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Stratigraphic coordinates

In order to define the first step for transformation to stratigraphic coordinates, I follow the predictive-painting algorithm (Fomel, 2010), which is based on the method of plane-wave destruction for measuring local slopes of seismic events (Fomel, 2002; Claerbout, 1992). By writing plane-wave destruction in the linear operator notation as:

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

where, $\mathbf{s}$ is a seismic section $\left(\mathbf{s} =\left[\mathbf{s}_1 \; \mathbf{s}_2 \; \ldots \;\mathbf{s}_N\right]^T\right)$, $\mathbf{r}$ is the destruction residual, and $\mathbf{D}$, the non-stationary plane-wave destruction operator, is
\mathbf{D} =
\mathbf{I} & 0...
...& - \mathbf{P}_{N-1,N} & \mathbf{I} \\
\end{displaymath} (4)

where $\mathbf{I}$ is the identity operator and $\mathbf{P}_{i,j}$ is the prediction of trace $j$ from trace $i$ determined by being shifted along the local slopes of seismic events. Local slopes are estimated by minimizing the destruction residual using a regularized least-squares optimization. The prediction of trace $\mathbf{s}_k$ from a distant reference trace $\mathbf{s}_r$ is $\mathbf{P}_{r,k}\,\mathbf{s}_r$, where
\mathbf{P}_{r,k} = \mathbf{P}_{k-1,k}\, \cdots\, \mathbf{P}_{r+1,r+2}\, \mathbf{P}_{r,r+1}\;.
\end{displaymath} (5)

This is a simple recursion, and $\mathbf{P}_{r,k}$ is called the predictive painting operator (Fomel, 2010). Predictive painting spreads the time values along a reference trace in order to output the relative geologic age attribute $\left(Z_0(x,y,z)\right)$. These painted horizons outputted by predictive painting are used as the first axis of the stratigraphic coordinate system. In the next step, following Karimi and Fomel (2014,2011), I find the two other axes, $X_0\left(x,y,z\right)$ and $Y_0\left(x,y,z\right)$, orthogonal to the first axis, $Z_0\left(x,y,z\right)$, by numerically solving the following gradient equations:
\mathbf{\nabla} Z_0 \cdot \mathbf{\nabla }X_0 = 0
\end{displaymath} (6)

\mathbf{\nabla} Z_0 \cdot \mathbf{\nabla} Y_0 = 0.
\end{displaymath} (7)

Equations 6 and 7 simply state that the $X_0$ and $Y_0$ axes should be perpendicular to $Z_0$. The boundary condition for the first gradient equation (equation 6) is
\end{displaymath} (8)

and the boundary condition for equation 7 is
\end{displaymath} (9)

These two boundary conditions mean that the stratigraphic coordinate system and the regular coordinate system $\left(x,y,z\right)$ meet at the surface $\left(z = 0\right)$. Note that the stratigraphic coordinates are designed for depth images (Mallet, 2014,2004). When applied to time-domain images, a scaling factor with dimensions of velocity-squared is added to equations 6 and 7, because the definition of the gradient operator becomes
\mathbf{\nabla} = \left(\frac{\partial}{\partial x},\frac{\p...
...rac{\partial}{\partial z}\frac{\partial z}{\partial t}\right).
\end{displaymath} (10)

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