Optimization of wavelet- and curvelet-based denoising algorithms by multivariate SURE and GCV

One of the most crucial challenges in seismic data processing is the reduction of noise in the data or improving the signal-to-noise ratio (SNR). Wavelet-and curvelet-based denoising algorithms have become popular to address random noise attenuation for seismic sections. Wavelet basis, thresholding function, and threshold value are three key factors of such algorithms, having a profound effect on the quality of the denoised section. Therefore, given a signal, it is necessary to optimize the denoising operator over these factors to achieve the best performance. In this paper a general denoising algorithm is developed as a multi-variant (variable) filter which performs in multi-scale transform domains (e.g. wavelet and curvelet). In the wavelet domain this general filter is a function of the type of wavelet, characterized by its smoothness, thresholding rule, and threshold value, while in the curvelet domain it is only a function of thresholding rule and threshold value. Also, two methods, Stein's unbiased risk estimate (SURE) and generalized cross validation (GCV), evaluated using a Monte Carlo technique, are utilized to optimize the algorithm in both wavelet and curvelet domains for a given seismic signal. The best wavelet function is selected from a family of fractional B-spline wavelets. The optimum thresholding rule is selected from general thresholding functions which contain the most well known thresholding functions, and the threshold value is chosen from a set of possible values. The results obtained from numerical tests show high performance of the proposed method in both wavelet and curvelet domains in comparison to conventional methods when denoising seismic data.