Discussions

SDRNAR is a new spectral decomposition approach that can decompose a 1D signal into different spectral components. With exact mathematical formulation, SDRNAR can control its behavior by selecting different parameters. SDRNAR differs from EMD in that SDRNAR can be manually manipulated rather than empirically data dependent. The proposed denoising algorithm is based on applying SDRNAR to each seismic trace. Although there are other options of direction and domain for applying SDRNAR, this paper focuses on the $t$-$x$ domain application of SDRNAR to each 1-D signal along temporal direction.

The proposed approach is more beneficial for post-stack or NMO corrected profiles where seismic events are mostly horizontal. We can understand this property by analyzing the implementation steps of SDRNAR. We apply SDRNAR to each trace of a profile with the same parameters. When seismic events are not horizontal, the optimal parameters used for each trace may not be equivalent. It can not be made possible for manually choosing the best parameters for each trace. However, an efficient and adaptive parameter selection algorithm for each trace can be developed in the future in order to handle the horizontal-events restriction. It can perform obviously better than those specific denoising approaches for horizontal events, e.g. mean filter, as it is applied based on the assumption of spatial coherency. There is danger in smoothing too much along spatial directions using other approaches. The proposed approach can be widely used in processing land post-stack data, where most of reflections are horizontal. The proposed approach can also be used to denoise microseismic trace, because many microseismic data can only be processed trace by trace.

The efficiency and performance for SDRNAR to decompose seismic traces into local monotonic components are better than that of EMD. Because of the explicit mathematical formulation of SDRNAR, we can use fast iterative solver to handle the under-determined equations involved in SDRNAR. However, without any mathematical model to support EMD, we can not use fast algorithm to apply EMD. The cost of SDRNAR is $O(NN_tN_{iter})$, where $N_t$ is the number of time samples and $N_{iter}$ is the number of shaping iterations (typically between 10 and 100). As can be seen in the examples, the decomposition of seismic traces achieved by EMD are not applicable for removing random noise, because of the serious mode-mixing problem. The separated components of SDRNAR, however, obtain excellent results for removing unpredictable random noise.