Imaging is often derived as the adjoint of modeling, where in the absence of explicit formulation for we seek an approximate inverse for . Mathematically, this means that we approximate an inverse of a matrix of very high order by the transpose (Hilbert adjoint) of . Claerbout 1999 points out that unless has no physical units, the units of the transpose solution do not match those of . Given the theoretical (least squares) solution , Claerbout suggests that the scaling units should be those of . He proposes a diagonal weighting function suggested by Bill Symes (private communication) that makes the image , where the weighting function is
In contrast to the scaled adjoint, the normalized solution is unitless. It therefore avoids the ambiguity of guessing approximate weights. The model represents a ratio of two images where the reference image is the output of an input vector with all components being equal to one. This is equivalent to a calibration by the response of a flat event. Similar approaches might exist in practice, often derived in heuristic ways, e.g., the DMO fold Slawson et al. (1995).