Estimating the spatial prediction filter

We use subscript k to represent the location of the trace, i.e., $W_{\rm k} = W(f,k\Delta x)$. If W_k is predictable from $(W_{\rm k-1}, W_{\rm k-2}, ..., W_{\rm k-N})$, we have

$$W_{\rm k} = \sum_{j=1}^N C_{\rm j} W_{\rm k-j}, \hspace*{0.2in}(k=2,3,...,N_2)$$ (21)

where N₂ is the length of the spatial axis.

Determination of the coefficients $C_{\rm j}$ can be treated as an optimization problem. Here we choose conjugate gradient code cgplus Claerbout (1994) as the solver. Originally, cgplus was used for real-valued optimization. We extend it to complex-valued optimization. Between the two gridding schemes, the most important difference is that

• Frequency-independent grids
  • Prediction filter is estimated in the rectangular f-x domain.
  • For each frequency slice, we need to solve an optimization problem separately. And the coefficients change from one slice to another.
• Frequency-dependent grids
  • Prediction filter is estimated in the pyramid domain.
  • We incorporate all the equations together and only solve one optimization problem. But the coefficients can be applied to all the frequencies within the pyramid.

Hopefully, this difference will save computing resources in 3-D. Also, we can expect that the result from FDG will more stable.