Data fitting goal

In this paper I do not use any regularization to solve the linear system in equation 3, since the objective of the study was to find out how well the Hessian operator could fit the different kind of data. Having regularization could obscure the results. However, in areas of poor illumination, this problem will have a large null space. The null space is partially caused by the fact that our survey can not have infinite extents and infinitely dense source and receiver grids. Any noise that exists within the null space can grow with each iteration until the problem becomes unstable.

The inversion was carried out in the subsurface-offset domain. The fitting goal used was

$$\begin{eqnarray} {\bf H}({\bf x, h};{\bf x',h'}) \hat{{\bf m}}({\bf x},{\bf h})-{\bf m}_{mig}({\bf x},{\bf h}) &\approx& 0, \end{eqnarray}$$ (8)

In the next sections I compare the numerical solution of the inversion problems stated in equation 8 in the imaging of the TRIP synthetic dataset.