The inverse problem

In seismic processing we often transform data into equivalent data using linear operators. Among these operators, we have the Fourier transform, the Radon transform, the migration operator etc. Some of these operators are unitary (Fourier transform), meaning that the input data are perfectly recoverable using the adjoint. Mathematically speaking, unitarity implies  

$${\bf H^T H} = {\bf I},$$ (3)

where H is the operator, ${\bf H^T}$ the adjoint, and I the identity matrix. Unfortunately, most of the operators are not unitary, meaning that one can't go back and forth between the model m and the data d without losing information or resolution. Mathematically speaking, non-unitarity implies  

$${\bf H^T H} \neq {\bf I}.$$ (4)

This loss of information can be overcome using inverse theory. The goal of inverse theory is to find a model m that optimally represents the input data d given an operator H and given a definition of optimality (minimum energy residual-l² for example):

$$f({\bf m}) = \Vert{\bf Hm-d}\Vert _2.$$ (5)