Predictive painting spreads information from a seed trace to its neighbors recursively by following the local dip (Fomel, 2010).
The spreading or “painting” process can be implemented using plane-wave construction filter (Fomel and Guitton, 2006).
The mathematical basis of this filter is a differential equation for local plane waves (Claerbout, 1992),
![$\displaystyle \frac{\partial P}{\partial x} + \sigma \frac{\partial P}{\partial t} =0 \;,$](img2.png) |
(1) |
where
is the wavefield and
is the local slope.
In the case of a constant slope, equation 1 has the simple general solution
![$\displaystyle P(t,x)=f(t-\sigma x) \; ,$](img5.png) |
(2) |
where
is an arbitrary waveform.
Equation 2 is nothing more than a mathematical description of a plane wave.
Assuming that the slope
varies in time and space, we can design a local operator to propagate trace
to trace
, and describe such prediction as
.
If
is a reference trace, spreading its information to a distant neighbor
(for example
) can be accomplished by simple recursion:
![$\displaystyle \mathbf{s}_k = \mathbf{A}_{k-1,k} \cdots \mathbf{A}_{r+1,r+2} \mathbf{A}_{r,r+1} \mathbf{s}_r \; .$](img14.png) |
(3) |
The recursive operator in Equation 3 is referred to as predictive painting (Fomel, 2010).
2019-05-06