The W Transform (2021)

Time-frequency spectral analysis methods such as the S transform cannot appropriately present low-frequency anomalies in seismic traces. That is because they generate a time-frequency spectrum with a low resolution in time at low frequencies. Professor Wang recently invented the W transform, to improve the time resolution of the spectrum at low frequencies, for an effective detection of seismic anomalies related to the hydrocarbon reservoirs in petroleum exploration and the abnormal features in near-surface geophysics. 

The W transform has three features: (1) The spectral energy is concentrated around the dominant frequency of a seismic waveform, rather than being shifted towards higher frequencies by the S transform; (2) The implementation is numerically stable as it avoids any potential frequency singularity in the S transform; (3) The Gaussian window function is defined using a nonstationary frequency weight, rather than using a stationary frequency weight in the S transform, because the dominant frequency is a time-dependent function. Therefore, the time-frequency spectrum generated by the W transform appropriately reflects seismic properties varying with the geological depth, and has an improved time resolution at low frequencies that it is suitable for characterizing reservoirs in petroleum geophysics and for detecting karsts in construction engineering.

This example shows four iso-frequency spectra at 12, 16, 20 and 24 Hz (from left to right). 

The first three spectral profiles at [12, 16, 20] Hz reveal a low-frequency shadow (LFS) at time 1.8 s (in the first trace on the left). The low-frequency shadow evidently disappears when the frequency is equal to or greater than 24 Hz.

The article is published in GEOPHYSICS (2021, vol. 86, no. 1), doi: 10.1190/GEO2020-0316.1