Brief review of predictive painting

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),

$\displaystyle \frac{\partial P}{\partial x} + \sigma \frac{\partial P}{\partial t} =0 \;,$ (1)

where $P(t,x)$ is the wavefield and $\sigma$ is the local slope. In the case of a constant slope, equation 1 has the simple general solution

$\displaystyle P(t,x)=f(t-\sigma x) \; ,$ (2)

where $f(t)$ is an arbitrary waveform. Equation 2 is nothing more than a mathematical description of a plane wave. Assuming that the slope $\sigma(t,x)$ varies in time and space, we can design a local operator to propagate trace $\mathbf{s}_i$ to trace $\mathbf{s}_j$, and describe such prediction as $\mathbf{A}_{i,j}$. If $\mathbf{s}_r$ is a reference trace, spreading its information to a distant neighbor $\mathbf{s}_k$ (for example $k>r$) can be accomplished by simple recursion:

$\displaystyle \mathbf{s}_k = \mathbf{A}_{k-1,k} \cdots \mathbf{A}_{r+1,r+2} \mathbf{A}_{r,r+1} \mathbf{s}_r \; .$ (3)

The recursive operator in Equation 3 is referred to as predictive painting (Fomel, 2010).