\email{gayeni@sep.stanford.edu}
\title{Lab 3: High-frequency imaging}
\author{TA: Gboyega Ayeni}  
\begin{center}
{Due Wednesday, February 3, 5:00 PM}
\end{center}
\begin{center}
{\bf Your Name: Ibrahim Badamasi Babangida} 
\end{center}

\begin{abstract}
In this exercise you will see how wave-propagation effects influence
 imaging in a medium with a complex velocity structure. We analyze the
implications of complex wave propagation on Kirchhoff migration.
\end{abstract}

\section{Introduction}
Kirchhoff methods account for about 99.9\% of commercial 3D prestack
imaging, and a large portion of this class.
As discussed previously, the main reason for this is its ability
to handle irregular (and sparse) 3D geometries, which leads to
practical, cost-effective implementations. 
\par
However, experience in 2-D has shown that wave-equation methods almost
always give preferable results to Kirchhoff methods, since they take
into account wave phenomena such as multiple scattering, frequency
dependent thin layer effects etc., whereas Green's functions for
Kirchhoff migration are usually calculated with infinite frequency
approximations.
\par
By comparing band-limited finite-difference wave propagation with
infinite frequency approximations, this lab illustrates why
wave-equation methods are often better than asymptotic
approximations, and so why the search for practical
wave-equation migration methods for 3D prestack data is a major area
of current research.

\section{Assignment}
Log into {\tt bad}. Edit {\tt paper.tex} to include your answers in this document. 
Type {\tt scons paper.pdf} to compile the Latex, and bring up an electronic version. 
Begin the exercise by changing the name on the assignment to yours.
\par
{\em Some of the questions ask you to mark the Figures in the paper,
so remember to copy your markings onto your final the version of the
paper that you hand in. Make sure you give the reasons for your answers.}

{\em Note that in this exercise, you will use {\tt make} to build your figures, whereas you will use {\tt scons} to build the paper.pdf! 
See your TA if you have any problems using either of the two build systems.}

\section{Finite-difference modeling}
Figure~\ref{fig:locations} shows a 2D section from a complicated velocity model
with a large salt intrusion. Also shown are the locations of four shot points.
Figures~\ref{fig:fdseis1} to~\ref{fig:fdseis4} show the
corresponding common source point gathers (CIP). 
%%
\plot{locations}{width=5.0in}{Shot locations.}
%%
To see the images more clearly on your computer screen, type
{\tt make fdseis*.view}, where {\tt *} is the shot location.
You should also watch the wave-propagation movies by typing  
{\tt make fdsnap*.view}. You can slow down the movie by pressing the {\em s} key
while keeping the cursor inside the xtpen window. You can speed it up
by pressiong the {\em f} key.

\subsection{Finite-difference modeling}
  Figure~\ref{fig:fdsnap1} shows a frame from the first wave-propagation 
  movie. Finite-differencing amounts to solving partial differential
  equations (in this case the wave-equation) by replacing the
  infinitesimal differentials with small but finite-differences.
%%
\plot{fdsnap1}{width=6.0in}{Snapshot of movie with source at location~1.} 
%%\plot{fdsnap2}{width=6.0in}{Snapshot of movie with source at location~2.} 
%%\plot{fdsnap3}{width=6.0in}{Snapshot of movie with source at location~3.} 
\plot{fdsnap4}{width=6.0in}{Snapshot of movie with source at location~4.} 
%%
	  \begin{enumerate} 
	  \item Was the modeling elastic or acoustic? How can you tell?. Be specific.
	  \item Identify the reflections coming from the base of the modeling grid both
	  on Figure~\ref{fig:fdsnap1} and Figure~\ref{fig:fdseis1}. Mark them in the
		figures.
	  \item High frequencies may become dispersive if the mesh is too coarse,
	  because the finite-difference convolution filter does not accurately
	  match the PDE at high spatial wavenumbers. Does this occur when the
	  velocity is too high, or too low? Why? Mark dispersion artifacts
	  on Figure~\ref{fig:fdseis1}.
	  \end{enumerate}
%%
\plot{fdseis1}{width=6.0in}{CIP gather at location~1.}
\plot{fdseis2}{width=6.0in}{CIP gather at location~2.}
\plot{fdseis3}{width=6.0in}{CIP gather at location~3.}
\plot{fdseis4}{width=6.0in}{CIP gather at location~4.}  
%%
\subsection{Wave phenomena}
	  \begin{enumerate}
	  \item In Figure~\ref{fig:fdseis3}, identify the diffraction from the
	  top of the salt, and estimate the velocity in the near surface.
	  \item By measuring propagation angles on Figure~\ref{fig:fdsnap1}, and
	  your answer above, estimate the velocity of the third (high velocity)
	  layer between $1.2-2.3$~km depth.
	  \item Energy can become trapped in low-velocity layers, causing
	  wavefield triplications and energy to arrive significantly later than
	  the first arrival. Find an example of this in movie number~4, and mark
	  the corresponding event on Figure~\ref{fig:fdsnap4}.
	  \end{enumerate}

\subsection{Common scatter point}
	  \begin{enumerate}
	  \item What is the principle of reciprocity?
	  \item Given the gather shown in Figure~\ref{fig:fdseis1} (generated by
	  placing a source at location~1), how would you create the contribution
	  from a scatterer at location~1, to a trace recorded on the surface at
	  $x=2000m$ with a source at $x=4000m$?
	  \end{enumerate}

\subsection{Specular reflections}
	  \begin{enumerate}
	  \item The Kirchhoff migration algorithm sums contributions within a
	  large aperture with many reflection angles.  
	  Why are raypath corresponding to {\em specular}
	  reflections the most important for imaging?
	  \item From the wave-propagation movie with the source at location~1,
	  estimate the surface location at which the zero-offset,
	  normal-incidence specular
	  reflection emerges at the surface. 
	  Mark this on the seismogram (Figure~\ref{fig:fdseis1}).
	  \item Repeat this process for Figures~\ref{fig:fdseis2}
	  through~~\ref{fig:fdseis4}.
	  \item Which specular reflections as defined in question 2
	  are affected by the complex velocity structure at the top of the salt.
	 \end{enumerate} 

\section{Solutions to the eikonal equation}
The eikonal equation is a high-frequency approximation of the
wave-equation, that can be solved rapidly and robustly.
Most of the eikonal equation solvers seek the first arrival. Howevever, 
other methods allow to compute other arrivals which can be later selected
using other criteria, like maximum energy or shortest ray-path. 
\par
We will investigate both alternatives, first analyzing an eikonal 
solver solution, and second the results of ray tracing.

\subsection{Eikonal solvers}
Eikonal solvers are ideal tools for computing traveltimes of first arrivals. 
Because of that, they are often used for recursive ray-based 
applications such as migration velocity analysis, and over structures 
where the first-arrivals contain most of the energy.
\par
Figure~\ref{fig:FME1} shows a contour map of traveltimes overlaying the 
finite-difference snapshot for the same propagation time. 
Other maps can be seen by typing {\tt make FME*.view}, 
where again {\tt *} is the shot point. You can also 
compare the traveltime picks with finite-difference wavefields 
in every common source-point gathers. Do this by typing {\tt make FMpicks*.view}.
%%
\plot{FME1}{width=6.0in}{Eikonal traveltime contours with source at location~1.}
%%\plot{FME2}{width=6.0in}{Eikonal traveltime contours with source at location~2.}
%%\plot{FME3}{width=6.0in}{Eikonal traveltime contours with source at location~3.}
%%\plot{FME4}{width=6.0in}{Eikonal traveltime contours with source at location~4.}
%%
\par
Replace Figure~\ref{fig:FME1} with the one for your favorite shot point, and mark 
your answers on the figure.

	\begin{enumerate}
	\item For each of the four shot points, assess whether the Eikonal
	solver does a good job of modeling the wavefields, especially the
	specular reflections.
	\item What criteria are you using to judge this?
	\item Which reflectors (if any) would you expect to be imaged better
	with a maximum amplitude ray-tracer.
	\end{enumerate}

\subsection{Ray tracing}
Similarly to the eikonal solvers, ray tracing methods provide solutions to 
the eikonal equation. The major difference is that ray tracing has the ability
to track not only the first arrival, but also the later arrivals that might carry
more of the wavefield energy.
\par
Figure~\ref{fig:HWT1} rays and wavefronts corresponding to the shot point no. 1
overlying the corresponding finite-difference snapshot. 
Other maps can be seen by typing {\tt make HWT*.view}, 
where again {\tt *} is the shot point. You can also compare directly the results of
finite-difference modeling, eikonal solution and ray tracing by typing
{\tt make Shot*.view}.
%%
\plot{HWT1}{width=6.0in}{Ray tracing traveltime results with source at location~1.}
%%\plot{HWT2}{width=6.0in}{Ray tracing traveltime results with source at location~2.}
%%\plot{HWT3}{width=6.0in}{Ray tracing traveltime results with source at location~3.}
%%\plot{HWT4}{width=6.0in}{Ray tracing traveltime results with source at location~4.}
%%
\par
Replace Figure~\ref{fig:HWT1} with the one for your favorite shot point, and mark 
your answers on the figure.

	\begin{enumerate}
	\item For each of the four shot points, assess whether ray tracing
	does a good job of modeling the wavefields, especially the
	specular reflections.
	\item What criteria are you using to judge this?
	\item Which events are better described by ray tracing? Why?
	\item Which events are better described by the eikonal solver? Why?
	\end{enumerate}

\subsection{Traveltime comparison}
Kirchhoff migration requires traveltime tables from which we can extract the total 
traveltime from the shot to the receiver for every point in the subsurface. Figures~
\ref{fig:3dfmesalt} and \ref{fig:3dhwtsalt} show two such tables for a shot point at
$6000$~m in the inline direction and $8000$~m in the crossline direction. One of the
figures correspond to an eikonal solver, while the other one corresponds to shortest 
ray-path ray tracing traveltimes. The velocity model is the SEG-EAGE salt model. You 
can take a look at the velocity model by typing {\tt make velosalt.H} and 
{\tt Grey < velosalt.H eout=1 bias=2000 | Ricksep}.

	\begin{enumerate}
	\item Which of the two images correspond to the eikonal solver and which 
	corresponds to ray tracing? Why? Change the caption of the figures.
	\item What criteria are you using to judge this?
	\item Which traveltimes would give better migration results? Why?
	\item Which tables would you use in migration? Explain.
	\item Experiment by changing the position of the source and of the slices that are 
	displayed in the image. Keep the image that best supports your answers to the
	preceding questions.
	\end{enumerate}
\plot{3dfmesalt}{width=5.0in}{Traveltime table.}
\plot{3dhwtsalt}{width=5.0in}{Traveltime table.}

\section{Kirchhoff migration}
Figure~\ref{fig:l7d_real_2dmig_3950_krc_fme} shows the result of 2-D Kirchhoff migration of a dataset given to SEP by ELF Aquitaine that closely resembles the synthetic used earlier.

	\begin{enumerate}
	\item Identify some of the artifacts on the section. Are they Kirchhoff migration
	artifacts or traveltime table artifacts. How can you tell?
	\item Do you like the image? Why? Explain using specific examples.
	\end{enumerate}
\plot{l7d_real_2dmig_3950_krc_fme}{width=6.0in}{2-D Kirchhoff migration}

\section{Optional questions - Extra credit}
Suggestions for those who are interested in imaging in complex media: 
\begin{enumerate}
\item How can you improve on ELF's migration results?
\item Ray-tracing methods often become unstable if velocity fields
contain sharp boundaries.  Therefore velocities are often heavily
smoothed before rays can be traced. How does smoothing affect the
finite-difference wave-fields? 
\item The common source point gather can be thought of as an exploding
reflector, so can be migrated with a zero-offset migration algorithm.
Try this out.
	\begin{enumerate}
	\item Does a maximum energy algorithm image the reflector at
	location~4 any better than first arrival?
	\item What about wavefield extrapolation?
	\end{enumerate}
\end{enumerate}

\section{All done}
When you are all done, print out a copy of this document ({\tt make
print}), and clean up your directory ({\tt make clean}).
