Seismic-wave imaging can be expressed as a least-squares inversion problem,
The solution of the inverse problem is expressed as follows:
Similarly, if we assume Hessian matrix is a unitary matrix, equation (), (), and () all degenerate to
Migration imaging avoids the matrix inversion by replacing the general inverse with a conjugate-transpose operator . The advantage of the processing is to change an ill-posed inverse problem into a well-posed wavefield backpropagation problem, which is quite stable and robust (, , , ). In fact, migration imaging mainly locates the reflector and gives only a qualitative estimate of the reflectivity. is the two-way or one-way propagator, which commonly is expressed in the form of the conjugate Green's function.
In fact, the quantitative estimation of the reflectivity should take advantage of inverse imaging. If we consider the reflectivity imaging as a weighting summation, equation () gives an unsuitable weight function. () discuss in detail about how to choose a suitable weight function.
The inverse of the Hessian matrix is just a deconvolution operator, which modifies the unsuitable weight function of the migration imaging. Therefore, equation () can give more accurate estimate of the reflectivity than can the migration imaging (equation ().
If equation () is rewritten as