What should we optimize?

Least-squares problems often present themselves as fitting goals such as

$$\begin{eqnarray} \bold 0 &\approx& \bold F \bold m - \bold d\\ \bold 0 &\approx& \bold m\end{eqnarray}$$ (58) (59)

To balance our possibly contradictory goals we need weighting functions. The quadratic form that we should minimize is

$$\min_m \quad (\bold F \bold m - \bold d)' \bold A'_n \bold A... ... F \bold m - \bold d) + \bold m' \bold A'_m \bold A_m \bold m$$ (60)

where $\bold A'_n \bold A_n$ is the inverse multivariate spectrum of the noise (data-space residuals) and $\bold A'_m \bold A_m$ is the inverse multivariate spectrum of the model. In other words, $\bold A_n$ is a leveler on the data fitting error and $\bold A_m$ is a leveler on the model. There is a curious unresolved issue: What is the most suitable constant scaling ratio of $\bold A_n$ to $\bold A_m$?