next up previous [pdf]

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

\frac{\partial P}{\partial x} +
\sigma\,\frac{\partial P}{\partial t} = 0\;,
\end{displaymath} (1)

where $P(t,x)$ is the wave field and $\sigma$ is the local slope, which may also depend on $t$ and $x$ (Claerbout, 1992). In the case of a constant slope, equation 1 has the simple general solution
P(t,x) = f(t - \sigma x)\;,
\end{displaymath} (2)

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

Let $\mathbf{s}$ represent a seismic section as a collection of traces: $\mathbf{s} =
\left[\mathbf{s}_1 \; \mathbf{s}_2 \; \ldots \;
\mathbf{s}_N\right]^T$, where $\mathbf{s}_k$ corresponds to $P(t,x_k)$ for $k=1,2,\ldots$ 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

\mathbf{r} = \mathbf{D\,s}\;,
\end{displaymath} (3)

where $\mathbf{r}$ is the destruction residual, and $\mathbf{D}$ is the destruction operator defined as
\mathbf{D} =
\mathbf{I} & 0...
...& - \mathbf{P}_{N-1,N} & \mathbf{I} \\
\end{displaymath} (4)

where $\mathbf{I}$ stands for the identity operator, and $\mathbf{P}_{i,j}$ describes prediction of trace $j$ from trace $i$. 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 $x$ direction. The dominant slopes are estimated by minimizing the prediction residual $\mathbf{r}$ 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, $\sigma_x(t,x,y)$ and $\sigma_y(t,x,y)$, and a pair of destruction operators, $\mathbf{D}_x$ and $\mathbf{D}_y$, 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 $k$ from trace $1$ is simply

\mathbf{P}_{1,k} = \mathbf{P}_{k-1,k}\,
\end{displaymath} (5)

If $\mathbf{s}_r$ is a reference trace, then the prediction of trace $\mathbf{s}_k$ is $\mathbf{P}_{r,k}\,\mathbf{s}_r$. I call the recursive operator $\mathbf{P}_{r,k}$ 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.

next up previous [pdf]

Next: Predictive painting in 2-D Up: Fomel: Predictive painting Previous: Introduction