Model fitting by least squares

Imaging

The example of dividing a function by itself $(1=F/F)$ might not seem to make much sense, but it is very closely related to estimation often encountered in imaging applications. It's not my purpose here to give a lecture on imaging theory, but here is an over-brief explanation.

Imagine a downgoing wavefield $D(\omega,x,z)$. Propagating against irregularities in the medium $D(\omega,x,z)$ creates by scattering an upgoing wavefield $U(\omega,x,z)$. Given $U$ and $D$, if there is a strong temporal correlation between them at any $(x,z)$ it likely means there is a reflector nearby that is manufacturing $U$ from $D$. This reflectivity could be quantified by $U/D$. At the Earth's surface the surface boundary condition says something like $U=D$ or $U=-D$. Thus at the surface we have something like $F/F$. As we go down in the Earth, the main difference is that $U$ and $D$ get time-shifted in opposite directions, so $U$ and $D$ are similar but for that time difference. Thus, a study of how we handle $F/F$ is worthwhile.

Model fitting by least squares