Fourier methods of seismic data regularisation

One Dimensional Example

The power of the ALFT is best observed using a 1-D example and examining how the input series and corresponding spectra evolve with iterations. A 1-D irregular axis is constructed, where the irregularity is limited to half a bin size about the corresponding `regular' position such that $x(ix) = ox + (ix-1)*dx + (rand-0.5)*dx$ , where $ix$ in the position index, $ox$ an arbitrary origin, $dx$ the receiver/trace spacing and $rand$ a random number between 0 and 1. This is representative of the problem for land data.

Using this axis a superposition of sine waves is created of varying frequencies, amplitude and phase. For the presented example two different waves are used for viewing simplicity.

input Figure 1. Part of the irregularly sampled sine wave superposition used as input data/

1ddata
Figure 2. A section of the DFT reconstructed sine wave and its corresponding spectrum.

1dalft
Figure 3. A section of the ALFT reconstructed sine wave and its corresponding spectrum.

As can be seen in Figure 3 the ALFT has perfectly reconstructed a regular signal from the irregular input, and the spectrum is free from leakage except for adjacent to the main peaks. In this oversimplified example only four iterations are really required. Also note the amplitude preservation between Figure 2 and Figure 3. For further insight a more complicated, higher dimensional example is required.

Fourier methods of seismic data regularisation