scholarly journals Growth of Forecast Errors from Covariances Modeled by 4DVAR and ETKF Methods

2011 ◽  
Vol 139 (5) ◽  
pp. 1505-1518 ◽  
Author(s):  
Chiara Piccolo
Keyword(s):  
Weather Forecasting ◽  
Initial Conditions ◽  
Forecast Error ◽  
Forecast Errors ◽  
Model Errors ◽  
Background Error ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Numerical Weather Forecasting ◽  
Ensemble Transform Kalman Filter

Numerical weather forecasting errors grow with time. Error growth results from the amplification of small perturbations due to atmospheric instability or from model deficiencies during model integration. In current NWP systems, the dimension of the forecast error covariance matrices is far too large for these matrices to be represented explicitly. They must be approximated. This paper focuses on comparing the growth of forecast error from covariances modeled by the Met Office operational four-dimensional variational data assimilation (4DVAR) and ensemble transform Kalman filter (ETKF) methods over a period of 24 h. The growth of forecast errors implied by 4DVAR is estimated by drawing a random sample of initial conditions from a Gaussian distribution with the standard deviations given by the background error covariance matrix and then evolving the sample forward in time using linearized dynamics. The growth of the forecast error modeled by the ETKF is estimated by propagating the full nonlinear model in time starting from initial conditions generated by an ETKF. This method includes model errors in two ways: by using an inflation factor and by adding model perturbations through a stochastic physics scheme. Finally, these results are compared with a benchmark of the climatological error. The forecast error predicted by the implicit evolution of 4DVAR does not grow, regardless of the dataset used to generate the static background error covariance statistics. The forecast error predicted by the ETKF grows more rapidly because the ETKF selects balanced initial perturbations, which project onto rapidly growing modes. Finally, in both cases it is not possible to disentangle the contribution of the initial condition error from the model error.

scholarly journals A Comparison of Hybrid Ensemble Transform Kalman Filter–Optimum Interpolation and Ensemble Square Root Filter Analysis Schemes

2007 ◽  
Vol 135 (3) ◽  
pp. 1055-1076 ◽  
Author(s):  
Xuguang Wang ◽  
Thomas M. Hamill ◽  
Jeffrey S. Whitaker ◽  
Craig H. Bishop
Keyword(s):  
Ensemble Size ◽  
Hybrid Scheme ◽  
Background Error ◽  
Background Error Covariance ◽  
Optimum Interpolation ◽  
Error Covariance ◽  
Analysis Scheme ◽  
Ensemble Transform Kalman Filter ◽  
Square Root Filter ◽  
Static Background

Abstract A hybrid ensemble transform Kalman filter (ETKF)–optimum interpolation (OI) analysis scheme is described and compared with an ensemble square root filter (EnSRF) analysis scheme. A two-layer primitive equation model was used under perfect-model assumptions. A simplified observation network was used, and the OI method utilized a static background error covariance constructed from a large inventory of historical forecast errors. The hybrid scheme updated the ensemble mean using a hybridized ensemble and static background-error covariance. The ensemble perturbations in the hybrid scheme were updated by the ETKF scheme. The EnSRF ran parallel data assimilation cycles for each member and serially assimilated the observations. The EnSRF background-error covariance was estimated fully from the ensemble. For 50-member ensembles, the analyses from the hybrid scheme were as accurate or nearly as accurate as those from the EnSRF, depending on the norm. For 20-member ensembles, the analyses from the hybrid scheme were more accurate than analyses from the EnSRF under certain norms. Both hybrid and EnSRF analyses were more accurate than the analyses from the OI. Further reducing the ensemble size to five members, the EnSRF exhibited filter divergence, whereas the analyses from the hybrid scheme were still better than those updated by the OI. Additionally, the hybrid scheme was less prone to spurious gravity wave activity than the EnSRF, especially when the ensemble size was small. Maximal growth in the ETKF ensemble perturbation space exceeded that in the EnSRF ensemble perturbation space. The relationship of the ETKF ensemble variance to the analysis error variance, a measure of a spread–skill relationship, was similar to that of the EnSRF ensemble. The hybrid scheme can be implemented in a reasonably straightforward manner in the operational variational frameworks, and the computational cost of the hybrid is expected to be much less than the EnSRF in the operational settings.

Scale-Dependent Background Error Covariance Localization: Evaluation in a Global Deterministic Weather Forecasting System

2018 ◽  
Vol 146 (5) ◽  
pp. 1367-1381 ◽  
Author(s):  
Jean-François Caron ◽  
Mark Buehner
Keyword(s):  
Data Assimilation ◽  
Weather Forecasting ◽  
Spatial Scales ◽  
Winter Period ◽  
Wind Fields ◽  
Horizontal Scale ◽  
Background Error ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Forecasting System

Abstract Scale-dependent localization (SDL) consists of applying the appropriate (i.e., different) amount of localization to different ranges of background error covariance spatial scales while simultaneously assimilating all of the available observations. The SDL method proposed by Buehner and Shlyaeva for ensemble–variational (EnVar) data assimilation was tested in a 3D-EnVar version of the Canadian operational global data assimilation system. It is shown that a horizontal-scale-dependent horizontal localization leads to implicit vertical-level-dependent, variable-dependent, and location-dependent horizontal localization. The results from data assimilation cycles show that horizontal-scale-dependent horizontal covariance localization is able to improve the forecasts up to day 5 in the Northern Hemisphere extratropical summer period and up to day 7 in the Southern Hemisphere extratropical winter period. In the tropics, use of SDL results in improvements similar to what can be obtained by increasing the uniform amount of spatial localization. An investigation of the dynamical balance in the resulting analysis increments demonstrates that SDL does not further harm the balance between the mass and the rotational wind fields, as compared to the traditional localization approach. Potential future applications for the SDL method are also discussed.

scholarly journals A Comparison of the Hybrid and EnSRF Analysis Schemes in the Presence of Model Errors due to Unresolved Scales

2009 ◽  
Vol 137 (10) ◽  
pp. 3219-3232 ◽  
Author(s):  
Xuguang Wang ◽  
Thomas M. Hamill ◽  
Jeffrey S. Whitaker ◽  
Craig H. Bishop
Keyword(s):  
Model Comparison ◽  
Additive Noise ◽  
Initial Conditions ◽  
Random Noise ◽  
Model Error ◽  
Forecast Errors ◽  
Model Errors ◽  
Hybrid Analysis ◽  
Error Covariance ◽  
Analysis Scheme

Abstract A hybrid analysis scheme is compared with an ensemble square root filter (EnSRF) analysis scheme in the presence of model errors as a follow-up to a previous perfect-model comparison. In the hybrid scheme, the ensemble perturbations are updated by the ensemble transform Kalman filter (ETKF) and the ensemble mean is updated with a hybrid ensemble and static background-error covariance. The experiments were conducted with a two-layer primitive equation model. The true state was a T127 simulation. Data assimilation experiments were conducted at T31 resolution (3168 complex spectral coefficients), assimilating imperfect observations drawn from the T127 nature run. By design, the magnitude of the truncation error was large, which provided a test on the ability of both schemes to deal with model error. Additive noise was used to parameterize model errors in the background ensemble for both schemes. In the first set of experiments, additive noise was drawn from a large inventory of historical forecast errors; in the second set of experiments, additive noise was drawn from a large inventory of differences between forecasts and analyses. The static covariance was computed correspondingly from the two inventories. The hybrid analysis was statistically significantly more accurate than the EnSRF analysis. The improvement of the hybrid over the EnSRF was smaller when differences of forecasts and analyses were used to form the random noise and the static covariance. The EnSRF analysis was more sensitive to the size of the ensemble than the hybrid. A series of tests was conducted to understand why the EnSRF performed worse than the hybrid. It was shown that the inferior performance of the EnSRF was likely due to the sampling error in the estimation of the model-error covariance in the mean update and the less-balanced EnSRF initial conditions resulting from the extra localizations used in the EnSRF.

scholarly journals Implementing Hybrid Background Error Covariance into the LETKF with Attenuation-based Localization: Experiments with a Simplified AGCM

2021 ◽  
Keyword(s):  
Weather Prediction ◽  
Hybrid Approach ◽  
Localization Length ◽  
Atmospheric Model ◽  
Forecast Errors ◽  
Observation Error ◽  
Background Error ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Optimal Localization

Abstract Recent numerical weather prediction systems have significantly improved medium-range forecasts by implementing hybrid background error covariance, for which climatological (static) and ensemble-based (flow-dependent) error covariance are combined. While the hybrid approach has been investigated mainly in variational systems, this study aims at exploring methods for implementing the hybrid approach for the local ensemble transform Kalman filter (LETKF). Following Kretchmer et al. (2015), the present study constructed hybrid background error covariance by adding collections of climatological perturbations to the forecast ensemble. In addition, this study proposes a new localization method that attenuates the ensemble perturbation (Z-localization) instead of inflating observation error variance (R-localization). A series of experiments with a simplified global atmospheric model revealed that the hybrid LETKF resulted in smaller forecast errors than the LETKF, especially in sparsely observed regions. Due to the larger ensemble enabled by the hybrid approach, optimal localization length scales for the hybrid LETKF were larger than those for the LETKF. With the LETKF, the Z-localization resulted in similar forecast errors as the R-localization. However, Z-localization has an advantage in enabling to apply different localization scales for flow-dependent perturbation and climatological static perturbations with the hybrid LETKF. The optimal localization for climatological perturbations was slightly larger than that for flow-dependent perturbations. This study proposes Optimal EigenDecomposition (OED) ETKF formulation to reduce computational costs. The computational expense of the OED ETKF formulation became significantly smaller than that of standard ETKF formulations as the number of climatological perturbations was increased beyond a few hundred.

scholarly journals Characteristics of Background Error Covariance of Soil Moisture and Atmospheric States in Strongly Coupled Land–Atmosphere Data Assimilation

2018 ◽  
Vol 57 (11) ◽  
pp. 2507-2529 ◽  
Author(s):  
Liao-Fan Lin ◽  
Zhaoxia Pu
Keyword(s):  
Soil Moisture ◽  
Surface Temperature ◽  
Surface Soil ◽  
Forecast Errors ◽  
Surface Soil Moisture ◽  
Background Error ◽  
Near Surface ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Temperature And Humidity

AbstractThis study characterizes the spatial and temporal variability of the background error covariance between the land surface soil moisture and atmospheric states for a better understanding of the potentials of assimilating satellite soil moisture data under a framework of strongly coupled land–atmosphere data assimilation. The study uses the Noah land surface model coupled with the Weather Research and Forecasting (WRF) Model and the National Meteorological Center (NMC) method for computing the land–atmosphere background error covariance from 2015 to 2017 over the contiguous United States. The results show that the forecast errors in top-10-cm soil moisture and near-surface air potential temperature and specific humidity are correlated and relatively large during the daytime in the summer. The magnitude of the error correlation between these three states is comparable. For example, in July, the error correlation averaged over all day- and nighttime samples is −0.13 for near-surface temperature and humidity, −0.20 for surface soil moisture and near-surface temperature, and 0.15 for surface soil moisture and near-surface humidity. During the summer, the forecast errors in surface soil moisture are correlated with those of atmospheric states up to the sigma pressure level of 0.9 (approximately 900 hPa for a sea level location) with domain-mean correlations of −0.15 and 0.1 for temperature and humidity, respectively. The results suggest that assimilation of satellite soil moisture data could provide cross-variable impacts comparable to those assimilating conventional near-surface temperature and humidity data. The forecast errors of soil moisture are only marginally correlated with those of the winds.

Impact of AMSU-A Radiances in a Column Ensemble Kalman Filter

2018 ◽  
Vol 146 (12) ◽  
pp. 3949-3976 ◽  
Author(s):  
Herschel L. Mitchell ◽  
P. L. Houtekamer ◽  
Sylvain Heilliette
Keyword(s):  
Covariance Matrix ◽  
Background Error ◽  
Error Covariance Matrix ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Microwave Sounding Unit ◽  
The Matrix ◽  
Microwave Sounding ◽  
Background Temperature ◽  
The Difference

Abstract A column EnKF, based on the Canadian global EnKF and using the RTTOV radiative transfer (RT) model, is employed to investigate issues relating to the EnKF assimilation of Advanced Microwave Sounding Unit-A (AMSU-A) radiance measurements. Experiments are performed with large and small ensembles, with and without localization. Three different descriptions of background temperature error are considered: 1) using analytical vertical modes and hypothetical spectra, 2) using the vertical modes and spectrum of a covariance matrix obtained from the global EnKF after 2 weeks of cycling, and 3) using the vertical modes and spectrum of the static background error covariance matrix employed to initiate a global data assimilation cycle. It is found that the EnKF performs well in some of the experiments with background error description 1, and yields modest error reductions with background error description 3. However, the EnKF is virtually unable to reduce the background error (even when using a large ensemble) with background error description 2. To analyze these results, the different background error descriptions are viewed through the prism of the RT model by comparing the trace of the matrix , where is the RT model and is the background error covariance matrix. Indeed, this comparison is found to explain the difference in the results obtained, which relates to the degree to which deep modes are, or are not, present in the different background error covariances. The results suggest that, after 2 weeks of cycling, the global EnKF has virtually eliminated all background error structures that can be “seen” by the AMSU-A radiances.

scholarly journals Model error in weather forecasting

2001 ◽  
Vol 8 (6) ◽  
pp. 357-371 ◽  
Author(s):  
D. Orrell ◽  
L. Smith ◽  
J. Barkmeijer ◽  
T. N. Palmer
Keyword(s):  
Prediction Models ◽  
Weather Forecasting ◽  
Initial Conditions ◽  
Forecast Error ◽  
Weather Prediction ◽  
Model Error ◽  
Local Model ◽  
State Dependent ◽  
Rapid Divergence ◽  
Dependent Model

Abstract. Operational forecasting is hampered both by the rapid divergence of nearby initial conditions and by error in the underlying model. Interest in chaos has fuelled much work on the first of these two issues; this paper focuses on the second. A new approach to quantifying state-dependent model error, the local model drift, is derived and deployed both in examples and in operational numerical weather prediction models. A simple law is derived to relate model error to likely shadowing performance (how long the model can stay close to the observations). Imperfect model experiments are used to contrast the performance of truncated models relative to a high resolution run, and the operational model relative to the analysis. In both cases the component of forecast error due to state-dependent model error tends to grow as the square-root of forecast time, and provides a major source of error out to three days. These initial results suggest that model error plays a major role and calls for further research in quantifying both the local model drift and expected shadowing times.

scholarly journals ROAD TEMPERATURE MODELLING WITHOUT IN-SITU SENSORS

2017 ◽  
Vol 12 (4) ◽  
pp. 241-247 ◽  
Author(s):  
Karol Opara ◽  
Jan Zieliński
Keyword(s):  
Energy Balance ◽  
Urban Areas ◽  
Initial Conditions ◽  
Forecast Error ◽  
Weather Data ◽  
Forecast Errors ◽  
The Road ◽  
In Situ Sensors ◽  
Pavement Temperature

Modelling of the pavement temperature facilitates winter road maintenance. It is used for predicting the glaze formation and for scheduling the spraying of the de-icing brine. The road weather is commonly forecasted by solving the energy balance equations. It requires setting the initial vertical profile of the pavement temperature, which is often obtained from the Road Weather Information Stations. The paper proposes the use of average air temperature from seven preceding days as a pseudo-observation of the subsurface temperature. Next, the road weather model is run with a few days offset. It first uses the recent, historical weather data and then the available forecasts. This approach exploits the fact that the energy balance models tend to “forget” their initial conditions and converge to the baseline solution. The experimental verification was conducted using the Model of the Environment and Temperature of Roads and the data from a road weather station in Warsaw over a period of two years. The additional forecast error introduced by the proposed pseudo-observational initialization averages 1.2 °C in the first prediction hour and then decreases in time. The paper also discusses the use of Digital Surface Models to take into account the shading effects, which are an essential source of forecast errors in urban areas. Limiting the use of in-situ sensors opens a perspective for an economical, largescale implementation of road meteorological models.

Development of Convective-Scale Static Background Error Covariance within GSI-Based Hybrid EnVar System for Direct Radar Reflectivity Data Assimilation

2021 ◽  
Author(s):  
Yongming Wang ◽  
Xuguang Wang
Keyword(s):  
Sampling Errors ◽  
Background Error ◽  
3 Dimensional ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Convective Scale ◽  
Cross Correlations ◽  
Horizontal Momentum ◽  
Reflectivity Data ◽  
Static Background

AbstractA convective-scale static background-error covariance (BEC) matrix is further developed to include the capability of direct reflectivity assimilation and evaluated within the GSI-based 3-dimensional variational (3DVar) and hybrid ensemble-variational (EnVar) methods. Specific developments are summarized as follows: 1) Control variables (CVs) are extended to include reflectivity, vertical velocity, and all hydrometeor types. Various horizontal momentum and moisture CV options are included. 2) Cross-correlations between all CVs are established. 3) A storm intensity-dependent binning method is adopted to separately calculate static error matrices for clear-air and storms with varying intensities. The resultant static BEC matrices are simultaneously applied at proper locations guided by the observed reflectivity. 4) The EnVar is extended to adaptively incorporate static BECs based on the quality of ensemble covariances.Evaluation and examination of the new static BECs are first performed on the 8 May 2003 Oklahoma City supercell. Detailed diagnostics and 3DVar examinations suggest zonal/meridian winds and pseudo-relative humidity are selected horizontal momentum and moisture CVs for direct reflectivity assimilation, respectively; inclusion of cross-correlations favors to spinup and maintain the analyzed storms; application of binning improves characteristics and persistence of the simulated storm. Relative to an experiment using the full ensemble BECs (Exp-PureEnVar), incorporating static BECs in hybrid EnVar reduces spinup time and better analyzes reflectivity distributions while the background ensemble is deficient in sampling errors. Compared to both pure 3DVar and Exp-PureEnVar, hybrid EnVar better predicts reflectivity distributions and better maintains strong mesocyclone. Further examination through the 20 May 2013 Oklahoma supercells confirms these results and additionally demonstrates the effectiveness of adaptive hybridization.

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