3D pyramid interpolation

PEF estimation and Missing data estimation

The missing data estimation algorithm presented here is different from what I presented in the previous paper (Shen, 2008). Missing data are fitted in the $f$-$\bf x$ domain to ensure better fitting of known data. Also, the 3D version of these algorithms use helical coordinates (Claerbout, 1999) to perform the convolution.

For PEF estimation, I try to solve the following problem assuming known pyramid data $\bf {m}$. Denoting convolution with m as operator $\bf {M}$, with ${\bf W}$ being a diagonal masking matrix that is $1$ where pyramid data can be used for PEF estimation and 0 elsewhere, I try to solve for the unknown PEF ${\bf a}$ using the following fitting goal(Claerbout, 1999):

$${\bf WM}{\bf a\approx 0, }$$ (4)

For missing-data estimation, I start with a known PEF $\bf A$, and try to solve the following least-squares problem (Claerbout, 1999):

$$\begin{array}{ll} {\bf K\left(Lm-d\right)\approx 0} \epsilon {\bf WAm}{\bf\approx 0,} \end{array}$$ (5)

where $\bf K$ is a diagonal masking matrix that is $1$ where data is known and 0 elsewhere, $\epsilon$ is a weight coefficient that reflects our confidence in the PEF, and $\bf {W}$ is the same as explained above.

3D pyramid interpolation