Predictive painting of 3-D seismic volumes |

Plane-wave destruction originates from a local plane-wave model for
characterizing seismic data (Fomel, 2002). The
mathematical basis is the local plane differential equation

where is an arbitrary waveform. Equation 2 is nothing more than a mathematical description of a plane wave. Assuming that the slope varies in time and space, one can design a local operator to propagate each trace to its neighbors.

Let represent a seismic section
as a collection of traces:
, where corresponds to
for A plane-wave destruction operator
(Fomel, 2002) effectively predicts each trace from its
neighbor and subtracts the prediction from the original trace. In the
linear operator notation, the plane-wave destruction operation can be
defined as

where stands for the identity operator, and describes prediction of trace from trace . Prediction of a trace consists of shifting the original trace along dominant event slopes. The prediction operator is a numerical solution of equation 1 for local plane wave propagation in the direction. The dominant slopes are estimated by minimizing the prediction residual using regularized least-squares optimization. I employ shaping regularization (Fomel, 2007a) for controlling the smoothness of the estimated slope fields. In the 3-D case, a pair of inline and crossline slopes, and , and a pair of destruction operators, and , are required to characterize the 3-D structure. Each prediction in 3-D occurs in either inline or crossline direction and thus conforms to equation 4. However, as explained below in the discussion of Dijkstra's algorithm, it is possible to arrange all 3-D traces in a sequence for further processing.

Prediction of a trace from a distant neighbor can be accomplished by
simple recursion. Predicting trace from trace is simply

Predictive painting of 3-D seismic volumes |

2013-03-02