\def\xiv{\mbox{\boldmath$\xi$}}
\chapter{Depth imaging: 2D synthetic}\label{chapter:DepthSyn}
\section{Introduction}
A 5D volume of seismic data is a voluminous and complicated volume 
to interpret correctly, that doesn't recapitulate human's 3D concept
of space.  Offset, moveout, traveltime, interference, and high 
dimensionality are nearly intractable parameters for an interpreter 
of large seismic surveys.  For transmission wavefields, and the 
gathers synthesized via correaltion,% with low signal-to-noise ratios
these interpretation problems can be insurmountable.  Under realistic
scenarios with real data explored in Chapter~\ref{chapter:TimeSynth},
the correlated gathers can easily be uninterpretable.  This chapter
investigates the extent to which the artifacts in the synthesized
gathers affect the image space after migration.
\par
Migration is a powerful data processing tool to collapse dimensionality
and sum reduntant information as a function of offset to subsurface
location.  Shot-profile depth migration is arguably the most powerful
algorithm to process seismic data and return an interpretable 
subsurface volume.  Fortuitously, recapitulating the seismic 
experiment in the computer during migration can reduce coherent 
noise events in the data volume that are aphysical.  Images can still
suffer from coherent noise, but it is often more recognisable
as implausible, and easily removed.  Interpreting a dense 3D image
volume is also much easier than a sparse 5D data volume.
\par
In the previous chapter, cross-correlation of transmission wavefields was
analyzed in terms of its efficacy in synthesizing shot gathers from a
conventional surface seismic active survey.  In this chapter, I present 
depth images produced with shot-profile migration using the correlation 
data volumes shown previously.  The best synthesized volume of data
was produced by summing the correlations from the transmission
wavefields from 225 impulsive sources across the bottom of the model
domain.  This volume still contains many artifacts.  Progressively
poorer correlation volumes were produced by using very few of the
transmission wavefields and by correlating the sum of transmission
wavefields.  The first section of this chapter will present the images
produced by migrating all of those synthesized gathers.  
\par
Next, I develop the processing of passive seismic data as a migration 
operation.  \longcite{me1} introduced the applicability of direct migration 
for transmission wavefields.  \longcite{meage} provided the mathematical
justification for zero-phase source functions.  \longcite{SAW} showed
results for the special case of imaging with teleseisms.  Direct
migration of transmission wavefields requires an imaging algorithm
composed of wavefield extrapolation and a correlation based imaging
condition.  Shot-profile \cite{GEO36-03-04670481} wave-equation depth
migration fulfills these requirements.
\par
Finally, I show images produced by direct migration of the transmission
wavefields used as input for the correlations presented in the previous
chapter and used as input for the migrations in the first section of this
chapter.
\section{Migration of correlations}
The first two panels in Figure~\ref{fig:mig.all} are two shot gathers from 
the volume of synthesized gathers produced by summing the correlations 
calculated from all 225 transmission wavefields as described in the 
previous chapter.  The right panel is the zero-subsurface offset image
produced by shot-profile migration of the entire volume from which the
gathers were extracted.  The image was processed with tpow=1.5 to diminish
a strong $z=0$~m flat event and boost the amplitude of the second 
reflector.  The event that partially obscures the second event is
a reflection from the bottom of the velocity model during 
modeling due to placing the sources just above the boundary.  This event 
highlights the potential problem of sources being distributed within the
volume of interest rather than around the domain.  Because the sources 
were all modeled at the same depth, this event is strong and 
continuous.  However, if the sources are distributed at random depths, 
this artifact will diminish in strength as more transmission wavefields 
from different source depths are included in the imaging 
process.  Despite the imperfectionsz in the synthesized 
gathers compared to active surface seismic gathers, the resulting image
is very clear.
\plot{mig.all}{width=\textwidth}{Representative gathers synthesized by 
summing correlations from all modeled transmission wavefields and the
depth image produced with shot profile migration of all such gathers.}
\par
The left column of Figure~\ref{fig:dat1200} shows the correlated gathers 
from a single shot at the left edge of the model domain whose generation 
was explained in the previous chapter.  The right column shows active 
data modeled at the same locations.  The correlated gather 
in the bottom-left image contains energy from only the late-time, 
high-order multiples modeled in the transmission wavefields.  These 
facts, and the gather's dissimilarity with the active data to its
right, would suggest that most of the energy within this 
synthesized gather is coherent noise.  In fact, the bottom-right
gather looks very much like a time-shifted version of the gather
above it.
\sideplot{dat1200}{width=\textwidth}{Correlated gathers from left and
  middle of synthesized data volume (left column) compared to active
  surface data at the same locations (right column).  Only a single
  transmission wavefield (source at bottom left edge of model domain)
  was used to produce gathers in the left column.}
\par
Figure~\ref{fig:mig.shot1200} shows two images produced from the 
volume of synthesized gathers produced with only a single source at the 
left edge of the model (left column Figure~\ref{fig:dat1200}).  For
comparison, the right panel was produced  
by migrating the first shot from the modeled surface seismic data 
volume (top-left panel Figure~\ref{fig:dat1200}).  The first panel migrated 
all gathers produced by 
correlation.  Clearly the right side of the model and down-right dipping
structure is not well imaged.  Surprisingly however, the entire domain 
is correctly imaged, if weak.  The middle panel was produced by migrating
the synthesized shot gathers from the right side of the model (such as
Figure~\ref{fig:dat1200} bottom-left).  Without the strong, well-imaged
left side of the domain, the gain for the figure is substantially boosted
which shows more of the background noise.  However, the noise is not
coherent, and the power of events is substantially above the noise
level.  Surprisingly, the poor quality gathers from the opposite side
of the model domain from the subsurface source still contain a
sufficient amount of correct subsurface information to correctly image
the structure on the right side of the domain.  The right most panel
shows how little of the domain is imaged with only a single 
surface shot (Figure~\ref{fig:dat1200} top-right) at zero subsurface
offset.
\plot{mig.shot1200}{width=\textwidth}{Left: Image produced by migrating
  all synthesized gathers from a single transmission wavefield due to
  an impulsive source at the left edge of the domain.  Center: Image
  produce by migrating right half of the total volume of synthesized
  gathers.  Right: Image produced by migrating a single shot from an
  surface seismic volume of shot gathers.}
\par
Figure~\ref{fig:mig.shot1200.3500} presents a suite of migrations using 
the correlation volumes produced in the previous chapter generated to 
explore the possible deleterious effects of processing two transmission
wavefields simultaneously.  The synthesized gathers presented
previously were obscured by the superposition of two correlation
results.  The gathers were not uninterpretable, but without knowing
that two sources  were contained within the transmission wavefield, it
is likely that the subsurface would be misinterpreted.  The structure
revealed in these depth images however is very easy to accurately
interpret and noise is easily identified as such.  The top row are
images produced by migrating the correlations of summed transmission
wavefields.  The left panel used impulsive wavefields, and the right
panel used randomized wavelets.  The same is true for the bottom row
of images, which show the migration of summed correlation volumes from
the two wavefields.  The bottom row thus represents a correctly processed
partial result if the two transmission wavefields were processed
independantly.
\sideplot{mig.shot1200.3500}{width=4in}{Images obtained from migrating
correlation volumes due to two sources (located below the left edge
and center of model domain).  Left column used impulsive sources,
while source functions were randomized for the right column.  Top row
was produced with summed transmission wavefields, while bottom row
from sum of correlations of transmission wavefields.}
\par
Despite the fact that the model was illuminated with only two
subsurface sources, all the structure is well imaged in
Figure~\ref{fig:mig.shot1200.3500}.  Much of the noise in the top-left
panel is eliminated in the bottom-left image by correctly processing
the transmission wavefields one at a time.  The images in the right
column are almost identical.  In fact, migrating the correlations from
the transmission wavefield modeled with a source in the middle of the
model domain produces a similar image.  Figure~\ref{fig:shot1200.3500.rand} 
shows that the random wavelet used for the source at the left edge of
the model also contains a nearly 2~s time delay.  Therefore, this
energy is not included in the image at such shallow depths. 
\par
Figure~\ref{fig:mig.all} showed that migrating the sum of correlations
produced from individual transmission wavefields generates a clean
interpretable image.  Figure~\ref{fig:mig.shotsmall} shows images
produced by migrating the correlations produced with wavefields that
contain more than two subsurface sources.  The top row contains 125
subsurface sources evenly distributed across the bottom of the model
domain.  The bottom row was produced with 30 sources located around
the center of the model domain.  The sources in the left column were
all impulsive.  The right column was produced by first adding a random
phase shift and wavelet coda to each transmission wavefield before
summing.
\sideplot{mig.shotsmall}{width=4in}{Images produced by migrating 
synthesized gathers of the summed transmission wavefields from many  
sources across the bottom of the model domaian.  Top row images
are migrations of correlations from the sum of 125 transmission 
wavefields due to sources across the bottom of the model 
domain.  Bottom row images are migrations of correlations from the 
sum of 30 transmission wavefields due to sources clustered 
around the center of of the model domain.}
\par
The top row images fully illuminate the structure, though steeply-dipping 
noise-trains are seen due to the sparse sampling of subsurface 
sources and the correlational interference of all the individual 
wavefields.  Similar features are in the top-left panel of
Figure~\ref{fig:mig.shot1200.3500}.  The addition of randomized wavelets in
the top-right panel does not substantially improve the image.  The
noise events are less structured, but there are some flat
semi-coherent events between the reflectors, and the second reflector
is less continuous.
\par
The bottom row images are of much poorer quality.  Since the 
subsurface sources used were so close together, the events are highly
correlable across the space-axis near the flat hyperbola tops in the
raw data. This leads to the strong, fat noise in the center of the
images.  A low-cut filter removes this energy to provide substantially
better images.  The randomized wavelets used in the input for the
bottom-right panel remove much of the low frequency noise in the
center of the neighboring image.  The noise is less structured, but
the overall quality of the image is improved.

\section{Direct migration}
\subsection{Theory}
Shot-profile migration uses one-way extrapolators to independently
extrapolate up-going, $U$, and down-going, $D$, energy through the
subsurface velocity model.  $U$, extrapolated acausally, is a single
shot-gather.  $D$, extrapolated causally, is a wavefield modeled to
mimic the source used in the experiment.   I will define the causal
extrapolation operator, $E^+$, and the acausal operator, $E^-$.  These
one-way operators can not extrapolate evanescent waves, but this
limitation is not crippling.  In this thesis, I use the split-step
Fourier plus interpolation (SSF-PI) \cite{SEG-1992-0917} for all
migration operations.
\par
Wavefields are extrapolated to progressively deeper levels,
$z$, by recursive application of $E^{\pm}$
\begin{eqnarray}
U_{z+1}({\bf k}_x; {\bf x}_{s},\omega) &=&
   E^{-} U_{z  }({\bf k}_x; {\bf x}_{s},\omega) \\
D_{z+1}({\bf k}_x; {\bf x}_{s},\omega) &=&
   E^{+} D_{z  }({\bf k}_x; {\bf x}_{s},\omega).
\end{eqnarray}
\par
The imaging condition combines the two wavefields to output a
subsurface image.  The zero-offset imaging condition for shot-profile
migration is defined as the zero lag of the cross-correlation of the
two wavefields 
\begin{equation}
i_z({\bf x})=\sum_{{\bf x}_{s}}\sum_\omega 
U_z  ({\bf x}; {\bf x}_{s}, \omega)
D_z^*({\bf x}; {\bf x}_{s},\omega) \; .
\label{ic} 
\end{equation}
The sum over frequency extracts the zero lag of the correlation.  The
sum over shots, ${\bf x}_s$, stacks the overlapping images from all the
individual shot-gathers.
\par
The important implication of the shot profile strategy is that the 
causal and acausal extrapolation operators applied to the down-going 
and up-coming energy are orthogonal operators on orthogonal 
components of the seismic experiment.  This is the reason that two 
one-way operators can be used in place of two-way operators to 
recapitulate the seismic experiment in the computer.  
\par
Transmission wavefields are defined as the superposition of the 
up-coming and down-going seismic energy.  As such, the down-going 
energy defined above as $D$ and the up-coming energy, $U$, is 
contained therein.  Therefore, propagating a transmission wavefield
causally will correctly extrapolate the down-going energy contained
therein for migration.  Similarly, propagating a transmission 
wavefield acausally will correctly extrapolate the up-coming 
energy for migration.  The correlation applied in the imaging
condition, equation~\ref{ic}, performs the trace comparison required
to synthesize active data from transmission data.  Performing this
operation in the image space simply moves the experiment down from the
acquisition surface.  This result is proved in terms of a 
redatuming experiment in \longcite{meeage}, and independently 
derived by \longcite{papa}.  The phase-encoding framework
explained in Chapter~\ref{chapter:Intro} was developed originally
as a migration algorithm.  Recognizing the similarities between a
transmission wavefield and a phase-encoded wavefield naturally leads
to direct migration.
\par
Table~\ref{flow} pictorially demonstrates how direct migration of
transmission wavefields fits into the framework of shot-profile 
migration to produce the $0^{th}$ and $1^{st}$ depth levels of 
the zero-offset image.  The correlation in the imaging condition
takes the place of preprocessing the transmission
wavefield.  The summations over shot locations and frequency in
equation~\ref{ic} are omitted to reduce complexity.  Note
however, that the sum over shot locations in equation~\ref{ic} takes
the place of the integral over source locations in
equation~\ref{rigor}.  Also, after the first extrapolation step, using
the two different phase-shift operators, the two transmission
wavefields are no longer identical, and can be redefined $U$ and $D$.
This is noted with superscripts on the $T$ wavefields at 
depth.  Synthesizing shot gathers by correlation followed by migration
is represented by movement from the top-right of the table to the
conventional data domain on the left, and then down via conventional
migration. 
%%\tabl{
\begin{table}
\begin{center}
\begin{tabular}{rclcrcl}
\multicolumn{3}{c}{shot-profile migration} & &
\multicolumn{3}{c}{transmission imaging}\\ 
\cline{1-3} \cline{5-7} 
$U_{z=0}({\bf x} ;{\bf x}_{s},\omega)$& 
$\otimes $ &
$D_{z=0}({\bf x};{\bf x}_{s},\omega)$& 
$ = $ &
$T_{z=0}({\bf x} ;\xiv,\omega)$ & $\otimes $& 
$T_{z=0}({\bf x} ;\xiv,\omega)$ \\

\multicolumn{1}{c}{$|$} & & \multicolumn{1}{c}{$|$} 
& & 
\multicolumn{1}{c}{$|$} & & \multicolumn{1}{c}{$|$} \\

\multicolumn{1}{c}{$E^{-}$} & &
\multicolumn{1}{c}{$E^{+}$} 
& &
\multicolumn{1}{c}{$E^{-}$} & & 
\multicolumn{1}{c}{$E^{+}$} \\

\multicolumn{1}{c}{$\downarrow$} & & 
\multicolumn{1}{c}{$\downarrow$} 
& &
\multicolumn{1}{c}{$\downarrow$} & & 
\multicolumn{1}{c}{$\downarrow$}\\

$U_{z=1}({\bf x};{\bf x}_{s},\omega)$& 
$\otimes $ &
$D_{z=1}({\bf x};{\bf x}_{s},\omega)$& 
$ = $ &
$T_{z=1}^-({\bf x};\xiv,\omega)$ & $\otimes $ & 
$T_{z=1}^+({\bf x};\xiv,\omega)$ \\
\end{tabular}
\end{center}
\caption{Equivalence of shot-profile migration of reflection data and
  direct migration of transmission wavefields, $T({\bf x},\xiv,t)$.
  ${\bf x}_s$ has a similar meaning to $\xiv$.  $\sum_{{\bf
  x}_s,\xiv}$ and $\sum_\omega$ produces the image $i_z({\bf x})$ for
  both methods.  Only first and second levels of the recursive process
  are depicted. } 
\label{flow} %}
\end{table}
The physics captured in the formulation of shot-profile migration
remains the same regardless of the temporal or areal characteristics
of the initial conditions in the wavefields.  The data records provide
boundary conditions for the experiment migration attempts to reverse
in the computer.  The extraction of only the zero lag of the
correlation for the image discards energy in the two wavefields that
is not collocated.  This includes energy that has been extrapolated in
the wrong direction (since the same data is used initially for both
$U$ and $D$ wavefields).  
\par
When considering the case of direct migration of transmission wavefields
due to multiple sources, the summation introduced within the right column 
of Table~\ref{flow} can also be included in the conventional migration
flow shown on the left column.  After summing across the source axis
of a conventional reflection seismic data volume, the appropriate
source wavefield, the sum of all the impulsive down-going wavefields,
is a horizontal planewave.  The direct migration of summed
transmission wavefields is very analogous to the horizontal planewave
migration strategy, though if the number of subsurface sources is 
small, a complete planewave will probably not be manufactured.  Further,
unless the sources fire simultaneously, they will not construct a
horizontal planewave no matter how many are included in the transmission
wavefield.
\par
Table~\ref{flow2} pictorially demonstrates how direct migration of
multiple-source transmission wavefields fits into the framework of plane-wave
migration analogous to Table~\ref{flow}.  Summing shot-gathers from an
active data set before migration is represented on the left.  The top 
row of the right column recognizes the implicit sum over the source axis 
if raw passive recordings can not be separated into individual
wavefields.  In this case, the combined direct arrival of the multiple
sources forms an unknown wavefront that will be used as the down-going
source energy.  For this reason, I label the approach wavefront imaging.
%%%%\setlength{\tabcolsep}{.03in}
%%\tabl{
\begin{table}
\begin{center}
\begin{tabular}{rclcrcl}
\multicolumn{3}{c}{plane-wave migration}& &
\multicolumn{3}{c}{wavefront imaging}\\ 
\cline{1-3} \cline{5-7} 
$\left(\sum_{{\bf x}_{s}}
U_{z=0}({\bf x};{\bf x}_{s},\omega)\right)$& 
$\otimes $ &
$\left(\sum_{{\bf x}_{s}}
D_{z=0}({\bf x}; {\bf x}_{s},\omega)\right)$& 
$ = $ &
$T_{z=0}({\bf x} , \omega )$ & $\otimes $& 
$T_{z=0}({\bf x} , \omega )$ \\

\multicolumn{1}{c}{$|$} & & \multicolumn{1}{c}{$|$} 
& & 
\multicolumn{1}{c}{$|$} & & \multicolumn{1}{c}{$|$} \\

\multicolumn{1}{c}{$E^{-}$} & &
\multicolumn{1}{c}{$E^{+}$} 
& &
\multicolumn{1}{c}{$E^{-}$} & & 
\multicolumn{1}{c}{$E^{+}$} \\

\multicolumn{1}{c}{$\downarrow$} & & 
\multicolumn{1}{c}{$\downarrow$} 
& &
\multicolumn{1}{c}{$\downarrow$} & & 
\multicolumn{1}{c}{$\downarrow$}\\

\multicolumn{1}{c}{$U_{z=1}({\bf x},\omega)$} & 
$\otimes $ &
\multicolumn{1}{c}{$D_{z=1}({\bf x},\omega)$} & 
$ = $ &
$T_{z=1}^-({\bf x} ,\omega)$ & $\otimes $ & 
$T_{z=1}^+({\bf x} ,\omega)$ \\
\end{tabular}
\end{center}
\caption{Equivalence between direct migration of passive field data and
  simultaneous migration of all shots in a reflection survey. Only first
  and second levels of the iterative process are depicted.  $\sum_\omega$ 
  produces the image $i_z$ for both methods. }
\label{flow2}%}
\end{table}
\subsection{Synthetic tests}
The modeled transmission wavefields used in previous examples are
again used below to produce images with the direct migration strategy
presented above.  In addition to direct migration of individual and
summed transmission wavefields, images are also presented using a 
horizontal planewave migration with the various volumes of correlated
shot gathers. 
\par
Figure~\ref{fig:dmigall} presents images produced with various
processing strategies, but containing all 225 transmission wavefields
from sources across the entire bottom of the model domain.  The
top-left image was produce by direct migration of the sum of all
225 modeled wavefields.  In this case, the minimum traveltime of the
direct arrival for each transmission wavefield is constant, and
therefore the summation builds a strong horizontal planewave as a
direct arrival.  The data for direct migration used to produce
the top-center image and long image on the right included the addition of
randomized source functions.  The right-most image used the same data,
but a deeper velocity model to image more of the deep artifacts.  The 
second reflector stands out more clearly, and the
reflection from the bottom of the model domain has been 
weakened in comparisson to the top-left imag.  Two coherent cross-talk 
events have been introduced however.  There is an event at $z=300$~m 
and a mirrored event below the flat-bottomed syncline and many strange 
events in the deep image.  Theese events are due to Fourier-domain 
wrap-around on the time axis.  The
time shift applied to the individual transmission wavefields
associated with the application of the randomized wavelet often shifts
late-time event around the time axis to early time.  Energy at these
times does not exist when the minimum traveltimes in each transmission
wavefield is aligned.  Since the source wavefield, $D$, is the same as
the data wavefield, $U$, for direct migration, the algorithm images an
inverted structure at deep depths when up- and down-going energy is
propogated innappropriately (in opposite directions).  The inverted
structure is not exactly mirrored however, even though the time mirror
would be symmetric.  Convolution of the time records with the velocity
model during migration leads to interesting combinations of the flat
events with the velocity model.
\par
The bottom figures (left and center) of Figure~\ref{fig:dmigall}
compare images produced by summing the direct migration images from
each individual transmission wavefield (left) to the image produced by
horizontal planewave migration of the sum of the correlations from
individual transmission wavefields.  The center image suffers from
steeply-dipping noise events, and the second layer shows weaker
amplitude compared to the artifact associated with the reflection from
the bottom of the model.  They are otherwise roughly equivalent in 
quality. 
\plot{dmigall}{width=\textwidth}{Left to right:  Direct migration of 
summed transmission wavefields, sum of direct migrations, planewave
migration of best possible synthesized shot gathers.}
\par
Figure~\ref{fig:dmig1200} compares the images produced by direct 
migration of a single transmission wavefield (due to a source at the 
left edge of the model domain) to the horizontal planewave migration
of the correlated shot gathers produced with the same transmission
wavefield.  The left panel, direct migration, is much stronger on the
left edge of the model where the predominance of energy is 
located.  The right image contains no valuable subsurface
information.  This result, interpreted with the correlated gathers in
Figure~\ref{fig:dat1200} is important.  No symmetric hyperbolas were
produced in the correlated gathers above.  The central, flat section
of a hyperbola is the portion of an event that would be imaged with 
a horizontal planewave migration.  The right panel of 
Figure~\ref{fig:dmig1200} shows that only the left edge of the model, 
directly above the impulsive subsurface source is correctly 
imaged.  However, the successful imaging of structure at the right 
edge of direct migration, as well as the structure shown in the center 
panel of Figure~\ref{fig:mig.shot1200} (which used only synthesized 
gathers  from the right side of the domain) proves that there is correct
subsurface information at large surface offset from the subsurface
source.  However, that information is carried in steeply-dipping 
ray parameter events that are not imaged with a horizontal planewave.
\sideplot{dmig1200}{width=4in}{Direct migration of a transmission
  wavefield due to an impulsive subsurface source at the left edge of
  the model domain (left) compared to the horizontal planewave
  migration of the correlated shot gathers produced with the same
  wavefield.}
\par
Figure~\ref{fig:dmig1200.3500} shows images produced with the various
migration strategies available when processing a transmission
wavefield due to two subsurface sources, one at the left edge and one
below the center of the model domain.  The top-left image is the
direct migration of the sum of the two impulsive transmission
wavefields.  The top-right image is the direct migration of summed
wavefields with randomized source wavelets.  The bottom-left image is
the sum of the images produced by direct migrations of the two
impulsive wavefields.  The bottom-right image is the horizontal
planewave migration using the correlations produced with the sum of
the two wavefields with randomized source functions.  The sum of
direct migrations is clearly the best image, suffering mostly just
partial illumination problems.  There is very little difference
between the upper two images (with and without randomized 
wavelets).  The horizontal planewave migration is completely
incorrect. 

\sideplot{dmig1200.3500}{width=4in}{Top, left to right: Direct
  migration of the sum two transmission wavefields without and with
  randomized source functions.  Bottom left: Sum of images produced by
  direct migration of individual transmission wavefields.  Bottom
  right: Horizontal planewave migration of the sum of correlations
  produced from the two wavefields, with randomized wavelets.}
\par
Figure~\ref{fig:dmigsmall} shows images produced with the sum of
transmission wavefields due to sources sparsely, but evenly 
distributed across the bottom of the model domain.  The left column
used wavefields with impulsive source functions, while the right 
column introduced randomness to the length, wavelet, and time delay
to the source function.  The top row was produced with direct 
migration, and the bottom row was produced with the horizontal 
planewave strategy beginning with synthesized shot gathers.

\sideplot{dmigsmall}{width=4in}{Images produced from a transmission
wavefield due 30 source clustered around the center of the bottom 
of the model domain.  Top row was produced by direct migration.  Bottom
row was produced by horizontal planewave migration of the correlated
gathers produced with the transmission wavefields used for direct 
migration above.  Left column used impulsive source functions, 
while the right column used randomized source functions.}
\par
Figure~\ref{fig:dmigsmallsmall} shows images produced with the sum of
transmission wavefields due to 30 sources clustered around the center
of the model domain.  The left column used wavefields with impulsive
source functions, while the right column introduced randomness to the
length, wavelet, and time delay to the source function.  The top row
was produced with direct migration, and the bottom row was produced
with the horizontal planewave strategy beginning with synthesized shot
gathers using the summed transmission wavefield.  The two direct
migration images are very comparable.  Again, steep, coherent noise on
the left is eliminated with the introduction of randomness, while the
coherence of the structure suffers marginally.  The two planewave
migrations are of very poor quality, though marginally interpretable.
However, the implied dip of the shallow reflector in the left image is
incorrect.

\sideplot{dmigsmallsmall}{width=4in}{Images using summed transmission
  wavefields due to 30 sources clustered around the center of the
  model domain.  Top row, left to right: Direct migration with
  impulsive sources and randomized sources.  Bottom row, left to
  right: Horizontal planewave migration using correlation without and
  with randomized source functions.}
%%\section{Migration of noisy data}

\section{Conclusions}
Migration is a powerful operator that substantially reduces the 
complexity and size of the volume of data to be analyzed and often
diminishes noise in the raw data or transforms it to a more obvious  
character.  Transmission wavefields are inherently imperfect data 
volumes.  Migration greatly simplifies the complexity and volume
of data to interpret, provided a reasonable accurate velocity
model is available.  
\par
Using transmission wavefields modeled only
across the bottom of a subsurface domain, the rigorous equation
to synthesize the conventional active seismic data volume is
not obeyed.  Without complete source coverage around the volume
of interest, artifacts will remain in the correlated 
gathers.  Synthetic experiments presented above show that many
of the artifacts within such synthesized gathers are minimized 
after depth migration.
\par
When multiple sources are captured in the same transmission 
wavefield during the recording of a passive seismic experiment,
the cross-talk associated with correlation can lead to 
misinterpretation of the shot gathers.  This misinterpretation 
of the subsurface is minimized, though not mitigated completly,
by migration.  Direct migration of transmissoin wavefields 
minimizes the artifacts in the image domain even more than the
migration of synthesized gathers when multiple sources are
processed simultaneously.
\par
At the limit of complete coverage around the volume of interest,
wavefront imaging (direct migration of the sum of all individual
transmission wavefields) produces an very good image of the 
subsurface structue.  At the same limit, horizontal planewave
migration of the correlated gathers from such wavefields produces
comparable images. If, however, the multiple sources captured 
in a transmission wavefield are not well distributed, the direct 
migration images are supperior to the migration of synthesized 
gathers or horizontal planewave migrations of the synthesized
gathers.


