Seislet Transform

The seislet transform is a digital wavelet-like transform adapted to seismic data. It uses a wavelet lifting scheme in which the input traces are divided into even and odd parts and . A prediction operator and an update operator form the residual and coarse approximation

			(1)
			(2)

The procedure is repeated at successive scales.

In two dimensions, prediction follows local plane-wave slopes so that the resulting multiscale basis functions align with seismic events. In one dimension, they follow locally sinusoidal components. The ordinary wavelet transform is recovered for zero slope in two dimensions or zero frequency in one dimension. Using several slope or frequency fields gives an overcomplete representation or tight frame rather than a single orthogonal basis (Fomel and Liu 2010).

The alignment makes seismic data sparse in the seislet domain and permits compression, denoising, and interpolation of missing traces. It is therefore an example of a sparsifying transform used in compressed sensing.