ROW NORMALIZATION, POSITIVE OPERATORS

Perhaps most of the operators in Geophysics are of the special type whose associated matrix contains no negative elements. Examples are stacking, migration, velocity analysis, DMO, tomography, integration, and linear interpolation. For such positive operators $\bold F$it can be informative to apply an input vector that has components that are all ones. Denote such a vector by $\bold 1$.The vector $\bold F \bold 1$ tells us the sum of the elements on any row of the associated matrix. We can define a diagonal matrix by spreading the vector $\bold F \bold 1$ on the diagonal of a matrix. Denote this by ${\bf diag}( \bold F \bold 1)$.We can smooth this diagonal and add a small threshhold $\epsilon \gt$so as to avoid any problem with inversion. Such a diagonal scaling operator might be denoted as $\bold D={\bf diag}( <\bold F \bold 1\gt +\epsilon )$.In various applications we might find it more useful to work with the row normalized operator $\bold D ^{-1}\bold F$than with the operator $\bold F$ itself.