\author{Gustav Kirchhoff}%%%%%%%%%%%%%%%%%%%%%%\title{Homework 5}\begin{abstract}In this lab, you will derive analytical expressions for geometricalspreading in layered media and experiment with creating angle gathersby wave equation migration.\end{abstract}\definecolor{frame}{rgb}{0.905,0.905,1.0}\lstset{language=Python,backgroundcolor=\color{frame},showstringspaces=false}\section{Theoretical part}In a layered isotropic medium with a $V(z)$ velocity distribution, thetraveltime and the offset of the reflected wave are given by\begin{eqnarray}  \label{eq:t}  t(p) & = & 2\,\int\limits_{0}^{H} \frac{d z}{V(z)\,\sqrt{1-p^2\,V^2(z)}} \\  \label{eq:x}  x(p) & = & 2\,\int\limits_{0}^{H} \frac{p\,V(z)\,d z}{\sqrt{1-p^2\,V^2(z)}}\;,\end{eqnarray}where $H$ is the reflector depth, and $p$ is the horizontal componentof the slowness vector in the reflection plane $p = \partialt/\partial x$. \begin{enumerate}\item Prove that the geometrical amplitude of the reflection  wave in the 3-D case is\begin{equation}  A(p) = \frac{R(p)\,V(0)\,\sqrt{V(0)\,\rho(0)}}{2\,\sqrt{1-p^2\,V^2(z)}}\,%    \left[\int\limits_{0}^{H} \frac{V(z)\,d z}{\sqrt{1-p^2\,V^2(z)}}\,      \int\limits_{0}^{H} \frac{V(z)\,d z}{\left(1-p^2\,V^2(z)\right)^{3/2}}    \right]^{-1/2}\;,\label{eq:a3}\end{equation}where $R(p)$ is the reflection coefficient.\textbf{Hint:} Rewrite equation~(\ref{eq:x}) as\begin{equation}  \label{eq:x3}  \mathbf{x}(\mathbf{p}) = 2\,\int\limits_{0}^{H} \frac{\mathbf{p}\,V(z)\,d z}{\sqrt{1-\mathbf{p} \cdot \mathbf{p}\,V^2(z)}}\;,\end{equation}where vectors $\mathbf{x}$ and $\mathbf{p}$ have two horizontalcomponents, and evaluate the result at $\mathbf{p}=\{p,0\}$. \item Derive the corresponding expression for the amplitude in the 2-D case.\begin{equation}  \label{eq:a2}  A(p) =\end{equation}\end{enumerate}\section{Computational part}\inputdir{hyper}Figure~\ref{fig:hyper,image} shows the model of a reflector and animage generated by wave-equation imaging of three shot gathersdepicted in Figure~\ref{fig:shots}. The shot gathers are generated byKirchhoff modeling.\multiplot{2}{hyper,image}{width=0.45\textwidth}{Reflector model (a)  and a wave-equation image (b) using three synthetic shot gathers.}\plot{shots}{width=0.9\textwidth}{Synthetic shot gathers generated by  Kirchhoff integral modeling. Analytical reflection traveltime curves  are overlaid on top of the reflections.}\begin{enumerate}\item Start by running\begin{verbatim}> cd ~/geo391> svn update\end{verbatim}    \item Change directory \begin{verbatim}> cd ~/geo391/hw5/hyper\end{verbatim}\item Run\begin{verbatim}> scons view\end{verbatim}to generate figures and display them on your screen.  \itemFigure~\ref{fig:cigs} shows three angle-domain common image gathersfor three image locations extracted from wave-equation migration.Explain the appearance of angle gathers.  Why does each shot gathergenerates only one point on an angle gather? You can use a sketch ofincident and reflection rays in the explanation.\plot{cigs}{width=0.9\textwidth}{Angle-domain common image gathers  for three image locations extracted from wave-equation migration.}\item Edit the \texttt{SConstruct} file to change the plane reflector  to a hyperbolic reflector (uncomment the line with \texttt{b}  parameter). Repeat the computations.\begin{enumerate}\item You will need to change the analytical traveltime formula in  \texttt{SConstruct} to match the new reflection traveltimes in  Figure~\ref{fig:shots}.  \textbf{Hint:} Use equations from Homework  3.\item Explain the change in the appearance of angle gathers. You can  use a sketch of incident and reflection rays in the explanation. \item Modify the \texttt{SConstruct} file to move the shot positions  at the surface so that the image point at $-0.5$~km is illuminated at  a reflection angle $\theta$ such that $\tan{\theta}=0.5$.\end{enumerate}\item After you are done, run\begin{verbatim}> scons lock > scons -c\end{verbatim}\item Edit the file  \verb#~/geo391/hw5/paper.tex# in your favorite editor and change thefirst line to have your name instead of Kirchhoff's. Run\begin{verbatim}> scons pdf\end{verbatim}and submit your result (file \texttt{paper.pdf}) on paper or bye-mail.\end{enumerate}{\tiny\lstinputlisting[frame=single]{hyper/SConstruct}}