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),

where is the wavefield and is the local slope. In the case of a constant slope, equation 1 has the simple general solution

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:

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

2019-05-06