Stratigraphic coordinates, a coordinate system tailored to seismic interpretation

Appendix A: Predictive painting

In the most general case, the predictive-painting method (Fomel, 2010) can be described as follows:

Local spatially-variable inline and cross-line slopes of seismic events are estimated by the plane-wave destruction method (Fomel, 2002). Plane-wave destruction originates from a local plane-wave model for characterizing seismic data, which is based on the plane-wave differential equation (Claerbout, 1992):

$$\frac{\partial P}{\partial x} +\sigma\frac{\partial P}{\partial t} =0.$$ (6)

Here $P\left( t,x \right)$ is the seismic wave-field at time $t$ and location $x$, and $\sigma$ is the local slope, which can be either constant or variable in both time and space. The local plane differential equation can easily be solved where the slope is constant, and it has a simple general solution

$$P\left(t,x\right)=f\left(t-\sigma x\right),$$ (7)

where $f\left(t\right)$ is an arbitrary waveform. Equation A-2 is just a mathematical description of a plane wave. In the case of variable slopes, a local operator is designed to propagate each trace to its neighbors by shifting seismic events along their local slopes.

By writing the plane-wave destruction operation in the linear operator notation, we have

$$\mathbf{r=Ds},$$ (8)

where $\mathbf{s}$ is a seismic section as a collection of traces $\left(\mathbf{s}=\left[s_1 s_2 \dots s_N\right]^T\right)$, $\mathbf{r}$ is the destruction residual, and $\tensor{D}$ is the destruction operator, defined as

$$\mathbf{D}=\left[\begin{array}{ccccc} \mathbf{I} & 0 & 0 & \... ... & \dots & -\mathbf{P}_{N-1,N} & \mathbf{I}\end{array}\right].$$ (9)

$\mathbf{I}$ is the identity operator, and $\mathbf{P}_{i,j}$ is an operator that predicts trace $j$ from trace $i$. By minimizing the prediction residual $\mathbf{r}$ using least-squares optimization and smooth regularization, the dominant slopes will be obtained. For $3D$ structure characterization, a pair of inline and crossline slopes, $\sigma_x(t,x,y)$ and $\sigma_y(t,x,y)$, and a pair of destruction operators, $\tensor{D}_x$ and $\tensor{D}_y$, are required. The prediction of trace $\mathbf{s}_k$ from reference trace $\mathbf{s}_r$ can be defined as $\mathbf{P}_{r,k}\mathbf{s}_r$, where

$$\mathbf{P}_{r,k} = \mathbf{P}_{k-1,k} \dots \mathbf{P}_{r+1,r+2} \mathbf{P}_{r,r+1}.$$ (10)

This is a simple recursion, and $\mathbf{P}_{r,k}$ is called the predictive-painting operator. After obtaining elementary prediction operators in equation A-4 by plane-wave destruction, predictive painting spreads the information contained in a seed trace to its neighbors by following the local slope of seismic events. In order to be able to paint all events in the seismic volume, one can use multiple references and average painting values extrapolated from different reference traces.

Stratigraphic coordinates, a coordinate system tailored to seismic interpretation