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Digital wavelet transform (DWT) is a well-known tool for characterizing piecewise-smooth signalsand is based on predicting smooth signals. Seismic signals are not smooth. However, they arepredictable. Seislet transform is a digital wavelet-like transform, which is tailored specifically forrepresenting seismic data. Its construction is based on the notion of signal prediction. Seislettransform uses predictions of sinusoids (applicable to seismic data in the F-X domain), plane waves(applicable to 2-D or 3-D seismic data in the T-X domain), or reflection events (applicable to prestackseismic data). We combine statistical or physical predictability of seismic signals with the liftingscheme of DWT to define the seislet transform.One can view the seislet transform as decomposition into multiscale orthogonal basis functionsaligned with seismic events. When multiple interfering events are present in the data, it is alsopossible to follow all of them simultaneously by turning the seislet basis into an overcompleterepresentation (a tight frame). Even though the seislet frame is overcomplete, it can be constrained tohave only a small number of significant coefficients and, therefore, to provide an optimally sparserepresentation. The sparsity is easily demonstrated by comparing the seislet transform and frame withthe classic transforms, such as Fourier and DWT. The classic DWT is equivalent to the seislettransform with a zero frequency (in 1-D) or zero slope (in 2-D).The sparsity of the transform domain provides not only an effective seismic data compression tool butalso a way for designing efficient data analysis algorithms. Traditional geophysical data analysistasks, such as signal-noise separation and data regularization, are conveniently formulated in thetransform domain, where the signal is sparse. When applied in the offset direction on prestack data,the seislet transform finds an additional application in optimal stacking of seismic records.

Seislet Transform and Seislet Frame: Tools for Compressive Representation of Seismic Data