Seismic random noise elimination according to the adaptive fractal conservation law

The fractal conservation law (FCL) is based on the Cauchy problem of the partial differential equation (PDE), which is modified by an anti-diffusive term of lower order. The analysis indicates that it can eliminate the high frequencies and preserve or amplify the low/medium frequencies. The performance of FCL depends on the threshold selected for the PDE. This threshold corresponds to the cut-off frequency of FCL in the frequency domain. Generally, the threshold is fixed. Thus, the FCL cannot track the signal beyond the cut-off frequency, and it removes the higher-frequency components of the signal. To solve this problem, an adaptive FCL filtering method is presented. The main purpose of this method is to select the optimal FCL threshold in each sample index such that it can adaptively track the rapid changes in the signal. In the adaptive FCL, we select FCL estimations with different thresholds and construct a convex hull of these estimations of each sample index. Consequently, we introduce a quadratic functional with respect to FCL estimation to ensure that we select the optimal estimation from the convex hull of each sample index. This leads to a box-constrained convex problem, which can be solved by the Viterbi algorithm.