A Fractal Conservation Law for Simultaneous Denoising and Enhancement of Seismic Data

In this letter, we have applied a new filtering method for the simultaneous noise reduction and enhancement of seismic signals using a fractal conservation law, which is simply a partial differential equation (PDE) modified by a nonlocal fractional antidiffusive term of lower order. By using an integral formula, which is based on Taylor-Poisson's formula and Fubini's theorem for an antidiffusive term, and then taking the fast Fourier transform of the PDE, the filter in the frequency domain can be derived. The study showed that this filter eliminates the high frequencies, amplifies the medium frequencies, and preserves the low frequencies. Thus, some features of the signal, such as relative maxima or minima, can be preserved or even amplified. Usually, these features are flattened by other denoising methods. The properties of this novel method are tested on both synthetic seismic data and real common shot point seismic records. Additionally, we compared this method with time-frequency peak filtering (TFPF), which has a good performance under low signal-to-noise ratio. The experimental results illustrate the superior performance of our method versus TFPF in the recovery of seismic events by removing random noise and enhancing valid signal.