Normalization of interpolation operators

Operator fold is defined as the impulse response of the operator to an input vector full of ones.

To understand this concept more clearly, it is best to begin with a very simple operator: nearest neighbour interpolation (${\bf A}_{\rm NNI}$) and its adjoint (binning). For this operator, the Hessian matrix, ${\bf A}_{\rm NNI}'\, {\bf A}_{\rm NNI}$, is exactly diagonal, and it would be possible to calculate the diagonal values accurately with any reasonable choice of reference model in equation (). A particularly simple choice of reference model is to fill the model-space with ones. Since the operator is nearest-neighbour interpolation, this results in a data-space full of ones too. Binning this data vector gives the fold of the operator in model-space Claerbout (1998a); and its inverse can be used directly as a weighting function for inverse nearest neighbour interpolation,

$$\begin{eqnarray} {\rm\bf diag} ({\bf A}_{\rm NNI}' \, {\bf 1})^{-1} & \equiv & ... ...\hspace{0.15in} {\bf 1} & = & (1 \;\; 1 \;\; 1\;\; ...\;\; 1)^T. \end{eqnarray}$$ (130)

For an example that is slightly more complex than nearest neighbour interpolation, consider linear interpolation. The Hessian matrix for linear interpolation, ${\bf A}_{\rm LI}'\, {\bf A}_{\rm LI}$ is tridiagonal rather than exactly diagonal; therefore, unless we know the true solution, any diagonal operator that we produce will be an approximation. This leads to a conundrum: is it better to find the diagonal of ${\bf A}_{\rm LI}'\, {\bf A}_{\rm LI}$, or another approximation that incorporates information about the off-diagonal elements?

Following the same approach as above, a vector full of ones seems a reasonable choice for a reference model; and as before, this generates a data vector full of ones. Applying the adjoint of linear interpolation to this data vector produces the model-space fold, ${\bf A}_{\rm LI}' \, {\bf 1}$,that can be used in the approximation,  

$${\rm\bf diag} ({\bf A}_{\rm LI}' \, {\bf 1})^{-1} \approx ({\bf A}_{\rm LI}'\, {\bf A}_{\rm LI})^{-1}.$$ (131)

The fold of the linear interpolation operator is not the diagonal of ${\bf A}_{\rm LI}'\, {\bf A}_{\rm LI}$, but rather the sum along the rows of ${\bf A}_{\rm LI}'$ Chemingui (1999), which for the case of any interpolator is equivalent to the row-sums of ${\bf A}_{\rm LI}'\, {\bf A}_{\rm LI}$ Biondi (1997). The distinction between taking the diagonal and the row-based sum is an important one because equation () is exact if the true model is a constant function, and will be a good approximation if the true model is smooth compared to the size of an impulse response of ${\bf A}_{\rm LI}'\, {\bf A}_{\rm LI}$ (two sample intervals). As long as this smoothness requirement is satisfied, equation () implies that a row-based normalization function is better than the diagonal itself.

Figures  and  strengthens this conclusion. The top panels show a simple regularly-sampled model consisting of a four events. The middle panels show data-points obtained by linear interpolation from this model, and the lower panels show the reestimated models after normalized binning with the adjoint of the linear interpolator. The thin solid-lines, the dashed-lines and the dotted-lines show the true model and the results of normalization by the matrix diagonal and operator fold, respectively.

For Figure , I sampled one hundred data points. Only where the model consists of an isolated single spike (at m₅) does the diagonal normalization out-perform operator fold. Elsewhere the amplitude of the original function is recovered more accurately by operator fold. For Figure , I only sampled 30 data points, leading to a less well-conditioned system than Figure . The row-based fold normalization scheme shows itself to be more robust in areas of poor coverage than the scheme that considers only the diagonal elements of the ${\bf A}_{\rm LI}'\, {\bf A}_{\rm LI}$ matrix.

 

linterp100 Figure 1 Comparison of normalizing adjoint linear interpolation with the row-sum of ${\bf A}' {\bf A}$ as opposed to its diagonal. Top panel shows the model. Center panel shows 100 interpolated data points. Lower panel shows reestimated data: dashed-line after diagonal normalization, dotted-line after column-based normalization. The solid-line shows the true model.

 

linterp30 Figure 2 Same as Figure , but with only 30 data points. The diagonal-based approach (dashed-line) is less robust in areas of poor coverage than the row-sum approach (dotted-line).