\chapter{Preconditioning in WEMVA}
\label{appendix:prec}

Inversion using WEMVA is intended to solely recover the low-wavenumber component of the subsurface. However, the high-wavenumber component, in other words, the reflectivity, often manifests its presence in the solution. The reason for that is because such a high-wavenumber component belongs to the model null space of the WEMVA operator. A similar problem arises when DSO is utilized in the extended domain and high-wavenumber artifacts are present in the gradient, thus affecting the final inverted model \citep{Vyas2010}. One popular way to combat such artifacts is preconditioning the gradient with B-splines \citep{Weibull2013}. On the other hand, the migration image in the WEMVA objective function might also required preconditioning to appropriately weight the amplitudes of the shallow part with respect to the deep part. In the following I show both preconditioning strategies (for the background model and for the image) that I applied in the numerical examples that illustrate this thesis.

\section{Preconditioning the background model}

Recall the WEMVA objective function for maximization of the image power,

\begin{equation*}
\Phi({\bf b})=-\frac{1}{2} \left\Vert {\bf I}({\bf b}) \right\Vert_2^2.
\end{equation*}

We can define ${\bf b}={\bf Bp}$, where ${\bf B}$ represents the B-spline interpolation operator \citep{Fomel.sep.105.sergey2, Tang.sepphd.144, Weibull2013}, and ${\bf p}$ represents the projection of the background model onto a coarse grid. Operator ${\bf B}$ maps the subsurface model from a coarse grid onto a fine grid.

Substituting in the objective function gives

\begin{equation}
\Phi({\bf p})=-\frac{1}{2} \left\Vert {\bf I}({\bf Bp}) \right\Vert_2^2. \label{eq:wemva-prec}
\end{equation}

This objective function is optimized with respect to the subsurface model projected onto the coarse grid, ${\bf p}$. Then, it is interpolated into the fine grid using operator ${\bf B}$. This procedure removes high-wavenumber components from the background model whilst retaining the expected low-wavenumber components. We can obtain the initial model ${\bf p}_0$ from ${\bf b}_0$ by iteratively solving the following normal equations:

\begin{equation*}
{\bf B}^{\text{T}}{\bf B} {\bf p}_0={\bf B}^{\text{T}} {\bf b}_0
\end{equation*}

\section{Preconditioning the image}

It is often the case that the migration image exhibits imbalanced amplitudes, e.g., high amplitudes the shallow part that decrease as going deeper. We can include a weighting factor that compensates for such an effect, thereby obtaining

\begin{equation}
\Phi({\bf p})=-\frac{1}{2} \left\Vert {\bf E}{\bf I}({\bf Bp}) \right\Vert_2^2. \label{eq:wemva-prec-w}
\end{equation}

where

\begin{equation*}
{\bf E}=\text{diag}({\bf z}^{\text{n}}),
\end{equation*}

and ${\bf z}$ is the vector that contains the depth steps, and $\text{n}$ is an integer positive value. In other words, the diagonal matrix operator ${\bf E}$ applies a depth-dependent amplitude compensation to the image.

\section{Gradient of preconditioned objective function}

We can calculate the gradient of the preconditioned objective function and obtain

\begin{equation}
\begin{aligned}
\nabla \Phi({\bf p}) &= -\bigg[ \frac{\partial {\bf E}{\bf I}({\bf Bp})}{\partial {\bf p}}  \bigg]^{\text{T}}{\bf E}{\bf I}({\bf Bp})
= - \bigg[ \frac{\partial {\bf I}({\bf Bp})} {\partial {\bf b}}  \frac{\partial {\bf b}}{\partial {\bf p}} \bigg]^{\text{T}} {\bf E}^{\text{T}} {\bf E}{\bf I}({\bf Bp}) \\ \\
&= -\big[ {\bf W}({\bf Bp}) {\bf B} \big]^{\text{T}} {\bf E}^{\text{T}} {\bf E}{\bf I}({\bf Bp})
= -{\bf B}^{\text{T}} {\bf W}({\bf Bp})^{\text{T}} {\bf E}^{\text{T}} {\bf E}{\bf I}({\bf Bp}).
\label{eq:grad-wemva-prec-w} 
\end{aligned}
\end{equation}

Note that the minus sign of the objective function ought to be preserved in the gradient.