2-D Adaptive Fractal Conservation Law for Seismic Random Noise Elimination

Abstract

The fractal conservation law (FCL) is a partial-differential-equation-based filtering approach. Analysis of the frequency response of the FCL indicates that it can eliminate high frequencies and preserve or amplify low/medium frequencies. Generally, the shape of the frequency response is fixed. Thus, the FCL cannot track the signal beyond the threshold, which corresponds to the cutoff frequency. Besides, the FCL recovers seismic events only along the time direction, thereby ignoring the coherence between neighboring traces. To resolve these shortcomings, this letter presents a novel spatiotemporal adaptive FCL method. We use a group of frequency responses of the FCL determined by different parameters to construct a convex hull of the filtering results. Then, we introduce an objective function based on the penalized least squares criterion with respect to FCL estimation on this convex hull and take the directional derivative as the penalty term. Thus, the correlations between the adjacent channels are taken into account in the algorithm. Therefore, the 2-D adaptive FCL is equivalent to a convex optimization problem with box constraints, which can be solved using the projected gradient descent algorithm. The application of gradient descent consists of taking the derivative of an objective function, which can be implemented quickly by means of the discrete cosine transform (DCT). The experimental results illustrate that our proposed algorithm has a higher output signal-to-noise ratio (SNR) than the 1-D adaptive FCL on some synthetic records and field seismic data.