Meaning of B'B

A matrix operation like $\bold B' \bold B$arises whenever we travel from one space to another and back again. The inverse of this matrix arises when we ask to return from the other space with no approximations. In general, $\bold B' \bold B$ can be complicated beyond comprehension, but we have seen some recurring features. In some cases this matrix turned out to be a diagonal matrix which is a scaling function in the physical domain. With banded matrices, the $\bold B' \bold B$matrix is also a banded matrix, being tridiagonal for $\bold B$ operators of both (22) and (21). The banded matrix for the derivative operator (22) can be thought of as the frequency domain weighting factor $\omega^2$.We did not examine $\bold B' \bold B$ for the filter operator, but if you do, you will see that the rows (and the columns) of $\bold B' \bold B$ are the autocorrelation of the filter. A filter in the time domain is simply a weighting function in the frequency domain.

The tridiagonal banded matrix for linearly-interpolated NMO is somewhat more complicated to understand, but it somehow represents the smoothing inherent to the composite process of NMO followed by adjoint NMO, so although we may not fully understand it, we can think of it as some multiplication in the spectral domain as well as some rescaling in the physical domain. Since $\bold B' \bold B$clusters on the main diagonal, it never has a ``time-shift'' behavior.