As proposed by Fomel and Liu (2010), the seislet transform can be
constructed using the lifting scheme (Sweldens, 1995).
The procedure of the lifting scheme is as follows:
- Arrange the input data as a sequence of records.
In the seislet transform, the input data are common shot gather or
seismic image.
- Divide the arranged records into even and odd components
and
. For the 2D seislet transform, each
trace is a component and the seismic data are split into two parts
according to trace indices.
- Find the residual
between the odd component and
its prediction from the even component
|
(1) |
where
is a prediction operator.
- Using the difference from the previous step to get a coarse
approximation
of the data by updating the even
component
|
(2) |
where
is an update operator.
- The coarse approximation
obtained from last step
becomes the new data, and previous steps are applied to the new data
at the next scale level.
The coarse approximation
from the final scale and residual
difference
from all scales form the result of the seislet
transform.
The prediction and update operators are defined, for example, by modifying
operators in the construction of CDF 5/3 biorthogonal
wavelets (Cohen et al., 1992) as follows:
|
(3) |
and
|
(4) |
where
and
can accomplish the
prediction of a trace from its left and right neighbors, respectively.
The predictions need to be applied at different scales.
For 2D seislet transform, different scales mean different distances
between traces.
More accurate higher-order formulations are possible.
Following the inverse lifting scheme from coarse scale to fine scale,
the inverse seislet transform can be constructed.
2024-07-04