    Predictive painting of 3-D seismic volumes  Next: Predictive painting in 2-D Up: Fomel: Predictive painting Previous: Introduction

# Destruction and prediction of plane waves

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 (1)

where is the wave field and is the local slope, which may also depend on and (Claerbout, 1992). In the case of a constant slope, equation 1 has the simple general solution (2)

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 (3)

where is the destruction residual, and is the destruction operator defined as (4)

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 (5)

If is a reference trace, then the prediction of trace is . I call the recursive operator predictive painting. Once the elementary prediction operators in equation 4 are determined by plane-wave destruction, predictive painting can spread information from a given trace to its neighbors recursively by following the local structure of seismic events. The next section illustrates the painting concept using 2-D examples.    Predictive painting of 3-D seismic volumes  Next: Predictive painting in 2-D Up: Fomel: Predictive painting Previous: Introduction

2013-03-02