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Elizabeth Satterfield
and
Istvan Szunyogh

Abstract

The performance of an ensemble prediction system is inherently flow dependent. This paper investigates the flow dependence of the ensemble performance with the help of linear diagnostics applied to the ensemble perturbations in a small local neighborhood of each model gridpoint location ℓ. A local error covariance matrix 𝗣 is defined for each local region, and the diagnostics are applied to the linear space defined by the range of the ensemble-based estimate of 𝗣. The particular diagnostics are chosen to help investigate the efficiency of in capturing the space of analysis and forecast uncertainties.

Numerical experiments are carried out with an implementation of the local ensemble transform Kalman filter (LETKF) data assimilation system on a reduced-resolution [T62 and 28 vertical levels (T62L28)] version of the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS). Both simulated observations under the perfect model scenario and observations of the real atmosphere in a realistic setting are used in these experiments. It is found that (i) paradoxically, the linear space provides an increasingly better estimate of the space of forecast uncertainties as the time evolution of the ensemble perturbations becomes more nonlinear with increasing forecast time; (ii) provides a more reliable linear representation of the space of forecast uncertainties for cases of more rapid error growth (i.e., for cases of lower predictability); and (iii) the ensemble dimension (E dimension) is a reliable predictor of the performance of in predicting the space of forecast uncertainties.

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Fan Han
and
Istvan Szunyogh

Abstract

A new morphing-based technique is proposed for the verification of precipitation forecasts for which the location error can be described by a spatial shift. An adaptation of the structural similarity index measure (SSIM) of image processing to the precipitation forecast verification problem, called the amplitude and structural similarity index (ASSIM), is also introduced. ASSIM is used to measure both the convergence of the new morphing algorithm, which is an iterative scheme, and the amplitude and structure component of the forecast error. The behavior of the proposed technique, which could also be applied to other forecast parameters with sharp gradients (e.g., potential vorticity), is illustrated with idealized and realistic examples. One of these examples examines the predictability of the location of precipitation events associated with winter storms. It is found that the functional dependence of the average magnitude of the location error on the forecast lead time is qualitatively similar to that of the root-mean-square error of the fields of the conventional atmospheric state variables (e.g., geopotential height). Quantitatively, the average magnitude of the estimated location error is about 40 km at initial time, 110 km at day 1, 250 km at day 3, and 750 km at week 1, and it eventually saturates at about week 2.

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Fan Han
and
Istvan Szunyogh

Abstract

This paper describes a morphing-based approach for the verification of precipitation forecasts. This approach employs a pyramid matching algorithm to morph the precipitation features in a forecast into features that match the related precipitation features in the verifying analysis (observations) as closely as possible. The algorithm computes an optical flow (vector field) that maps the original forecast features into the morphed forecast features. The optical flow also provides quantitative information about the error in the location of the forecast features. This information, combined with information about the error in the prediction of the total precipitation over the verification domain, is used to quantify the structure error in the precipitation forecast. The proposed approach has three novel aspects compared to the published morphing-based verification strategies. First, it imposes a constraint on the pyramid matching algorithm to prevent overconvergence toward strong precipitation features during morphing. Second, it introduces an objective criterion for the selection of the subsampling parameter to avoid splitting or distorting features due to an arbitrary maximum displacement limit. Third, the proposed definitions of the location and structure errors are new. The behavior of the proposed multivariate verification metrics is investigated by applications to both idealized and numerical forecast examples.

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Elizabeth Satterfield
and
Istvan Szunyogh

Abstract

The ability of an ensemble to capture the magnitude and spectrum of uncertainty in a local linear space spanned by the ensemble perturbations is assessed. Numerical experiments are carried out with a reduced resolution 2004 version of the model component of the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS). The local ensemble transform Kalman filter (LETKF) data assimilation system is used to assimilate observations in three steps, gradually adding more realistic features to the observing network. In the first experiment, randomly placed, noisy, simulated vertical soundings, which provide 10%% coverage of horizontal model grid points, are assimilated. Next, the impact of an inhomogeneous observing system is introduced by assimilating simulated observations in the locations of real observations of the atmosphere. Finally, observations of the real atmosphere are assimilated.

The most important findings of this study are the following: predicting the magnitude of the forecast uncertainty and the relative importance of the different patterns of uncertainty is, in general, a more difficult task than predicting the patterns of uncertainty; the ensemble, which is tuned to provide near-optimal performance at analysis time, underestimates not only the total magnitude of the uncertainty, but also the magnitude of the uncertainty that projects onto the space spanned by the ensemble perturbations; and finally, a strong predictive linear relationship is found between the local ensemble spread and the upper bound of the local forecast uncertainty.

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Christina R. Holt
,
Istvan Szunyogh
, and
Gyorgyi Gyarmati

Abstract

This study investigates the benefits of employing a limited-area data assimilation (DA) system to enhance lower-resolution global analyses in the northwest Pacific tropical cyclone (TC) basin. Numerical experiments are carried out with a global analysis system at horizontal resolution T62 and a limited-area analysis system at resolutions from 200 to 36 km. The global and limited-area DA systems, which are both based on the local ensemble transform Kalman filter algorithm, are implemented using a unique configuration, in which the global DA system provides information about the large-scale analysis and background uncertainty to the limited-area DA system. The limited-area analyses of the storm locations are, on average, more accurate than those from the global analyses, but increasing the resolution of the limited-area system beyond 100 km has little benefit. Two factors contribute to the higher accuracy of the limited-area analyses. First, the limited-area system improves the accuracy of the location estimates for strong storms, which is introduced when the background is updated by the global assimilation. Second, it improves the accuracy of the background estimate of the storm locations for moderate and weak storms. Improvements in the steering flow analysis as a result of increased resolution are modest and short lived in the forecasts. Limited-area track forecasts are more accurate, on average, than global forecasts, independently of the strength of the storms up to five days. This forecast improvement is due to the more accurate analysis of the initial position of storms and the better representation of the interactions between the storms and their immediate environment.

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Michael A. Herrera
,
Istvan Szunyogh
, and
Joseph Tribbia

Abstract

This paper employs local linear, spatial spectral, and Lorenz curve–based diagnostics to investigate the dynamics of uncertainty in global numerical weather forecasts in the NH extratropics. The diagnostics are applied to ensembles in the THORPEX Interactive Grand Global Ensemble (TIGGE). The initial growth of uncertainty is found to be the fastest at the synoptic scales (zonal wavenumbers 7–9) most sensitive to baroclinic instability. At later forecast times, the saturation of uncertainties at the synoptic scales and the longer sustainable growth of uncertainty at the large scales lead to a gradual shift of the wavenumber of the dominant uncertainty toward zonal wavenumber 5. At the subsynoptic scales, errors saturate as predicted by Lorenz’s classic theory. While the ensembles capture the general characteristics of the uncertainty dynamics efficiently, there are locations where the predicted magnitude and structure of uncertainty have considerable time-mean errors. In addition, the magnitude of systematic errors in the prediction of the uncertainty increases with increasing forecast time. These growing systematic errors are dominated by errors in the prediction of low-frequency changes in the large-scale flow.

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Zhao-Xia Pu
,
Eugenia Kalnay
,
Joseph Sela
, and
Istvan Szunyogh

Abstract

A quasi-inverse linear method has been developed to study the sensitivity of forecast errors to initial conditions for the National Centers for Environmental Prediction’s (NCEP) global spectral model. The inverse is approximated by running the tangent linear model (TLM) of the nonlinear forecast model with a negative time step, but reversing the sign of friction and diffusion terms, in order to avoid the computational instability that would be associated with these terms if they were run backward. As usually done using the adjoint model integrations, the quasi-inverse TLM is started at the time of the verified forecast error and integrated backward to the corresponding initial time.

First, a numerical experiment shows that this quasi-inverse linear estimation is able to trace back the differences between two perturbed forecasts from the NCEP ensemble forecasting system and recover with good accuracy the known difference between the two forecasts at the initial time. This result shows that both the linear estimation and the quasi-inverse linear estimation are quite close to the nonlinear evolution of the perturbation in the nonlinear forecast model, suggesting that it should be possible to apply the method to the study of the sensitivity of forecast errors to initial conditions. The authors then calculate the perturbation field at the initial time (estimate the initial error) by tracing back a 1-day forecast error using the TLM quasi-inverse estimation. As could be expected from the previous experiment, when the estimated error is subtracted from the original analysis, the new initial conditions lead to an almost perfect 1-day forecast. The forecasts beyond the first day are also considerably improved, indicating that the initial conditions have indeed been improved.

In the remainder of the paper, this quasi-inverse linear method is compared with the adjoint sensitivity method (Rabier et al., Pu et al.) for medium-range weather forecasting. The authors find that both methods are able to trace back the forecast error to perturbations that improve the initial conditions. However, the forecast improvement obtained by the quasi-inverse linear method is considerably better than that obtained with a single adjoint iteration and similar to the one obtained using five iterations of the adjoint method, even though each adjoint iteration requires at least twice the computer resources of the quasi-inverse TLM estimation. Whereas the adjoint forecast sensitivities are closely related to singular vectors, the quasi-inverse linear perturbations are associated with the bred (Lyapunov) vectors used for ensemble forecasting at NCEP (Toth and Kalnay). The features of the two types of perturbations are also compared in this study. Finally, the possibility of the use of the sensitivity perturbation to improve future forecast skill is discussed, and preliminary experiments encourage further testing of this rather inexpensive method for possible operational use.

The model used in this study is the NCEP operational global spectral model at a resolution of T62/L28. The corresponding TLM, and its adjoint, are based on an adiabatic version of the model but include both horizontal and vertical diffusion.

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Aleksey V. Zimin
,
Istvan Szunyogh
,
Brian R. Hunt
, and
Edward Ott

Abstract

Previously developed techniques that have been used to extract envelopes of Rossby wave packets are based on the assumption of zonally propagating waves. In this note a method that does not require such an assumption is proposed. The advantages of the new technique, both on analytical and real-world examples, are demonstrated.

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Seung-Jong Baek
,
Istvan Szunyogh
,
Brian R. Hunt
, and
Edward Ott

Abstract

Model error is the component of the forecast error that is due to the difference between the dynamics of the atmosphere and the dynamics of the numerical prediction model. The systematic, slowly varying part of the model error is called model bias. This paper evaluates three different ensemble-based strategies to account for the surface pressure model bias in the analysis scheme. These strategies are based on modifying the observation operator for the surface pressure observations by the addition of a bias-correction term. One estimates the correction term adaptively, while another uses the hydrostatic balance equation to obtain the correction term. The third strategy combines an adaptively estimated correction term and the hydrostatic-balance-based correction term. Numerical experiments are carried out in an idealized setting, where the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) model is integrated at resolution T62L28 to simulate the evolution of the atmosphere and the T30L7 resolution Simplified Parameterization Primitive Equation Dynamics (SPEEDY) model is used for data assimilation. The results suggest that the adaptive bias-correction term is effective in correcting the bias in the data-rich regions, while the hydrostatic-balance-based approach is effective in data-sparse regions. The adaptive bias-correction approach also has the benefit that it leads to a significant improvement of the temperature and wind analysis at the higher model levels. The best results are obtained when the two bias-correction approaches are combined.

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Soojin Roh
,
Marc G. Genton
,
Mikyoung Jun
,
Istvan Szunyogh
, and
Ibrahim Hoteit

Abstract

Current ensemble-based Kalman filter (EnKF) algorithms are not robust to gross observation errors caused by technical or human errors during the data collection process. In this paper, the authors consider two types of gross observational errors, additive statistical outliers and innovation outliers, and introduce a method to make EnKF robust to gross observation errors. Using both a one-dimensional linear system of dynamics and a 40-variable Lorenz model, the performance of the proposed robust ensemble Kalman filter (REnKF) was tested and it was found that the new approach greatly improves the performance of the filter in the presence of gross observation errors and leads to only a modest loss of accuracy with clean, outlier-free, observations.

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