next up previous print clean
Next: Seabeam: theory to practice Up: LEVELED INVERSE INTERPOLATION Previous: Test results for leveled

Analysis for leveled inverse interpolation

Here we see how the interpolation beyond aliasing was done. interpolation ! beyond aliasing The first ``statement of wishes'' is that the observational data $\bold d$should result from a linear interpolation $\bold L$ of the uniformly sampled model space $\bold m$; that is, $ \bold 0 \approx \bold L \bold m - \bold d $.Expressing this as a change $\Delta \bold m$ gives the fitting goal in terms of the model change, $\bold 0 \approx\bold L \Delta\bold m+(\bold L \bold m-\bold d)=\bold L \Delta\bold m + \bold r $.The second wish is really an assertion that a good way to find missing parts of a function (the model space) is to solve for the function and its PEF at the same time. We are merging the fitting goal ([*]) for irregularly sampled data with the fitting goal (42) for finding the prediction-error filter.

      \begin{eqnarray}
\bold 0 &\approx& \bold r_d \eq
\bold L \Delta \bold m + (\bold...
 ...ld K \Delta \bold a +
(\bold A\bold m \ {\rm or}\
 \bold M\bold a)\end{eqnarray} (48)
(49)
Writing this out in full for 3 data points and 6 model values on a uniform mesh and a PEF of 3 terms, we have  
 \begin{displaymath}
\left[ 
\begin{array}
{cccccc\vert ccc}
 .8 & .2 & . & . & ....
 ...r_{m6} \\  r_{m7}
 \end{array} \right] 
\quad \approx \ \bold 0\end{displaymath} (50)
where rm is the convolution of the filter at and the model mt, where rd is the data misfit $ \bold r = \bold L\bold m - \bold d $,and where $\bold K$ was defined in equation (11). fitting goal ! nonlinear

Before you begin to use this nonlinear fitting goal, you need some starting guesses for $\bold m$ and $\bold a$.The guess $\bold m = 0$ is satisfactory (as explained later). For the first guess of the filter, I suggest you load it up with $\bold a = (1,-2,1)$ as I did for the examples here.


next up previous print clean
Next: Seabeam: theory to practice Up: LEVELED INVERSE INTERPOLATION Previous: Test results for leveled
Stanford Exploration Project
4/27/2004