A large amount of sampling noise which exists in the ensemble-based background error variance need be reduced effectively before being applied to operational data assimilation system. Unlike the typical Gaussian white noise, the sampling noise is scaled and space-dependent, thus making its energy level on some scales much larger than the average. Although previous denoising methods such as spectral filtering or wavelet thresholding have been successfully used for denoising Gaussian white noise, they are no longer applicable for dealing with this kind of sampling noise. One can use a different threshold for each scale, but it will bring a big error especially on larger scales. Another modified method is to use a global multiplicative factor, alpha, to adjust the filtering strength based on the optimization of trade-off between removal of the noise and averaging of the useful signal. However, tuning alpha is not so easy, especially in real operational numerical weather prediction context. It motivates us to develop a new nearly cost-free filter whose threshold can be automatically calculated. According to the characteristics of sampling noise in background error variance, a heterogeneous filtering method similar to wavelet threshold technology is employed. The threshold, T-A, determined by iterative algorithm is used to estimate the truncated remainder whose norm is smaller than T-A. The standard deviation of truncated remainder term is regard as first guess of sampling noise. Non-Guassian term of sampling noise, whose coefficient modulus is above T-A, is regarded as a small probability event. In order to incorporate such a coefficient into the domain of [-T, T], a semi-empirical formula is used to calculate and approach the ideal threshold. Investigations are first conducted in a one-dimensional (1D) framework: several methods such as spectral filter, primal wavelet filter, optimal wavelet filter, and proposed filter are used to denoise the scale and space-dependent sampling noise in variance estimations. Their validity and performance are compared and examined with different ensemble sizes. Results show that the proposed method can efficiently filter out a large amount of sampling noise efficiently and improve the estimation accuracy of background error variance. Compared with previous filters, the modified threshold can help to reduce some residual noise arising from the scales where the noise level is above the average level, and the filtering performance of the new method is improved by almost 13.28%. Application to the real ensemble four-dimensional variational data assimilation system is then considered. Results show that the wavelet threshold method outperforms the spectral method. Modified threshold can enhance denosing without deforming the signal of interest. A new nearly cost-free filter is proposed to reduce the scale and space-dependent sampling noise in ensemble-based background error variance. It is able to remove most of the sampling noises, while extracting the signal of interest. Compared with those of primal wavelet filter and spectral filter, the performance and efficiency of proposed method are improved in 1D framework and real data assimilation system experiments. Further work should focus on the sphere wavelets, which is appropriate for analysing and reconstructing the signals on the sphere in global spectral models.
