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Each offset $x_j$ allows us to make a different estimate of the earth model $\mathbf z_j$. There are two possiblities:

i.e. $\mathbf z_j \ne \mathbf z_{j+1}$.

\hat {\mathbf z}_j \quad =\quad \mathbf H_j' \mathbf L' \mathbf d_j
\end{displaymath} (3)

\hat {\overline{\mathbf z}}_j \quad =\quad \overline{\mathbf H}_j' \mathbf L' \mathbf d_j
\end{displaymath} (4)

We should plot $\hat{\mathbf z}_j$ as a function of $j$. We should also plot $\hat {\mathbf z}_j - \hat {\mathbf z}_{j-1}$ as a function of $j$ and see if we can find any higher temporal frequencies.