Inversion of the Hessian

Using the results above, the least-squares estimate of ${\bf m}$ in equation () is derived. Assuming that ${\bf C_d}=\sigma_d^2{\bf I}$, the fitting goal is

$${\bf 0} \approx {\bf r_d} = {\bf Lm}-{\bf d},$$ (97)

with ${\bf L}=({\bf L_s} \;\; {\bf L_n})$ and ${\bf m'}=({\bf m_s} \;\; {\bf m_n})$.The normal equations are given by  

$$\left( \begin{array} {cc} {\bf L_s'L_s} & {\bf L_s'L_n} \\... ...gin{array} {c} {\bf L_s'd} \\ {\bf L_n'd}\end{array}\right),$$ (98)

where ${\bf m_s}$ and ${\bf m_n}$ are the unknowns. The least-square estimate $\hat{{\bf m_s}}$ of ${\bf m_s}$ can be derived from the bottom row of equation (). The least-square estimate $\hat{{\bf m_n}}$ of ${\bf m_n}$ can be derived from the top row of equation (). We have, then,

$$\begin{eqnarray} \hat{{\bf m_s}}&=&({\bf L_s'L_s}-{\bf L_s'L_n}({\bf L_n'L_n})^{... ... {\bf L_s'L_n})^{-1}{\bf L_n'L_s}({\bf L_s'L_s})^{-1}{\bf L_s'd},\end{eqnarray}$$ (99) (100)

which can be simplified as follows:

$$\begin{eqnarray} \hat{{\bf m_s}}&=&({\bf L_s'}({\bf I}-{\bf L_n}({\bf L_n'L_n})... ...bf L_n'}({\bf I}-{\bf L_s}({\bf L_s'L_s})^{-1}{\bf L_s'}){\bf d}.\end{eqnarray}$$ (101) (102)

${\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}$ is the coherent noise resolution matrix, whereas ${\bf L_s}({\bf L_s'L_s})^{-1}{\bf L_s'}$ is the signal resolution matrix Tarantola (1987). Denoting $\overline{{\bf R_s}} = {\bf I}-{\bf L_s}({\bf L_s'L_s})^{-1} {\bf L_s'}$ and $\overline{{\bf R_n}} = {\bf I}-{\bf L_n}({\bf L_n'L_n})^{-1} {\bf L_n'}$ yields the following simplified expression for $\hat{{\bf m_s}}$ and $\hat{{\bf m_n}}$: 

$$\left( \begin{array} {c} \hat{{\bf m_s}} \\ \hat{{\bf m_n... ..._s}L_n})^{-1}{\bf L_n'\overline{R_s}}\end{array}\right){\bf d}.$$ (103)

By property of the resolution operators, ${\bf \overline{R_n}}$ and ${\bf \overline{R_s}}$ perform noise and signal filtering, i.e.,

$$\begin{array} {ccc} {\bf \overline{R_n}d}&=&{\bf \overline... ... {\bf \overline{R_s}d}&=&{\bf \overline{R_s} n}, \end{array}$$ (104)

if the noise and signal are well predicted by the noise and signal modeling operators. Nemeth (1996) demonstrates that the inverse of the Hessian in equation () is well conditioned if the noise and signal operators are orthogonal, meaning that they predict distinct parts of the model space without overlapping. If overlapping occurs, a model regularization term can improve the signal/noise separation.