A new extension of seismic instantaneous frequency using a fractional time derivative

The seismic instantaneous frequency attribute has been applied to interpret a depositional layer. Instead of the standard instantaneous frequency, the Caputo fractional differential operator has been demonstrated as a more suitable mathematical tool of nonlinear analysis for the depositional layer. Based on the Caputo operator, we propose a fractional extension of instantaneous frequency attribute to detect thin layers of sandstone formations. From numerical analysis of a 25-Hz Ricker wavelet, we found that the fractional instantaneous frequency of 0.99 order keeps an approximation of standard instantaneous frequency and enhances the negative frequency (anomalous frequency spike), which help in more precise geological interpretation. A three-layer wedge model and a case history from the Bohai Bay Basin showed that the 0.99 order instantaneous frequency provides more depositional sandstone information than the standard instantaneous frequency. The anomalous frequency spike, at the merging point of the top and the base reflections or at the tip of the stratum, of 0.99 order instantaneous frequency is more prominent than that of the standard one. The results help to interpret the thickness changes of thin sandstone formations. In practical application, the 0.99 order instantaneous frequency has revealed several deposition phases and lateral heterogeneity of a fluvial reservoir system. Therefore, the 0.99 order instantaneous frequency is a valuable seismic attribute for reservoir characterization.