Chris Snyder

Distance-Dependent Filtering of Background Error Covariance Estimates in an Ensemble Kalman Filter

The usefulness of a distance-dependent reduction of background error covariance estimates in an ensemble Kalman filter is demonstrated. Covariances are reduced by performing an elementwise multiplication of the background error covariance matrix with a correlation function with local support. This reduces noisiness and results in an improved background error covariance estimate, which generates a reduced-error ensemble of model initial conditions. The benefits of applying the correlation function can be understood in part from examining the characteristics of simple 2 3 2 covariance matrices generated from random sample vectors with known variances and covariance. These show that noisiness in covariance estimates tends to overwhelm the signal when the ensemble size is small and/or the true covariance between the sample elements is small. Since the true covariance of forecast errors is generally related to the distance between grid points, covariance estimates generally have a higher ratio of noise to signal with increasing distance between grid points. This property is also demonstrated using a twolayer hemispheric primitive equation model and comparing covariance estimates generated by small and large ensembles. Covariances from the large ensemble are assumed to be accurate and are used a reference for measuring errors from covariances estimated from a small ensemble. The benefits of including distance-dependent reduction of covariance estimates are demonstrated with an ensemble Kalman filter data assimilation scheme. The optimal correlation length scale of the filter function depends on ensemble size; larger correlation lengths are preferable for larger ensembles. The effects of inflating background error covariance estimates are examined as a way of stabilizing the filter. It was found that more inflation was necessary for smaller ensembles than for larger ensembles.

A Hybrid Ensemble Kalman Filter-3D Variational Analysis Scheme

A hybrid ensemble Kalman filter‐three-dimensional variational (3DVAR) analysis scheme is demonstrated using a quasigeostrophic model under perfect-model assumptions. Four networks with differing observational densities are tested, including one network with a data void. The hybrid scheme operates by computing a set of parallel data assimilation cycles, with each member of the set receiving unique perturbed observations. The perturbed observations are generated by adding random noise consistent with observation error statistics to the control set of observations. Background error statistics for the data assimilation are estimated from a linear combination of time-invariant 3DVAR covariances and flow-dependent covariances developed from the ensemble of short-range forecasts. The hybrid scheme allows the user to weight the relative contributions of the 3DVAR and ensemble-based background covariances. The analysis scheme was cycled for 90 days, with new observations assimilated every 12 h. Generally, it was found that the analysis performs best when background error covariances are estimated almost fully from the ensemble, especially when the ensemble size was large. When small-sized ensembles are used, some lessened weighting of ensemble-based covariances is desirable. The relative improvement over 3DVAR analyses was dependent upon the observational data density and norm; generally, there is less improvement for data-rich networks than for data-poor networks, with the largest improvement for the network with the data void. As expected, errors depend on the size of the ensemble, with errors decreasing as more ensemble members are added. The sets of initial conditions generated from the hybrid are generally well calibrated and provide an improved set of initial conditions for ensemble forecasts.

Assimilation of Simulated Doppler Radar Observations with an Ensemble Kalman Filter

Assimilation of Doppler radar data into cloud models is an important obstacle to routine numerical weather prediction for convective-scale motions; the difficulty lies in initializing fields of wind, temperature, moisture, and condensate given only observations of radial velocity and reflectivity from the radar. This paper investigates the potential of the ensemble Kalman filter (EnKF), which estimates the covariances between observed variables and the state through an ensemble of forecasts, to assimilate radar observations at convective scales. In the basic experiment, simulated observations are extracted from a reference simulation of a splitting supercell and assimilated using the EnKF and the same numerical model that produced the reference simulation. The EnKF produces accurate analyses, including the unobserved variables, after roughly 30 min (or six scans) of radial velocity observations. Additional experiments, in which forecasts are made from the ensemble-mean analysis, reveal that forecast errors grow significantly in this simple system, so that the ability of the EnKF to track the reference solution is not simply because of stable system dynamics. It is also found that the covariances between radial velocity and temperature, moisture, and condensate are important to the quality of the analyses, as is the initialization chosen for the ensemble members prior to assimilating the first observations. These results are promising, especially given the ease of implementing the EnKF. A number of important issues remain, however, including the initialization of the ensemble prior to the first observation, the treatment of uncertainty in the environmental sounding, the role of error in the forecast model (particularly the microphysical parameterizations), and the treatment of lateral boundary conditions.

Obstacles to High-Dimensional Particle Filtering

Abstract Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian, and of unit variance and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel, and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and ...

Impacts of Initial Estimate and Observation Availability on Convective-Scale Data Assimilation with an Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) uses an ensemble of short-range forecasts to estimate the flow-dependent background error covariances required in data assimilation. The feasibility of the EnKF for convective-scale data assimilation has been previously demonstrated in perfect-model experiments using simulated observations of radial velocity from a supercell storm. The present study further explores the potential and behavior of the EnKF at convective scales by considering more realistic initial analyses and variations in the availability and quality of the radar observations. Assimilation of simulated radial-velocity observations every 5 min where there is significant reflectivity using 20 ensemble members proves to be successful in most realistic observational scenarios for simulated supercell thunderstorms, although the same degree of success may not be readily expected with real observations and an imperfect model, at least with the present EnKF implementation. Even though the filter converges toward the truth simulation faster from a better initial estimate, an experiment with the initial estimate of the supercell displaced by 10 km still yields an accurate estimate of the storm for both observed and unobserved variables within 40 min. Similarly, radial-velocity observations below 2 km are certainly beneficial to capturing the storm (especially the detailed cold pool structure), but in their absence the assimilation scheme can still achieve a comparably accurate estimate of the state of the storm given a slightly longer assimilation period. An experiment with radar observations only above 4 km fails to assimilate the storm properly, but, with the addition of a hypothetical surface mesonet taking wind and temperature observations, the EnKF can again provide a good estimate of the storm. The supercell can also be successfully assimilated in the case of radar observations only below 4 km (such as those from the ground-based mobile radars). More frequent observations can help the storm assimilation initially, but the benefit diminishes after half an hour. Results presented here indicate that the vertical resolution and the uncertainty of observations, for the typical range of most of the observational radars, would have little impact on the overall performance of the EnKF in assimilating the storm.

Prediction of Landfalling Hurricanes with the Advanced Hurricane WRF Model

Abstract Real-time forecasts of five landfalling Atlantic hurricanes during 2005 using the Advanced Research Weather Research and Forecasting (WRF) (ARW) Model at grid spacings of 12 and 4 km revealed performance generally competitive with, and occasionally superior to, other operational forecasts for storm position and intensity. Recurring errors include 1) excessive intensification prior to landfall, 2) insufficient momentum exchange with the surface, and 3) inability to capture rapid intensification when observed. To address these errors several augmentations of the basic community model have been designed and tested as part of what is termed the Advanced Hurricane WRF (AHW) model. Based on sensitivity simulations of Katrina, the inner-core structure, particularly the size of the eye, was found to be sensitive to model resolution and surface momentum exchange. The forecast of rapid intensification and the structure of convective bands in Katrina were not significantly improved until the grid spacing ap...

Wind and temperature retrievals in the 17 May 1981 Arcadia, Oklahoma, supercell: Ensemble Kalman filter experiments

Abstract 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. Radial-velocity 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 dual-Doppler wind syntheses, and in situ measurements from an instrumented tower. Observation errors make it more difficult to assess EnKF performance than in previous storm-scale 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 v...

Effects of Moist Convection on Mesoscale Predictability

In a previous study by the authors, it was shown that the problematic numerical prediction of the 24‐25 January 2000 snowstorm along the east coast of the United States was in some measure due to rapid error growth at scales below 500 km. In particular they found that moist processes were responsible for this strong initial-condition sensitivity of the 1‐2-day prediction of mesoscale forecast aspects. In the present study they take a more systematic look at the processes by which small initial differences (‘‘errors’’) grow in those numerical forecasts. For initial errors restricted to scales below 100 km, results show that errors first grow as small-scale differences associated with moist convection, then spread upscale as their growth begins to slow. In the context of mesoscale numerical predictions with 30-km resolution, the initial growth is associated with nonlinearities in the convective parameterization (or in the explicit microphysical parameterizations, if no convective parameterization is used) and proceeds at a rate that increases as the initial error amplitude decreases. In higherresolution (3.3 km) simulations, errors first grow as differences in the timing and position of individual convective cells. Amplification at that stage occurs on a timescale on the order of 1 h, comparable to that of moist convection. The errors in the convective-scale motions subsequently influence the development of meso- and larger-scale forecast aspects such as the position of the surface low and the distribution of precipitation, thus providing evidence that growth of initial errors from convective scales places an intrinsic limit on the predictability of larger scales.

A Hybrid ETKF–3DVAR Data Assimilation Scheme for the WRF Model. Part I: Observing System Simulation Experiment

Abstract A hybrid ensemble transform Kalman filter–three-dimensional variational data assimilation (ETKF–3DVAR) system for the Weather Research and Forecasting (WRF) Model is introduced. The system is based on the existing WRF 3DVAR. Unlike WRF 3DVAR, which utilizes a simple, static covariance model to estimate the forecast-error statistics, the hybrid system combines ensemble covariances with the static covariances to estimate the complex, flow-dependent forecast-error statistics. Ensemble covariances are incorporated by using the extended control variable method during the variational minimization. The ensemble perturbations are maintained by the computationally efficient ETKF. As an initial attempt to test and understand the newly developed system, both an observing system simulation experiment under the perfect model assumption (Part I) and the real observation experiment (Part II) were conducted. In these pilot studies, the WRF was run over the North America domain at a coarse grid spacing (200 km) t...

Boundary Conditions for Limited-Area Ensemble Kalman Filters

Abstract One aspect of implementing a limited-area ensemble Kalman filter (EnKF) involves the specification of a suitable ensemble of lateral boundary conditions. Two classes of methods to populate a boundary condition ensemble are proposed. In the first class, the ensemble of boundary conditions is provided by an EnKF on a larger domain and is approximately a random draw from the probability distribution function for the forecast (or analysis) on the limited-area domain boundary. The second class perturbs around a deterministic estimate of the state using assumed spatial and temporal covariance relationships. Methods in the second class are relatively flexible and easy to implement. Experiments that test the utility of these methods are performed for both an idealized low-dimensional model and limited-area simulations using the Weather Research and Forecasting (WRF) model; all experiments employ simulated observations under the perfect model assumption. The performance of the ensemble boundary condition ...