Stratigraphic coordinates, a coordinate system tailored to seismic interpretation

Theory

In order to define the first step for transformation to stratigraphic coordinates, we follow the predictive-painting algorithm (Fomel, 2010), which is reviewed in appendix A. Predictive painting spreads the time values along a reference trace into the seismic volume to output the relative geologic age attribute $\left(Z_0(x,y,z)\right)$. The painted horizons output by predictive painting are used as the first axis of our stratigraphic coordinate system. Several alternative methods exist to track horizons automatically in a seismic volume and produce horizon cubes (Hoyes and Cheret, 2011; Wolak et al., 2013). We choose predictive painting because of its simplicity and efficiency.

In the next step, we 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:

$$\mathbf{\nabla} Z_0 \cdot \mathbf{\nabla }X_0 = 0$$ (1)

and

$$\mathbf{\nabla} Z_0 \cdot \mathbf{\nabla} Y_0 = 0.$$ (2)

Equations 1 and 2 simply state that the $X_0$ and $Y_0$ axes should be perpendicular to $Z_0$. We can define the boundary condition for the first gradient equation (equation 1) as

$$X_0\left(x,y,0\right)=x$$ (3)

and the boundary condition for equation 2 as

$$Y_0\left(x,y,0\right)=y.$$ (4)

These two boundary conditions mean that the stratigraphic coordinate system and the regular coordinate system $\left(x,y,z\right)$ become equivalent at the surface $\left(z = 0\right)$.

The stratigraphic coordinates are originally designed for depth images. When applied to time-domain images, the definition of the gradient operator becomes

$$\mathbf{\nabla} = \left(\frac{\partial}{\partial x},\frac{\p... ...rac{\partial}{\partial z}\frac{\partial z}{\partial t}\right),$$ (5)

so a scaling factor with dimensions of velocity-squared is needed in equations 1 and 2.

Stratigraphic coordinates, a coordinate system tailored to seismic interpretation