Research On The Adaptive Filtering Technology In The Flow-dependent Background Error Covariance

A coordinated and reasonable initial field is essential to improve the numerical forecasts quality,and the data assimilation method makes the background field into a more accurate and reasonable initial field by assimilating the observations.The background error covariance matrix describes the distribution of the background error,which not only determines the weight of the background field in the assimilation process,but also has important effects on the assimilation system such as information propagation,smoothing,dynamic balance relationships and flow-dependent properties.The ensemble data assimilation has spurious correlation problem in the estimated flow-dependent background error covariance because the ensemble members are much smaller than the mode state dimension.Currently,covariance localization methods are widely used in operational systems to alleviate various problems caused by under-sampling in ensemble data assimilation.However,traditional covariance localization methods require empirical adjustment of the localization radius,which introduces empirical errors and increases the computational cost by repeated adjustment.Although some adaptive localization methods have been developed,these are designed based on a specified ensemble data assimilation system and make specific assumptions based on the corresponding assimilation system.In this paper,we propose an adaptive covariance localization method for a variety of ensemble data assimilation systems to address the spurious correlation problem.The adaptive localization method proposed in this paper has two key implementation steps,the localization weighting function and the locaization radius.Firstly,the characteristics of the sampling error of the ensemble Kalman filter are analyzed,and the spurious correlation in the background error covariance estimates is assessed qualitatively.Under the consideration of the property that the background error covariance is a semi-positive definite matrix,and the Gaspari-Cohn function is proposed to be used as the localization weight function for the Schur product.This is because this type of Gaussian function has a positive effect in reducing the sampling noise and improving the matrix dissatisfaction rank and eigenvalues when Schuring product with the error covariance matrix.Second,based on the optimal linear filtering and the information of second-order moments and fourth-order moments of the ensemble background estimate,the relationship between the covariance localization radius threshold and the correlation statistics of the ensemble is constructed.This enables real-time updating of the localization function based on the flow-dependent of the ensemble background members.The adaptive localization method is improved in the data assimilation of the atmospheric system,and the spatial angle averaging is used to estimate the ensemble correlation expectation of high-dimensional variables.The improved adaptive covariance localization method has convergence in the expectation estimation process and the obtained radius threshold is more stable.It is applicable to a variety of ensemble data assimilation systems such as ensemble Kalman filter and its variants,ensemble variational data assimilation and hybrid assimilation,and can be written as an adaptive processing module in the traditional localization method,which is easily portable and run repeatedly.The adaptive localization method is based on the flow-dependent information of the ensemble background error,which can better reflect the real-time state of the weather scenario,enhance the robustness of the localization method in eliminating the spurious correlation,and save the computational resources wasted in adjusting the localization function repeatedly for different weather phenomena.At the same time,the adaptively updated localization function not only has flow-dependent characteristics,but also has a close connection with the observed variables and model levels,and the analysis incremental information fits well with the variable correlation change characteristics,which improves the assimilation analysis results and enhances the analysis and forecast of typhoon”Dujuan”.The simplification and approximation of the background error information by the background error covariance model of the variational assimilation system through the control variable transformation greatly improves the feasibility of the background error covariance matrix information operation.The standard deviation information of the analysis control variables is one of the most important statistical parameters in the background error covariance model,whose distribution information represents the error distribution of the mode variables and determines the contribution of the background field to the final analysis field in the assimilation process.The introduction of the ensemble-based flow-dependent background error standard deviation into the covariance matrix model in the variational data assimilation is an important direction in the development of the hybrid data assimilation methods.In response to the fact that the current standard deviation estimate suffers from sampling noise due to the finite samples,an anisotropic filtering method is proposed to improve the accuracy of the sampled estimates.By studying the characteristics of the signal and noise in the estimated values,the structure shows that there are significant differences in the spatial distribution and structure,and the estimated information of different mode variables has unique physical characteristics,while the noise distribution characteristics in the estimated values of different variables have similarity.The noise can be reduced by increasing the number of samples,but there is a contradiction between the accuracy of the estimated values and the computational cost in the operational implementation.A local weight filtering method based on distance weights and similarity weights is proposed after fully studying the characteristic differences of comparing estimated signal and noise.The method achieves the expansion of sample realizations by referring to the information of the estimated values of the neighbouring points in the filtering process,which has a good suppression effect on the estimation noise,significantly improves the signal-to-noise ratio of the filtered small sample estimates.The model variable estimates have a strong local variation property,and the local feature distribution is also crucial to the assimilation.Due to the introduction of similarity weight,the local weight filtering method can maintain the local characteristics of the estimated signal well during the filtering process,and performs better than the spectral truncation method and spatial averaging method currently used in the system.Moreover,the local weight filtering method is suitable to be written as a filtering program module in the operational system implementation,which can be run directly in the process of background error statistics estimation with high parallelism.