Appendix B: Predictive Painting

We adopt predictive painting to interpolate well log data along seismic structure. Predictive painting is defined using plane-wave destruction filters that measure the local slopes of seismic events (Fomel, 2002). The plane-wave destruction operator can be written in linear operator notation

$$\boldmath {r} = \boldmath {D}\boldmath {s}$$ (22)

where s is a group of seismic traces from a seismic image ( $\textbf{s} = [\textbf{s}_1 \textbf{s}_2 ... \textbf{s}_N]^T$), r is the destruction residual, and $\textbf{\textit{D}}$, the destruction operator, is defined as

$$\boldmath {D} = \begin{bmatrix} \boldmath {I} & 0 & 0 & \dots & 0... ... & \ddots & \vdots \\ 0 & 0 & \dots & -P_{N-1,N} & \boldmath {I} \end{bmatrix}$$ (23)

where $I$ is the identity operator and $P_{i,j}$ describes the prediction of trace j from trace i by shifting along the local slope of the seismic data. Slopes can be estimated in this way by minimizing the prediction residual operator r using regularized least-squares optimization. The prediction of one trace from another trace (Fomel, 2010) can be defined as

$$\boldmath {s_k} = \boldmath {P_{r,k}}\boldmath {s_r}$$ (24)

where $s_k$ is the unknown trace and $s_r$ is the reference trace. The predictive painting operator is defined as:

$$\boldmath {P_{1,k}} = \boldmath {P_{k-1,k}}\dots\boldmath {P_{2,3}}\boldmath {P_{1,2}}$$ (25)

Predictive painting spreads information along local seismic structures to generate volumes of well log data from a single well log reference trace providing a method to predict an expected log profile in a location with no well log data.