    Structure-constrained relative acoustic impedance using stratigraphic coordinates  Next: example Up: Method Previous: Method

## 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: (3)

where, is a seismic section , is the destruction residual, and , the non-stationary plane-wave destruction operator, is (4)

where is the identity operator and is the prediction of trace from trace 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 from a distant reference trace is , where (5)

This is a simple recursion, and 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 . 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, and , orthogonal to the first axis, , by numerically solving the following gradient equations: (6)

and (7)

Equations 6 and 7 simply state that the and axes should be perpendicular to . The boundary condition for the first gradient equation (equation 6) is (8)

and the boundary condition for equation 7 is (9)

These two boundary conditions mean that the stratigraphic coordinate system and the regular coordinate system meet at the surface . 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 (10)    Structure-constrained relative acoustic impedance using stratigraphic coordinates  Next: example Up: Method Previous: Method

2015-05-06