Analysis for leveled inverse interpolation

Here we see how the interpolation beyond aliasing was done.

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 (36) for finding the prediction-error filter.

$$\begin{eqnarray} \bold 0 &\approx& \bold L \Delta \bold m + (\bold L \bold m - \... ...ld K \Delta \bold a + (\bold A\bold m \ {\rm or}\ \bold M\bold a)\end{eqnarray}$$ (42) (43)

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  

$$\left[ \begin{array} {rrrrrr\vert rrr} .8 & .2 & . & . & .... ... \\ r_{m7} \\ 0 \end{array} \right] \quad \approx \ \bold 0$$ (44)

where r_m is the convolution of the filter a_t and the model m_t, where r_d is the data misfit $\bold r = \bold L\bold m - \bold d$,and where $\bold K$ was defined in equation (11).

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.