The W transform

Time-frequency spectral analysis methods such as the S transform cannot appropriately present low-frequency anomalies in seismic traces because they generate a time-frequency spectrum with a low resolution in time at low frequencies. I have developed the W transform to improve the time resolution of the spectrum at low frequencies, for an effective detection of seismic anomalies related to hydrocarbon reservoirs in petroleum exploration and 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 toward higher frequencies by the S transform; (2) the implementation is numerically stable because 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 represents seismic properties varying with geologic depth, and it has an improved time resolution at low frequencies that makes it suitable for characterizing reservoirs in petroleum geophysics and for detecting karsts in construction engineering.