\chapter{Problem in linearizing the image norm}
\label{appendix:lin}
The linear objective function \eqref{eq:lwivu-zeroh} has a serious drawback in the second functional. The key point is that first performing the linearization of $\mathbf{I}(\mathbf{b})$ and plugging the expression into the norm $||\cdot||_2^2$, is not the same as the linearization of $||\mathbf{I}(\mathbf{b})||_2^2$. In fact, doing the former completely changes the properties of $||\mathbf{I}(\mathbf{b})||_2^2$.

Let us include another term in the expansion of the migration image

\begin{equation}
    \mathbf{I}(\mathbf{b}_0 + \mathbf{\Delta b}) = \mathbf{I}(\mathbf{b}_0) + \mathbf{W}(\mathbf{b}_0)\mathbf{\Delta b} + \mathbf{Z}(\mathbf{b}_0, \mathbf{\Delta b}) + O(||\mathbf{\Delta b}||^3),
\label{eq:linwemva_ext}
\end{equation}

where $\mathbf{Z}(\mathbf{b}_0, \mathbf{\Delta b})$ incorporates all the second order terms in the expansion. Plugging \eqref{eq:linwemva_ext} into the function $||\mathbf{I}(\mathbf{b}_0 + \mathbf{\Delta b})||_2^2$ gives

\begin{equation}
\begin{split}
    ||\mathbf{I}(\mathbf{b}_0 + \mathbf{\Delta b})||_2^2 &=
    \mathbf{I}(\mathbf{b}_0)^\mathbf{T}\mathbf{I}(\mathbf{b}_0) + 2 \mathbf{\Delta b}^{\mathbf{T}} \mathbf{W}(\mathbf{b}_0)^{\mathbf{T}}\mathbf{I}(\mathbf{b}_0) \\
    & + 2 \mathbf{Z}(\mathbf{b}_0, \mathbf{\Delta b})^\mathbf{T}\mathbf{I}(\mathbf{b}_0) + \mathbf{\Delta b}^{\mathbf{T}} \mathbf{W}(\mathbf{b}_0)^{\mathbf{T}} \mathbf{W}(\mathbf{b}_0) \mathbf{\Delta b} + O(||\mathbf{\Delta b}||^3) \\
    &= \left \Vert \mathbf{I}(\mathbf{b}_0) + \mathbf{W}(\mathbf{b}_0)\mathbf{\Delta b} \right \Vert_2^2 + 2 \mathbf{Z}(\mathbf{b}_0, \mathbf{\Delta b})^\mathbf{T}\mathbf{I}(\mathbf{b}_0) + O(||\mathbf{\Delta b}||^3).
\end{split}
\label{eq:expan1}
\end{equation}

Notice the additional term $2 \mathbf{Z}(\mathbf{b}_0, \mathbf{\Delta b})^\mathbf{T}\mathbf{I}(\mathbf{b}_0)$, which is second order and is not accounted for by the second term in the objective function \eqref{eq:lwivu-zeroh}. Therefore, the approximation of such term of is not order accurate, which is problematic in this case as the curvature of the function is flipped near $\mathbf{\Delta b}^{\text{true}}$, being no longer concave. In fact, the approximation $\left \Vert \mathbf{I}(\mathbf{b}_0) + \mathbf{W}(\mathbf{b}_0)\mathbf{\Delta b} \right \Vert_2^2$ is actually a convex function with a constant Hessian that is at least positive semidefinite. As consequence, there is no upper bound or maximum close to the correct background model perturbation, as we observe in the numerical results in Chapter 2 (see Figure \ref{fig:Ch2/Flat-test-wemva-alpha,Ch2/SigsA-test-wemva-alpha}). This implies that one can move along any arbitrary direction with strictly positive curvature, and increase the objective function $\left \Vert \mathbf{I}(\mathbf{b}_0) + \mathbf{W}(\mathbf{b}_0)\mathbf{\Delta b} \right \Vert_2^2$ indefinitely, which is clearly not what is desired.