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189
Toward a nonlinear ensemble filter for highdimensional systems
 J. Geophysical ResearchAtmospheres
, 2003
"... [1] Many geophysical problems are characterized by highdimensional, nonlinear systems and pose difficult challenges for realtime data assimilation (updating) and forecasting. The present work builds on the ensemble Kalman filter (EnsKF), with the goal of producing ensemble filtering techniques app ..."
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Cited by 47 (6 self)
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[1] Many geophysical problems are characterized by highdimensional, nonlinear systems and pose difficult challenges for realtime data assimilation (updating) and forecasting. The present work builds on the ensemble Kalman filter (EnsKF), with the goal of producing ensemble filtering techniques applicable to nonGaussian densities and highdimensional systems. Three filtering algorithms, based on representing the prior density as a Gaussian mixture, are presented. The first, referred to as a mixture ensemble Kalman filter (XEnsF), models local covariance structures adaptively using nearest neighbors. The XEnsF is effective in a threedimensional system, but the required ensemble grows rapidly with the dimension and, even in a 40dimensional system, we find the XEnsF to be unstable and inferior to the EnsKF for all computationally feasible ensemble sizes. A second algorithm, the locallocal ensemble filter (LLEnsF), combines localizations in physical as well as phase space, allowing the update step in highdimensional systems to be decomposed into a sequence of lowerdimensional updates tractable by the XEnsF. Given the same prior forecasts in a 40dimensional system, the LLEnsF update produces more accurate state estimates than the EnsKF if the forecast distributions are sufficiently
Assimilation of Simulated Polarimetric Radar Data for a Convective Storm Using the Ensemble Kalman Filter. Part I: Observation Operators for Reflectivity and Polarimetric Variables
 MONTHLY WEATHER REVIEW VOLUME 136
, 2006
"... A radar simulator for polarimetric radar variables, including reflectivities at horizontal and vertical polarizations, the differential reflectivity, and the specific differential phase, has been developed. This simulator serves as a test bed for developing and testing forward observation operators ..."
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Cited by 43 (31 self)
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A radar simulator for polarimetric radar variables, including reflectivities at horizontal and vertical polarizations, the differential reflectivity, and the specific differential phase, has been developed. This simulator serves as a test bed for developing and testing forward observation operators of polarimetric radar variables that are needed when directly assimilating these variables into stormscale numerical weather prediction (NWP) models, using either variational or ensemblebased assimilation methods. The simulator takes as input the results of highresolution NWP model simulations with ice microphysics and produces simulated polarimetric radar data that may also contain simulated errors. It is developed based on calculations of electromagnetic wave propagation and scattering at the S band of wavelength 10.7 cm in a hydrometeorcontaining atmosphere. The Tmatrix method is used for the scattering calculation of raindrops and the Rayleigh scattering approximation is applied to snow and hail particles. The polarimetric variables are expressed as functions of the hydrometeor mixing ratios as well as their corresponding drop size distribution parameters and densities. The presence of wet snow and wet hail in the melting layer is accounted for by using a new, relatively simple melting model that defines the water fraction in the melting snow or hail. The effect of varying density due to the melting snow or hail is also included. Vertical cross sections and profiles of the polarimetric variables for a simulated mature multicellular squallline system and a supercell storm show that polarimetric signatures of the bright band in the stratiform region and those associated with deep convection are well captured by the simulator.
Ensemblebased atmospheric data assimilation
, 2004
"... Ensemblebased data assimilation techniques are being explored as possible alternatives to current operational analysis techniques such as 3 or 4dimensional variational assimilation. Ensemblebased assimilation techniques utilize an ensemble of parallel data assimilation and forecast cycles. The ..."
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Cited by 42 (2 self)
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Ensemblebased data assimilation techniques are being explored as possible alternatives to current operational analysis techniques such as 3 or 4dimensional variational assimilation. Ensemblebased assimilation techniques utilize an ensemble of parallel data assimilation and forecast cycles. The backgrounderror covariances are estimated using the forecast ensemble and are used to produce an ensemble of analyses. The backgrounderror covariances are flow dependent and often have very complicated structure, providing a very different adjustment to the observations than are seen from methods such as 3 dimensional variational assimilation. Though computationally expensive, ensemblebased techniques are relatively easy to code, since no adjoint nor tangentlinear models are required, and previous tests in simple models suggest that dramatic improvements over existing operational methods may be possible. A review of the ensemblebased assimilation is provided here, starting from the basic concepts of Bayesian assimilation. Without some simplification, full Bayesian assimilation is computationally impossible for model states of large dimension. Assuming normality of error statistics and linearity of error growth, the state and its error covariance may be predicted optimally using Kalman filter (KF) techniques. The ensemble Kalman filter (EnKF) is then described. The EnKF is an approximation to the KF in that backgrounderror covariances are estimated from a finite ensemble of forecasts. However, no assumptions about linearity of error growth are made. Recent algorithmic variants on the standard EnKF are also described, as well as methods for simplifying the computations and increasing the accuracy. Examples of ensemblebased assimilations are provided in simple and more realistic dynamical systems.
Accounting for the Error due to Unresolved Scales in Ensemble Data Assimilation: A Comparison of Different Approaches
 3132 MONTHLY WEATHER REVIEW VOLUME
, 2005
"... Insufficient model resolution is one source of model error in numerical weather predictions. Methods for parameterizing this error in ensemble data assimilations are explored here. Experiments were conducted with a twolayer primitive equation model, where the assumed true state was a T127 forecast ..."
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Cited by 42 (4 self)
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Insufficient model resolution is one source of model error in numerical weather predictions. Methods for parameterizing this error in ensemble data assimilations are explored here. Experiments were conducted with a twolayer primitive equation model, where the assumed true state was a T127 forecast simulation. Ensemble data assimilations were performed with the same model at T31 resolution, assimilating imperfect observations drawn from the T127 forecast. By design, the magnitude of errors due to model truncation was much larger than the error growth due to initial condition uncertainty, making this a stringent test of the ability of an ensemblebased data assimilation to deal with model error. Two general methods, “covariance inflation ” and “additive error, ” were considered for parameterizing the model error at the resolved scales (T31 and larger) due to interaction with the unresolved scales (T32 to T127). Covariance inflation expanded the background forecast members ’ deviations about the ensemble mean, while additive error added specially structured noise to each ensemble member forecast before the update step. The method of parameterizing this model error had a substantial effect on the accuracy of the ensemble data assimilation. Covariance inflation produced ensembles with analysis errors that were no lower than the analysis errors from threedimensional variational (3DVar) assimilation, and for the method to avoid filter
Curseofdimensionality revisited: Collapse of the particle filter in very large scale systems.
"... It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate that the maximum of the weights associated with the sample ..."
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Cited by 35 (0 self)
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It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate that the maximum of the weights associated with the sample ensemble converges to one as both the sample size and the system dimension tends to infinity. Specifically, under fairly weak assumptions, if the ensemble size grows subexponentially in the cube root of the system dimension, the convergence holds for a single update step in statespace models with independent and identically distributed kernels. Further, in an important special case, more refined arguments show (and our simulations suggest) that the convergence to unity occurs unless the ensemble grows superexponentially in the system dimension. The weight singularity is also established in models with more general multivariate likelihoods, e.g. Gaussian and Cauchy. Although presented in the context of atmospheric data assimilation for numerical weather prediction, our results are generally valid for highdimensional particle filters. 1 1
A multicase comparative assessment of the ensemble Kalman filter for assimilation of radar observations. Part I: Stormscale analyses.” Monthly Weather Review
, 2009
"... The effectiveness of the ensemble Kalman filter (EnKF) for assimilating radar observations at convective scales is investigated for cases whose behaviors span supercellular, linear, andmulticellular organization. The parallel EnKF algorithm of the Data Assimilation Research Testbed (DART) is used fo ..."
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Cited by 30 (4 self)
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The effectiveness of the ensemble Kalman filter (EnKF) for assimilating radar observations at convective scales is investigated for cases whose behaviors span supercellular, linear, andmulticellular organization. The parallel EnKF algorithm of the Data Assimilation Research Testbed (DART) is used for data assimilation, while the Weather Research and Forecasting (WRF)Model is employed as a simplified cloud model at 2km horizontal grid spacing. In each case, reflectivity and radial velocity measurements are utilized from a single Weather Surveillance Radar1988 Doppler (WSR88D) within the U.S. operational network. Observations are assimilated every 2 min for a duration of 60 min and correction of folded radial velocities occurs within the EnKF. Initial ensemble uncertainty includes random perturbations to the horizontal wind components of the initial environmental sounding. The EnKF performs effectively and with robust results across all the cases. Over the first 18–30 min of assimilation, the rms and domainaveraged prior fits to observations in each case improve significantly from their initial levels, reaching comparable values of 3–6 m s21 and 7–10 dBZ. Representation of mesoscale uncertainty, albeit in the simplest form of initial sounding perturbations, is a critical part of the assimilation system, as it increases ensemble spread and improves filter performance. In addition, assimilation of ‘‘no precipitation’ ’ observations (i.e., reflectivity observations with values small enough to indicate the absence of precipitation) serves to suppress spurious convection in ensemble members. At the same time, it is clear that the assimilation is far from optimal, as the ensemble spread is consistently smaller than what would be expected from the innovation statistics and the assumed observationerror variance. 1.
Local ensemble Kalman filtering in the presence of model bias. Tellus 58A
, 2006
"... We modify the local ensemble Kalman filter (LEKF) to incorporate the effect of forecast model bias. The method is based on augmentation of the atmospheric state by estimates of the model bias, and we consider different ways of modeling (i.e. parameterizing) the model bias. We evaluate the effectiven ..."
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Cited by 29 (7 self)
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We modify the local ensemble Kalman filter (LEKF) to incorporate the effect of forecast model bias. The method is based on augmentation of the atmospheric state by estimates of the model bias, and we consider different ways of modeling (i.e. parameterizing) the model bias. We evaluate the effectiveness of the proposed augmented state ensemble Kalman filter through numerical experiments incorporating various model biases into the model of Lorenz and Emanuel. Our results highlight the critical role played by the selection of a good parameterization model for representing the form of the possible bias in the forecast model. In particular, we find that forecasts can be greatly improved provided that a good model parameterizing the model bias is used to augment the state in the Kalman filter. 1.
System Design and Evaluation of Coupled Ensemble Data Assimilation for Global Oceanic Climate Studies
, 2007
"... A fully coupled data assimilation (CDA) system, consisting of an ensemble filter applied to the Geophysical Fluid Dynamics Laboratory’s global fully coupled climate model (CM2), has been developed to facilitate the detection and prediction of seasonaltomultidecadal climate variability and climate ..."
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Cited by 29 (6 self)
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A fully coupled data assimilation (CDA) system, consisting of an ensemble filter applied to the Geophysical Fluid Dynamics Laboratory’s global fully coupled climate model (CM2), has been developed to facilitate the detection and prediction of seasonaltomultidecadal climate variability and climate trends. The assimilation provides a selfconsistent, temporally continuous estimate of the coupled model state and its uncertainty, in the form of discrete ensemble members, which can be used directly to initialize probabilistic climate forecasts. Here, the CDA is evaluated using a series of perfect model experiments, in which a particular twentiethcentury simulation—with temporally varying greenhouse gas and natural aerosol radiative forcings—serves as a “truth ” from which observations are drawn, according to the actual ocean observing network for the twentieth century. These observations are then assimilated into a coupled model ensemble that is subjected only to preindustrial forcings. By examining how well this analysis ensemble reproduces the “truth,” the skill of the analysis system in recovering anthropogenically forced trends and natural climate variability is assessed, given the historical observing network. The assimilation successfully reconstructs the twentiethcentury ocean heat content variability and trends in most locations. The experiments highlight the importance of maintaining key physical relationships among model fields, which are
Ensemble squareroot filters
, 2003
"... Ensemble data assimilation methods assimilate observations using statespace estimation methods and lowrank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. ..."
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Cited by 26 (2 self)
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Ensemble data assimilation methods assimilate observations using statespace estimation methods and lowrank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. This transformation may be preformed stochastically by treating observations as random variables, or deterministically by requiring that the updated analysis perturbations satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are implementations of Kalman squareroot filters. The nonuniqueness of the deterministic transformation used in squareroot Kalman filters provides a framework to compare three recently proposed ensemble data assimilation methods. 2
supercell: ensemble Kalman filter experiments
 2004: Wind and thermodynamic retrievals in the 17
, 1981
"... The feasibility of using an ensemble Kalman filter (EnKF) to retrieve the wind and temperature fields in an isolated convective storm has been tested by applying the technique to observations of the 17 May 1981 Arcadia, Oklahoma, tornadic supercell. Radialvelocity and reflectivity observations from ..."
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Cited by 23 (3 self)
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The feasibility of using an ensemble Kalman filter (EnKF) to retrieve the wind and temperature fields in an isolated convective storm has been tested by applying the technique to observations of the 17 May 1981 Arcadia, Oklahoma, tornadic supercell. Radialvelocity and reflectivity observations from a single radar were assimilated into a nonhydrostatic, anelastic numerical model initialized with an idealized (horizontally homogeneous) base state. The assimilation results were compared to observations from another Doppler radar, the results of dualDoppler wind syntheses, and in situ measurements from an instrumented tower. Observation errors make it more difficult to assess EnKF performance than in previous stormscale EnKF experiments that employed synthetic observations and a perfect model; nevertheless, the comparisons in this case indicate that the locations of the main updraft and mesocyclone in the Arcadia storm were determined rather accurately, especially at midlevels. The magnitudes of vertical velocity and vertical vorticity in these features are similar to those in the dualDoppler analyses, except that the lowlevel updraft is stronger in the EnKF analyses than in the dualDoppler analyses. Several assimilationscheme parameters are adjustable, including the method of initializing the ensemble, the inflation factor applied to perturbations, the magnitude of the assumed observationerror variance, and the degree of localization of the filter. In the Arcadia storm experiments, in which observations of a mature storm were