observed and model blocking climates is the result of a drift in the underlying climate (for example, the positions of the jet streams) in the MRF forecasts. If so, the climate drift of the MRF must he further reduced in order to produce more accurate medium-range forecasts of blocking events.