METHOD: INVERSE LINEAR INTERPOLATION WITH KNOWN FILTER

The theory of inverse linear interpolation with a known filter is given in Applications of Three-Dimensional Filtering Claerbout (1994), section 2.6, and can be easily extended to the 2-D case. The algorithm is based on the following procedure. To invert the data vector ${\bf d}$ given on an irregular grid for a regularly sampled model ${\bf m}$, we run the conjugate-gradient solver on the system of equations  

$${\bf d \approx Lm\;, }$$ (1)

 

$${\bf 0 \approx \epsilon \, Am\;. }$$ (2)

Equation (1) formulates the basic assumption of the method, stating that the data is related to the model with a linear interpolation operator $\bold{L}$. The next equation (2) is required to constrain an underdetermined part of the inverse problem. Minimizing the output power of the model filtered by some roughening filter $\bold{A}$ is a way to smooth the model components that are not determined by equation (1). Laplacian filter of the form  

$${\bf A}= \begin{array} {\vert c\vert c\vert c\vert} \hline .... .... \ \hline 1 & -4 & 1 \ \hline . & 1 & . \ \hline\end{array}$$ (3)

is a conventional choice for smoothing in two dimensions.