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\chapter{Depth imaging passive seismic data}\label{chapter:DepthSyn}
\section{Summary}
Migrating the synthesized shot gathers produced in
Chapter~\ref{chapter:TimeSynth} produces interpretable images even when
the synthesized gathers look noisy or completely incorrect.  Incomplete
distribution of subsurface sources surrounding the domain of interest 
and simultaneous processing of multiple subsurface sources are
the most likely weaknesses in processing data from a passive recording
campaign.  These problems cause coherent noise events in the
synthesized gathers that do not conform to the kinematics of the
active reflection seismic experiment.  Migration 
minimizes the effect of the noise and
correctly images subsurface structure even when events in the gathers
do not appear to conform to the conventional experiment.  For this
reason, I believe migration should be a near mandatory processing step
for transmission wavefields.  Direct migration of raw
passive data produces accurate subsurface images without first
synthesizing shot gathers.  The images produced by direct migration
are almost indistinguishable from those produced by migrating the
synthesized shot gathers with the synthetic tests presented in
Chapter~\ref{chapter:TimeSynth}.  Under most scenarios, direct
migration saves substantial computational cost.  Any migration
strategy that maintains separate up-coming and down-going wavefields
and employs a correlation based imaging condition to combine them at
depth can be employed to migrate transmission wavefields. 

\section{Introduction}
A 5D volume of seismic data is a very large and complicated volume 
to interpret, that doesn't recapitulate humans' 3D concept
of space.  Offset, moveout, traveltime, interference, and high 
dimensionality make interpreting the subsurface in the data domain a
challenging task.  For passive data composed of transmission
wavefields due to unknown sources with incomplete subsurface
distribution, the gathers synthesized via correlation may be
impossible to interpret correctly.  Under scenarios expected to be
realistic with field data, shown in Chapter~\ref{chapter:TimeSynth},
small violations of the assumptions behind correlation
processing of transmission wavefields can quickly make the synthesized
gathers 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 redundant information as a function of offset to subsurface
location.  Shot-profile depth migration is a robust
algorithm to process seismic data and return an interpretable 
subsurface volume.  Fortuitously, migration can also reduce coherent 
noise events in the data volume that do not comply with the kinematics
of simple reflections.  Images still suffer from coherent noise, but
it is often more recognizable as implausible, and easily ignored or
removed.  Interpreting a dense 3D image volume is also much easier
than a sparse 5D data volume. 
\par
In Chapter~\ref{chapter:TimeSynth}, 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 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 presents the images produced by migrating
those synthesized gathers.   
\par
Next, I develop the processing of passive seismic data as a migration 
operation: Direct migration.  \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 teleseismic events.  Direct
migration of transmission wavefields requires an imaging algorithm
composed of wavefield extrapolation and a correlation based imaging
condition \cite{artman:SI177}.  Shot-profile wave-equation depth migration 
\cite{GEO36-03-04670481} fulfills these requirements.  Images produced by direct
migration are presented to show the validity of the new method.

\section{Migration of synthesized gathers}
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 
Chapter~\ref{chapter:TimeSynth}.  The virtual shot locations for the
gathers are indicated by vertical lines.  The right panel is the
zero subsurface-offset image produced by shot-profile migration of the
entire volume from which the gathers were extracted.  All images
produced in this chapter are zero subsurface-offset images.  The delta 
function in equation~\ref{rigor} results in a strong flat event at 
$z=0$~m in the image.   To diminish that event and boost the amplitude
of the second reflector, the image was processed with a
depth-dependent gain function.  All images in this chapter are
presented with this same post-processing step.  The event  
that partially obscures the second reflection is the first order multiple
between the surface and the first reflector.  Despite the
imperfections in the synthesized gathers compared to active surface
seismic gathers, the resulting image is very clear.  Because most of
the noise events in the synthesized gather do not conform to the
kinematics of the reflection experiment from which the migration
algorithm is designed, their effects are minimal in the image domain.
\activeplot{mig\DOT all}{width=\textwidth}{CR}{Representative gathers synthesized by 
summing correlations from all modeled transmission wavefields
(zero-offset trace under black line) and the
depth image produced with shot profile migration of all such gathers.}
\par
The left column of Figure~\ref{fig:dat1200} (panels \pan{a}, \pan{c}, \pan{e})
are correlated gathers from a single source beneath the left edge of 
the model domain whose generation was explained in 
Chapter~\ref{chapter:TimeSynth}.  The right column (panels \pan{b}, 
\pan{d}, \pan{f})
shows active data modeled at the same (virtual) shot locations.  The 
correlated gathers suffer many problems that make them obviously poor
reproductions of the surface data modeled to the right.  The gathers
are a partial result of the summation over subsurface source required
by the development of correlation processing to synthesize active
surface data.  Produced with only a single transmission wavefield, the 
synthesized gathers are not just poor quality, but produce completely 
wrong kinematics in many places.  The synthesized gather with 
virtual shot location directly above the subsurface source is the closest
to correctly reproducing an active gather.  At far offset the events are 
incorrect, but the near-offset traces in this synthesized shot gather are 
accurate.  Synthesized gathers with virtual source locations further away 
from the horizontal coordinate of the subsurface source become less like 
the shot gathers which one is trying to synthesize via correlation.  Most
of the energy in panels \pan{c} and \pan{e} is from the high order multiples 
in the transmission wavefield.  There is seemingly no correspondence between
synthesized gathers and the modeled active gathers presented in panels
\pan{d} and \pan{f}.  Most of the events in these synthesized gathers
dip to the right.  There is little to no flat or left-dipping energy in the 
synthesized gathers.  Such results would likely be discarded if produced 
from field data since they do not obviously conform to active seismic
shot gathers.
%%\sideplot{dat1200}{height=8in}{Left column: Synthesized 
\activesideplot{dat1200}{width=3in}{CR}{Left column: Synthesized 
  gathers from left and middle of synthesized data volume produced with a 
  single transmission wavefield due to a source below the left edge of the 
  model domain.  Right column: active surface data at the same (virtual) 
  shot locations.}
\par
Figure~\ref{fig:mig.shot1200} shows images produced from the
volume of synthesized gathers using a transmission wavefield due to a
single source beneath the left edge of the model (left column
Figure~\ref{fig:dat1200}).  Panel \pan{a} is the migration of all gathers
produced by correlation, which images the subsurface structure well above the 
noise level.  Structure on the right side of the model and
down-right dipping structure is diffuse and weak.  However, the entire
domain is correctly imaged.  Panel \pan{b} was produced by migrating
the synthesized shot gathers from only the right side of the model
(such as Figure~\ref{fig:dat1200}\pan{c},\pan{e}).  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.  The negative (black) part of the wavelet below
the reflector is lost, but the positive (white) peak is in the same
location ($z=200$~m) as in panel \pan{a}.  Despite the correlated
gathers on this half of the domain looking completely incorrect
(bottom Figure~\ref{fig:dat1200}), using them as data for migration has
correctly imaged the subsurface structure.  
\par
Zero subsurface-offset images produced with a single shot gather from
active or passive surveys image flat reflections across a very limited
lateral distance.  Figure~\ref{fig:mig.shot1200}\pan{c} shows the
image resulting from the migration of a single synthesized shot gather
with virtual source location $x=4000$~m.  It is impossible to
interpret structure and noise in this image.  Panel \pan{d} is the
migration of 20 synthesized shot gathers, beginning with virtual
source location $x=4000$~m.  Destructive interference of the noise
within the image produced with each individual shot gather begins to
build the two subsurface reflectors seen above in panels
\pan{a},\pan{b}.  These images show that although the synthesized
gathers with virtual source locations at large horizontal distance
from the subsurface source look completely incorrect
(Figure~\ref{fig:dat1200}\pan{c},\pan{e}) they do contain appropriate
events to image the subsurface structure during migration.
%%\sideplot{mig.shot1200}{width=4in}{a) Image produced by migrating
\activesideplot{mig\DOT shot1200}{width=3in}{CR}{a) Image produced by migrating
  all synthesized gathers from a single transmission wavefield due to
  an impulsive source at the left edge of the domain.  b) Image
  produce by migrating right half of the total volume of synthesized
  gathers. c) Migration of synthesized gather with virtual source
  location $x=4000$~m.  d) Migration of 20 synthesized gathers from
  virtual shot locations $x=4000-4380$~m.}
\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 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 (independent correlation).
%%\sideplot{mig.shot1200.3500}{width=4in}{Images obtained by 
\activesideplot{mig\DOT shot1200\DOT 3500}{width=3in}{CR}{Images obtained by 
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, and the bottom row
by summing the correlations of single 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 panel
\pan{a} is eliminated in panel \pan{c} 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 due to a single source below the middle of the
model domain produces a similar image.  Figure~\ref{fig:vshot1200.3500.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.shotsmall1} shows images
produced by migrating the correlations produced with wavefields that
contain more than two subsurface sources.  Both panels were produced
with the synthesized correlations from the sum of 25 transmission
wavefields due to sources evenly distributed across the bottom of the model
domain.  The sources used to generate panel \pan{a} were all
impulsive.  The image in panel \pan{b} was produced by first adding a
random phase shift and wavelet coda to each transmission wavefield before
summing.  The subsurface structure is fully illuminated, though steeply-dipping 
noise-trains are seen due to the sparse sampling of subsurface 
sources and the correlation interference of all the individual 
wavefields.  Similar features are in visible in 
Figure~\ref{fig:mig.shot1200.3500}\pan{a}.  The addition of randomized 
wavelets in panel \pan{b} does not substantially differentiate the result 
from the image in panel \pan{a}.  The noise events are less structured and 
the second reflector is marginally less continuous in panel \pan{b}.
%%\sideplot{mig.shotsmall1}{width=4in}{Images produced by migrating 
\activesideplot{mig\DOT shotsmall1}{width=3in}{CR}{Images produced by migrating 
synthesized gathers of a summed transmission wavefield due to 25 
sources across the bottom of the model domain.  Panel \pan{a} used
impulsive wavelets, while Panel \pan{b} was produced after introducing
randomized source functions.} 
\par
Figure~\ref{fig:mig.shotsmall2} was produced by 
migrating the synthesized shot gathers from a single transmission wavefield
due to 30 subsurface sources clustered around the center of the model 
domain.  These images have less artifacts than the previous images, but do not 
illuminate the structure as well due to the limited lateral distribution of 
subsurface sources.  Since the subsurface sources used were so close
together, the events are highly correlatable across the space-axis
near the flat hyperbola tops in the raw data.  This leads to the
strong, low frequency noise in the center of the images.  A low-cut
filter can remove this energy to provide better imaging of the second
reflector, but the filter cut-off must be carefully chosen since the
frequency content of the noise events is only marginally below that of
the real events.  The randomized wavelets introduced to the input for
the image in panel \pan{b} removes much of the low frequency noise in
the center of panel \pan{a}.  The noise is less structured, and the
signal to noise ratio of the result is improved. 
%%\sideplot{mig.shotsmall2}{width=4in}{Images produced by migrating 
\activesideplot{mig\DOT shotsmall2}{width=3in}{CR}{Images produced by migrating 
synthesized gathers of summed transmission wavefields from 30 sources
clustered around the center of of the model domain.  Panel \pan{a} used
impulsive wavelets, while Panel \pan{b} was produced after introducing
randomized source functions.} 

\section{Direct migration}
\subsection{Theory}
Shot-profile migration uses one-way extrapolators to independently
extrapolate up-going, $R$, and down-going, $S$, energy through the
subsurface velocity model.  $R$, extrapolated acausally, is a single
shot gather.  $S$, 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 or overturning waves, 
but this limitation does not adversely affect imaging simple structure
with mild dip.  In this thesis, I use the split-step
Fourier plus interpolation (SSF-PI) \cite{SEG-1992-0917} extrapolator
for all migration operations.
\par
Wavefields are extrapolated to progressively deeper levels,
$z$, by recursive application of $E^{\pm}$
\begin{eqnarray}
R_{z+1}({\bf x}; {\bf x}_s,\omega) &=&
   E_z^{-}({\bf x},\omega) R_{z}({\bf x}; {\bf x}_s,\omega) \\
S_{z+1}({\bf x}; {\bf x}_s,\omega) &=&
   E_z^{+}({\bf x},\omega) S_{z}({\bf x}; {\bf x}_s,\omega).
\end{eqnarray}
\par
With two wavefields at each depth, an imaging condition is invoked to 
extract energy for the image, $i$, that is collocated within the two
wavefields.  The zero-offset imaging condition for shot-profile
migration is defined as the zero lag of the cross-correlation of the
two wavefields at every depth
\begin{equation}
i_z({\bf x})=\sum_{{\bf x}_{s}}\sum_\omega 
R_z  ({\bf x}; {\bf x}_{s}, \omega)
S_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.  The important implication of the shot profile 
strategy is that the causal and acausal extrapolation operators are applied 
to independent (down-going and up-coming energy) 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 field
experiment in the computer.  
\par
Direct migration capitalizes on the
correlation in the imaging condition, equation~\ref{ic}, to satisfy
the one in equation~\ref{rigor} after extrapolating (or redatuming) the
wavefield to levels progressively further away from the acquisition
surface.  The imaging condition performs the trace comparison required
to synthesize active data from transmission data as explained in
Chapter~\ref{chapter:Intro}.  Performing this operation in the image
space simply moves the correlation processing into the subsurface from
the acquisition plane.  This result is proved in terms of a redatuming
experiment in \longcite{meage}, and independently derived in
\longcite{papa}. 
\par
A simple framework within which to view correlation processing of
transmission wavefields is phase-encoded migration 
\cite{GEO65-02-04260436,biondo3d}.  The motivation for phase encoding 
is to manipulate the independent shot gathers in a predictable manner
such that the entire data volume can be processed (in this case
migrated) simultaneously after a summation to reduce 
dimensionality.  The reduction in dimensionality similarly reduces the
cost of the processing.  To that end, each shot gather is 
convolved with a unique function that will minimize cross talk
interference between the shots during a summation over the shot axes
in a 3D seismic survey.  Random time series are often used as effective
encoding functions.  Plane wave migration strategies can also be viewed
as a phase encoding migration operation \cite{liu:S129}.  Consider a set 
of arbitrary encoding functions $A({\bf x};{\bf x}_s,\w)$ for each
shot ${\bf x}_s$ over the map coordinates ${\bf x}$ with which shot
records are convolved.  Normally, source functions for active seismic
data are bandlimited (in time) impulses at the source location and
zero elsewhere, $\delta({\bf x}-{\bf x}_s,\w)$.   After 
convolving the impulses with encoding functions the source 
wavefield, $\hat{S}$, required for simultaneous migration is
\begin{equation}\label{source}
\hat{S}_{z=0}({\bf x},\w)=\sum_{{\bf x}_s} 
A({\bf x};{\bf x}_s,\w)\delta({\bf x}-{\bf x}_s,\w)\;.
\end{equation}
The five dimensional data volume of shot gathers, $R({\bf x},{\bf x}_s,\w)$, 
is similarly convolved with encoding functions and summed to reduce 
dimensionality
\begin{equation}\label{encode}
\hat{R}_{z=0}({\bf x},\w)=\sum_{{\bf x}_s} 
A({\bf x};{\bf x}_s,\w)R_{z=0}({\bf x};{\bf x}_s,\w)\;.
\end{equation}
The image is produced at each depth level $z$ with the same 
correlational imaging condition as equation~\ref{ic}, but does not
require the sum over ${\bf x}_s$, and the encoded wavefields $\hat{S}$,
$\hat{R}$ are used in place of $S,\;R$.  If the encoding functions
do not correlate at all, migrating the sum of the shot gathers,
$\hat{R}$, would reproduce the subsurface image desired at a
substantial cost savings.  Unfortunately, perfect encoding functions
do not exist, and steps must be taken to minimize the cross talk
introduced during summation.  This is normally done by summing the
results of $j$ encoded migrations with different function sets, 
$A_j$.  In this manner, the cross talk incurred by each migration will
be out of phase and stacking the images will reduce their effects.
\par
For passive seismic records, we can assume unknown and
arbitrarily complicated encoding functions arrive from the
subsurface.  Therefore a transmission record $T({\bf x},\w)$ recorded
by an array at the surface is $\hat{R}_{z=0}$.  To recover 
$R_{z=0}({\bf x};{\bf x}_s,\w)$, we need to undo the convolution in
equation~\ref{encode}.  For a single source the correlation of
$\hat{R}$ will reproduce the kinematics of $R$.  The autocorrelation
of $A$ becomes a scalar amplitude and $R^2$ has the same kinematics
as the reflection data $R$.  Unfortunately, after the summation in the
equation, this is not possible.  To avoid this problem, we must
correlate wavefields before source summation.  The rigorous
development of the correlation theory developed in the literature,
equation~\ref{rigor}, also relies on processing data due to a single
source, $\xi$.  The individual transmission wavefields indexed by
$\xi$ play the same role as a suite of encoded gathers $\hat{R}_j$ 
produced with functions $A_j$.  Viewing a suite of transmission
wavefields as phase-encoded shot gathers provides an simple
explanation for need to sum the results of the correlations rather
than correlate summed wavefields.  
\par
%%The phase encoding experiment requires convolving both $S$ and $R$
%%with encoding functions.  The source wavefield $\tilde{S}$ is the 
%%sum of all the encoding functions.  Calculating $S$ numerically 
%%from $\tilde{S}$ by correlation is unneccessary.  The
%%auto-correlations of the encoding functions, $a({\bf x},\xi,\w)$, is a
%%scalar amplitude term, and we are left with $\delta({\bf x}-{\bf
%%  x}_s,\w)$ which can be easily modeled.  Therefore,
%%if sufficient source energy is captured in the transmission wavefields
%%and the resulting transmission wavefields are correlated and stacked,
%%the appropriate source functions for migration are impulses at every
%%recording station \cite{gerry}. If the transmission data are
%%incomplete or improperly processed however, it is impossible to
%%estimate the appropriate source wavefield required for migration.
%%\par

\par
While phase-encoding can provide a framework for the analysis of
passive seismic records, the particular methodology was developed as a
migration algorithm.  Instead of convolving conventional gathers with
a synthetic encoding function for migration,  a
transmission wavefield is data convolved with a subsurface source
function.  In this case, the Earth provides the encoding function and
the data is recorded by the geophone array as $\hat{R}$.  Recognizing
the similarities between a transmission wavefield and a phase-encoded
wavefield naturally leads to direct migration of transmission wavefields.  
\par
Transmission wavefields are defined as the superposition of the 
up-coming and down-going seismic energy convolved with some unknown
function: $T=A(S+R)$.  Therefore, all the
boundary conditions for the wavefields needed to perform a migration,
$\hat{R}$ and $\hat{S}$, are recorded directly in the data.  Propagating
a transmission wavefield acausally will correctly
extrapolate the up-coming energy contained therein during
migration.  Similarly, propagating a transmission wavefield causally
will correctly extrapolate the down-going energy.  Any spatial/temporal
energy source (such as an array of Vibroseis trucks) can be used in a 
seismic experiment as long as the source characteristics are modeled
correctly for the wavefield $S$ used to migrate the data.  For direct
migration, we do not need to model the source wavefield because we
record it.
\par
The imaging condition for direct migration is a hybrid of the
standard form presented in equation~\ref{ic}, and the one appropriate
for phase-encoded gathers.  Because the transmission wavefield is
parameterized as a function of individual subsurface source, $\xi$,
the direct migration imaging condition retains a summation over the
source term.  Therefore the imaging condition for direct migration is
\begin{equation}\label{imageT}
i_z({\bf x})= \sum_\xi E_z^-({\bf x},\omega) T_{z-1}({\bf x},\omega,\xi) \;
                       [E_z^+({\bf x},\omega) T_{z-1}({\bf x},\omega,\xi)]^* \;,
\end{equation}
which in practice is an identical operation to the conventional
algorithm.  The only difference between a direct migration of passive
data and a conventional shot profile migration is the initial
condition of the down-going wavefield.  Instead of modeling a source
wavefield, the conjugate of the data wavefield is used for the
down-going wavefield in the 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 the correlation in equation~\ref{rigor} to synthesize 
shot gathers.  The summations over shot locations and frequency in
table~\ref{flow} are omitted to reduce complexity.  Note
however, that the sum over subsurface source locations in
equation~\ref{imageT} takes the place of the integral over subsurface sources in
equation~\ref{rigor}.  Also, after the first extrapolation step, using
the two different propagation operators, the two transmission
wavefields are no longer identical (and can be redefined $R$ and $S$).
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} 
$R_{z=0}({\bf x} ;{\bf x}_{s},\omega)$& 
$\otimes $ &
$S_{z=0}({\bf x};{\bf x}_{s},\omega)$& 
$ = $ &
$T_{z=0}({\bf x} ;\xi,\omega)$ & $\otimes $& 
$T_{z=0}({\bf x} ;\xi,\omega)$ \\

\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}{$\downarrow$}\\

$R_{z=1}({\bf x};{\bf x}_{s},\omega)$& 
$\otimes $ &
$S_{z=1}({\bf x};{\bf x}_{s},\omega)$& 
$ = $ &
$T_{z=1}^-({\bf x};\xi,\omega)$ & $\otimes $ & 
$T_{z=1}^+({\bf x};\xi,\omega)$ \\
\end{tabular}
\end{center}
\caption{Equivalence of shot-profile migration of reflection data and
  direct migration of transmission wavefields, $T({\bf x},\xi,t)$.
  ${\bf x}_s$ has a similar meaning to $\xi$.  $\sum_{{\bf
  x}_s,\xi}$ 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}
\par
The principle difference between the phase-encoded sum of all shots
and and a transmission wavefield, is that the transmission wavefield
is defined as a function of subsurface source,
$\xi$, while equations~\ref{source} and~\ref{encode} are defined as
functions of surface source locations ${\bf x}_s$.  However,
whether the sources are initiated at the surface or anywhere else is
unimportant as long as the source wavefield used for migration mimics
the spatial and temporal characteristics of the source energy that
gives rise to subsequent reflections from the subsurface.  The appropriate
source energy is the direct arrival from the subsurface.  Therefore, 
$\xi$ is merely a wavefield
index rather than a vector location containing the required information 
to model $S({\bf x},{\bf x}_s,\w)=\delta({\bf x}-{\bf x}_s,\w)$.  Summing the
images produced with each transmission wavefield due to all available
sources $\xi$ builds the complete subsurface image and cancels
artifacts associated with incomplete illumination.
\par
The source wavefield, $\hat{S}$, required to migrate phase encoded gathers
is the encoding function with which the gathers were 
convolved.  Transmission wavefields contain that function and all the events
represented by $\hat{R}$, which introduces noise to individual 
images.  The extraction of only the zero lag of the correlation at the imaging
condition discards energy in the two wavefields that is not
collocated.  This includes much of the energy that has been
extrapolated in the wrong direction (since the same data is used
initially for both $S$ and $R$ wavefields).  However, some cross talk
will be introduced since this approach strictly violates the phase-encoded 
migration strategy.  cross talk events in direct migration images are
progressively eliminated by summing results from many individual
transmission wavefields.  This same cancellation happens while summing
correlation volumes when synthesizing shot gathers at the acquisition 
surface or summing images produced with a suite of encoding 
functions.  If the number of subsurface sources recorded is small, some
cross talk artifacts will remain in the final image.  If the
subsurface sources do not completely surround the volume of interest,
artifacts in the image are due to incomplete illumination.
\subsection{Multiple subsurface sources}
When considering the case of direct migration of transmission wavefields
due to multiple sources, a summation over $\xi$ is implied in field data used
as data on the right column of Table~\ref{flow}.  This can happen when 
subsurface sources are not well separated in time, or are too weak to be
apparent in the raw data to facilitate separation by time windowing.  If we 
assume that we can not separate passive records into individual transmission
wavefields, field data, $T_f$, is most likely a partial sum over subsurface 
source
\begin{equation}
T_f({\bf x},\w)=\sum_\xi^n T({\bf x},\xi,\w)\;,
\end{equation}
where $n$ is an unknown number of sources that were active during a time
window of the passive data.  Processing relatively short time windows
independently will tend to minimize $n$, though can not address the 
possibility that multiple sources may be firing simultaneously as can
be expected if the ambient sonic energy is caused by cultural noise or 
reservoir compaction, for example.  
\par
Source summation is explicitly included in the conventional migration flow 
shown on the left column of Table~\ref{flow2}.  The implicit sum over 
subsurface sources for the transmission wavefields on the right column is 
indicated by not parameterizing the wavefield as a function of $\xi$.  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 plane wave.  The direct
migration of summed transmission wavefields is very analogous to the
horizontal plane-wave migration strategy, though if the number of
subsurface sources is small, a complete plane wave will probably not
be manufactured.  Further, unless the sources fire simultaneously,
they will not construct a horizontal plane wave no matter how many are
included in the transmission wavefield.  In this case, the combined
arrivals of the multiple sources forms an unknown wavefront, or series
of wavefronts, that is the source function to be used as the
down-going source energy.  For this reason, I label the approach
wavefront imaging. 
\par
%%%%\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} 
$\left(\sum_{{\bf x}_{s}^n}
R_{z=0}({\bf x};{\bf x}_{s},\omega)\right)$& 
$\otimes $ &
$\left(\sum_{{\bf x}_{s}^n}
S_{z=0}({\bf x}; {\bf x}_{s},\omega)\right)$& 
$ = $ &
$T_{z=0}({\bf x} , \omega )$ & $\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}{$R_{z=1}({\bf x},\omega)$} & 
$\otimes $ &
\multicolumn{1}{c}{$S_{z=1}({\bf x},\omega)$} & 
$ = $ &
$T_{z=1}^-({\bf x} ,\omega)$ & $\otimes $ & 
$T_{z=1}^+({\bf x} ,\omega)$ \\
\end{tabular}
\end{center}
\caption{Equivalence between direct migration of passive field data and
  simultaneous migration of many 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}
Figure~\ref{fig:directsum} shows direct arrivals from 
subsurface sources that will reflect from the free surface to 
provide the down-going energy to investigate the subsurface.  These
are the wavefront source functions to which I am referring with the
name ``wavefront imaging.'' Panel \pan{a}
is a single direct arrival from a source below the center of the model
domain.  Panel \pan{b} is the sum of direct arrivals from 30 subsurface sources 
clustered over 600~m beneath the center of the domain.  Arbitrary
time delays have been added to each source.  For clarity, no random
wavelet coda was added to the arrivals.  Panel \pan{c} shows the
wavefront sum of 25 subsurface sources evenly distributed across the
entire bottom of the model domain.  As more subsurface sources explode close
together in time, destructive interference progressively simplifies  
the resulting superposition to generate a plane wave at the limit.  The 
direct arrival is actually only a part of the appropriate source
wavefield required to image the subsurface.  It is the principle
component, but multiple reflections image subsurface reflectors as
down-going energy for the next higher-order multiple in the
transmission wavefield during migration.  This is why the bad
looking synthesized gathers in Figure~\ref{fig:dat1200} were able to
image the subsurface correctly in Figure~\ref{fig:mig.shot1200}.  The
importance of the multiple reflections in the source wavefield will
also be discussed in the interpretation of the images in
Figure~\ref{fig:dmig1200}.  
\activeplot{directsum}{width=\textwidth}{CR}{a) Direct arrival of energy from a 
  source below the center of the model domain.  b)  Wavefronts due to 
  the direct arrivals from 30 subsurface sources spanning 600~m
  around the center of the domain.  c) Direct arrivals of 25
  subsurface sources evenly distributed across the bottom of the
  domain.  Arbitrary time delays were added to each direct arrival
  wavefront in panels  b) and c).} 
\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, images are also
produced using the direct arrival from subsurface sources for the
down-going source wavefield and horizontal plane-wave migration.  Some
figures from above are presented again next to their direct migration
counterpart to facilitate comparisons between the two approaches.
\par
Figure~\ref{fig:dmig1200} shows images produced with several processing
strategies all beginning with the transmission wavefield due to a single 
impulsive source at the left edge of the model domain.  Panel \pan{a} is the 
direct migration result.  As can be expected, events are much stronger
on the left edge of the model above the subsurface source.  The 
entire domain is imaged fairly well, and the imperfections are easily 
identified.  Because the right side of the domain is so weak however,
it is probable that this structure would be below the noise threshold 
in a field experiment with weak, unidentifiable sources.
\activeplot{dmig1200}{width=\textwidth}{CR}{a) Direct migration of a transmission
  wavefield due to an impulsive subsurface source at the left edge of
  the model domain. b) Migration of synthesized gathers produced with the 
  same transmission wavefield.  c) Image produced with the
  transmission wavefield as data and only the direct arrival as the
  source.} 
\par
Figure~\ref{fig:dmig1200}\pan{b} is the same image shown in 
Figure~\ref{fig:mig.shot1200}\pan{a} (migration of synthesized
gathers) for comparison to the direct migration result, 
panel \pan{a}.  Panel \pan{b} shows a slightly
different image than the direct migration to its left.  The images
were gained identically.  There is marginally less cross-hatched noise
on the right side of the image in panel \pan{b}, but the reflectors
are not as strong as in panel \pan{a} on the right side of the
domain.  The migration of correlations, \pan{b}, enjoys a slightly
higher frequency content.  All events, artifacts, and edge effects in
the imaging results are identifiable in both images.  Both images are
incomplete results since only a single transmission wavefield was
processed.  The image produced by direct migration of 280
traces is over two orders of magnitude less expensive to produce than migrating
the $280^2=78,400$ traces in the volume of synthesized gathers.  This
cost difference does not include the cost of calculating the
correlations required to compute the synthesized gathers, which is
almost the same as the cost of a single direct migration.
\par
Figure~\ref{fig:dmig1200}\pan{c} was produced using the same transmission 
wavefield due to a single subsurface source below the left edge of the 
domain used for input in panels \pan{a} and \pan{b}.  This image was created
using the total transmission wavefield for the the up-going (data) 
wavefield.  The down-going (source) wavefield was only the direct
arrival from the subsurface source.  This scenario is directly
analogous to imaging with teleseismic arrivals \cite{SAW}.  The image
has less cross talk noise than panel \pan{a}.  However, the right edge
of the model and the second reflector are not imaged as well.  Recall
that the synthesized gathers on the right edge of the domain produced
with this transmission wavefield, Figure~\ref{fig:dat1200}\pan{c},
contain energy from only high-order multiples.  Therefore, the
appropriate source wavefield for migrating a single transmission
wavefield is not simply the direct arrival, but should also include
primary and multiple events. Using the entire transmission
wavefield as a source function produces cross talk in an image due to
a single transmission wavefield.  However, using only the direct
arrival as a source function does not fully utilize all of the events
in the transmission 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.  Panel \pan{a} shows the
direct migration of the sum of the two impulsive transmission
wavefields.  The result in panel \pan{b} includes randomized source
functions with the same summed transmission wavefield.  The two
transmission wavefields were imaged independently, by direct
migration, and summed to produce the result in panel \pan{c}.  Panel 
\pan{d} is the result of migrating the correlations produced with the
summed transmission wavefield using randomized wavefields
(Figure~\ref{fig:mig.shot1200.3500}\pan{d}).  The sum of 
direct migrations, panel \pan{c}, is clearly the best image, as
dictated by theory.  The main problem in the image 
is partial illumination.  There is very little difference 
between the upper two images (with and without randomized 
wavelets).  The comparison between panels \pan{b} and \pan{d} shows
that there is very little difference between the direct migration,
\pan{b}, and the migration of the synthesized correlations, 
\pan{d}.  There is a small phase roll between the two images, but 
the two methods provide roughly equivalent imaging capability.  The
left side of the second reflector is a little more interpretable in
\pan{b}.
%%\sideplot{dmig1200.3500}{width=4in}{a,b) Direct migration of two
\activesideplot{dmig1200\DOT 3500}{width=3in}{CR}{a,b) Direct migration of two
  summed transmission wavefields without and with randomized source
  functions.  c) Sum of images produced by direct migration of
  individual transmission wavefields.  d) Migration of synthesized
  gathers produced from the summed transmission wavefield.}
\par
Figure~\ref{fig:dmigall} presents images produced with various
processing strategies, but always using all 225 transmission wavefields
from sources across the entire bottom of the model domain.  Panel \pan{a}
was produced 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 wavefront very close to a horizontal plane wave.  The
data for direct migration used to produce 
the image in panel \pan{b} and long image in panel \pan{e} included
the addition of randomized source functions.  Panel \pan{e} is the
same as panel \pan{b} with an extended depth axis to show more of the
deep artifacts.  The second reflector stands out more clearly, and the
multiple has been weakened in comparison to panel \pan{a}.  Some
coherent cross talk events have been introduced however.  There is an
event at $z=300$~m, a mirrored event below the flat-bottomed syncline,
and many strange events in the deep image.  The deep events are due to
Fourier-domain wrap-around on the time axis.  These can be simply
removed by truncating the depth axis below the deepest reflector of
interest similarly to truncating the correlation lag axis after the
two-way traveltime to the deepest reflector of interest when
synthesizing shot gathers.
\activeplot{dmigall}{width=\textwidth}{CR}{a) Direct migration of the
  transmission wavefield due to all 225 subsurface sources.  b,e) Direct
  migration of summed transmission wavefields including randomized
  wavelets.  c) Sum of images produced by direct migration of
  individual transmission wavefields.  d) Horizontal plane-wave
  migration of synthesized gathers produced by correlating individual
  transmission wavefields.}
\par
Figure~\ref{fig:dmigall}\pan{c} shows the sum of the direct migrations
of individual transmission wavefields.  This result is very high
quality and conforms to the requirement of independently processing
each transmission wavefield.  Panel \pan{d} shows the image produced
by horizontal plane-wave migration of the gathers individually correlated
(Figures~\ref{fig:mig.all}\pan{a},\pan{b}).  This image suffers from
steeply-dipping noise events, and the reflectors are weaker compared
to panels \pan{b} and \pan{c} despite all images being gained to the
same level.  The amplitudes in panels \pan{a} and \pan{d} are close to
the same level.  The source summations applied to the data used for
these images discards almost all energy but vertically propagating
plane waves.  This limitation on the spectrum of plane wave
constituents diminishes the quality of the imaging results.
\par
Figure~\ref{fig:allcomp} presents comparisons between direct
migration and the migration of synthesized gathers using modeled
transmission wavefields below the entire domain.  The top row of
images was produced by independent processing of wavefields due to
impulsive sources.  The bottom row used a single transmission
wavefield as input, which was produced by summing all the modeled
wavefields due to randomized source functions.  The left column images
are direct migration results, and the right column images are
migrations of synthesized shot gathers.  Panel \pan{b} is a
reproduction of Figure~\ref{fig:mig.all}\pan{c}.  The direct migration
result in panel \pan{a} is nearly indistinguishable from the migration
of synthesized gathers in panel \pan{b}.  There are some minor
amplitude differences and the migration of synthesized gathers,
\pan{b}, seems to have a slightly broader wavelet.  This minor
difference could be caused by creating a source wavelet for use in the
down-going wavefield, $S$, with slightly different characteristics
than the wavelet resulting from the correlations.  I used a standard
Ricker wavelet for all migrations conforming to the conventional shot
profile migration.  The number of modeled transmission wavefields used
to produce image \pan{a} was almost the same as the number of virtual
shots (and receiver) locations to produce image \pan{b}.  Thus, the
cost of the two migrations is roughly equivalent.  However,
synthesizing the gathers used as input for the migration to produce
panel \pan{b} requires substantial preprocessing that is not required
for direct migration. 
%%\sideplot{allcomp}{width=4in}{Left column shows direct migrations,
\activesideplot{allcomp}{width=3in}{CR}{Left column shows direct migrations,
  while right column shows migrations of synthesized gathers.  Top row
  processed impulsive transmission wavefields independently, while the
  bottom row used a single transmission wavefield generated by summing
  the transmission wavefields due to randomized subsurface sources
  distributed densely below the entire model domain.}
\par
Figures~\ref{fig:allcomp}\pan{c} and \pan{d} compare direct migration
(left) to the migration of synthesized gathers (right) using a single
transmission wavefield due to subsurface sources below the entire
domain.  These experiments represent processing the passive data
from an entire data collection campaign simultaneously.  The image in
panel \pan{d} took 123 minutes to produce, while the direct migration 
image took only 0.41 minutes: a cost ratio of 300.  Again, the images
are almost indistinguishable.  All the main features in the 
images are directly comparable, though the events in the migration of
correlations look to be of marginally lower frequency content.  
\par
Minimizing the number of subsurface sources active within a single
transmission wavefield reduces the amount of cross talk introduced to
the final image.  The rows of Figure~\ref{fig:allcomp} compare images
produced by processing 225 transmission wavefields due to single
subsurface sources to those produced by processing a single
transmission wavefield due to 225 subsurface sources.  Figure~\ref{fig:migsums}
presents images produced with a scenario between those two end-member
cases.  Figure~\ref{fig:migsums}\pan{a} is a transmission wavefield
due to five subsurface sources.  The 225 available transmission
wavefields were summed in groups of five to generate 45 similar
transmission wavefields.  The five subsurface sources in each summed
wavefield were all spaced 1125~m apart.  All transmission wavefields
were convolved with random source functions before summation.  Panel
\pan{b} was produced by direct migration of the 45 summed transmission 
wavefields.  For comparison, panel \pan{c} is the result of summing the
images produced by direct migration of all 225 individual transmission
wavefields with randomized source functions.  There are some weak
steeply-dipping noise events crossing the bottom of the second
syncline in \pan{b}.  The similarity between the images suggests that the
assumption that transmission wavefields due to single sources must be
processed independently is more restrictive than it needs be in
practice.  The phase-encoding development of processing passive
seismic data provides the justification for this observation.  The
wavelet functions applied to the subsurface sources before
summation are meant to model the differentiation of individual sources
that can be expected in a real experiment.  These natural encoding
functions decorrelate the events within the transmission wavefields
such that the cross talk induced by combining the wavefields is
dampened.  Spatial separation of the subsurface sources is also an
effective natural encoding function.
\activeplot{migsums}{width=\textwidth}{CR}{a) Transmission wavefield due to five
  subsurface sources with randomized wavelet functions.  b) Sum of 45
  images produced with direct migration of wavefields due to five
  subsurface sources with randomized wavelets.  c) Sum of 225 images
  produced with direct migration of wavefields due to single
  subsurface sources 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 and wavelet, and time delays
to the source functions.  The top row, \pan{a} and \pan{b}, was produced
by direct migration: Source and data wavefields were identical.  The
bottom row, \pan{c} and \pan{d}, was produced 
using the sum of all direct arrivals as the source wavefield and the
same data wavefield as used to create the images above them.  Both
migration strategies incur the same computational cost, as they
differ only in the initial conditions of the source wavefield.  The
top row used the data itself as the source wavefield, while the bottom
row results required knowledge of the up-coming source energy to model
an appropriate source wavefield.  Without the multiples in the source
wavefield however, the bottom row of images do not image the
structure as well as the images produced by direct migration above
them.
%%\sideplot{dmigsmall}{width=4in}{Images produced from a transmission
\activesideplot{dmigsmall}{width=3in}{CR}{Images produced from a transmission
wavefield due 30 source clustered around the center of the bottom 
of the model domain.  Top row produced by direct migration.  Bottom
row produced using the direct arrivals for the source wavefield
in the migration.  Wavefields used for results in the left column had
impulsive wavelets, while those used for the right column had randomized
wavelets.}
\par 
Figure~\ref{fig:smallcomp} shows comparisons between direct
migration and the migration of synthesized gathers.  It reproduces the
images from Figure~\ref{fig:mig.shotsmall1} and the top row of
Figure~\ref{fig:dmigsmallsmall}.  The transmission wavefield 
used for this example was the sum of 25 modeled transmission
wavefields due to sources distributed evenly beneath the model
domain.  The left column used data due to impulsive wavelets, while
the right column used data due to randomized wavelets.  The top row
were produced by migrating the correlations.  The bottom row
was produced by direct migration.  The two rows of images are largely
comparable.  There is a minor phase roll between the images produced
with the two different processing algorithms.  The second reflector,
is more continuous and less effected by noise in the direct migration
images, panels \pan{c} and \pan{d}.  
%%\sideplot{smallcomp}{width=4in}{Comparison of processing
\activesideplot{smallcomp}{width=3in}{CR}{Comparison of processing
  strategies to image a transmission wavefield due to 25 sources
  distributed evenly across the bottom of the domain.  Top: Migration
  of correlations.  Bottom: Direct migration of the summed 
  wavefield.  Wavefields used for results in the left column had
  impulsive wavelets, while those used for the right column had
  randomized wavelets.}
\par
Figure~\ref{fig:dmigsmallsmall} shows images produced with the sum of
transmission wavefields due to 30 subsurface sources clustered around
the center of the model domain.  This experiment represents the
situation in field data were the predominance of subsurface source
energy is localized in a small area, such as might be expected in the
case of induced fracturing.  The left column used wavefields with impulsive
source functions, while the right column introduced randomness to the
length and wavelet, and time delays to the source functions.  The top row, 
\pan{a} and \pan{b}, was produced by direct migration, and the bottom
row was produced using the summed direct arrivals as the source
wavefield for the migration.  Panel \pan{b} shows a better result,
with the introduction of randomized source function, than panel
\pan{a}.  There is less energy in the noise, and the structure is better
illuminated at the edges of the model domain.  Panel \pan{b} shows a
result with a modeled data volume that can readily be expected to
provide insight into real situations encountered with field data
recordings. 
%%\sideplot{dmigsmallsmall}{width=4in}{Images using summed transmission
\activesideplot{dmigsmallsmall}{width=3in}{CR}{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 plane-wave migration using synthesized gathers
  without and with randomized source functions.}
\par
Figure~\ref{fig:dmigsmallsmall}, in contrast to Figure~\ref{fig:dmigsmall} 
above, shows a much more marked difference
between the images in the two rows.  For the case of a localized
cluster of subsurface sources, using the direct arrivals as the source
wavefield for migration (\pan{c} and \pan{d}) has produced much worse
images.  When all the subsurface sources contain the same wavelet
function and are perfectly in phase, the result using the summed direct
arrivals as the source function for migration, \pan{c}, images a tiny
fraction of the model domain.  The randomized source functions used to
produce panel \pan{d} move the individual transmission wavefields up
and down the time axis such that the destructive interference leading
to the poor result in panel \pan{c} is not as severe.  Direct
migration (panels \pan{a},\pan{b}) provides the highest quality images
with passive data due to a localized cluster of subsurface sources as
compared to using only the direct arrivals as the source function for
migration.
\par
Figure~\ref{fig:smallsmallcomp} shows comparisons between direct
migration and the migration of synthesized gathers.  It reproduces the
images from Figure~\ref{fig:mig.shotsmall2} and the top
row of Figure~\ref{fig:dmigsmallsmall}.  The summed
transmission wavefield used for these images was the result of 30
subsurface sources below the center of the model domain.  The results
in the left column used data due to impulsive wavelets and those in
the right column used randomized wavelets.  These images roughly
reinforce the observations made in analysis of
Figure~\ref{fig:smallcomp}.  The migration of correlations 
(\pan{a},\pan{b}) are marginally poorer than the direct migration
results (\pan{c},\pan{d}).  The ringing noise in the center of panel
\pan{a} is not as problematic in panel \pan{b}.  The most realistic
data scenario with respect to potential field data (randomized
wavelets) results in little noticeable difference in the quality
of the images produced with the two imaging methodologies.  As noted
above in reference to Figure~\ref{fig:dmig1200}, however, the direct
migration result was 300 times less expensive to compute.
%%\sideplot{smallsmallcomp}{width=4in}{Comparison of processing
\activesideplot{smallsmallcomp}{width=3in}{CR}{Comparison of processing
  strategies to image a transmission wavefield due to 25 sources
  distributed evenly across the bottom of the domain.  Top: Migration
  of correlations.  Bottom: Direct migration of the wavefield.  Left
  column summed wavefields as modeled, while right column convolved
  each wavefield with a random wavelet before summing.}

\section{Discussion}
When passive data does not complete the requirements of the rigorous
development of seismic interferometry, equation~\ref{rigor}, the
synthesized gathers may be misinterpreted or uninterpretable.  The
results may even be misleading enough that they are discarded.  For
the case of imaging the subsurface with only a single 
subsurface source, Figure~\ref{fig:mig.shot1200} showed that
synthesized gathers that seem completely devoid of reflections
corresponding to the kinematics of active seismic shot gathers
still contribute significantly to imaging the subsurface via 
migration.  The synthesized gathers with virtual shot locations at far
horizontal distance from the subsurface source were completely devoid
of interpretable events and do not provide a useful data volume with
which to assess the quality of the partial result.
\par 
The cost of synthesizing shot gathers compared to the cost of
direct migration of a single transmission wavefield is almost the same
for the modeled data presented above.  For the same 
computational cost, the rows of Figure~\ref{fig:disccomp} show the
two results possible when processing a single transmission wavefield.  The
left column shows a single synthesized shot
gather, where the virtual source location is indicated by the vertical
line.  The right column shows direct migration results.  The top row
are results obtained by processing a transmission wavefield which
contained subsurface sources clustered around the center of the
model domain. The subsurface sources were evenly distributed
beneath the domain in the transmission wavefield used to produce the
bottom panels.  
%%\sideplot{disccomp}{width=4in}{Synthesized shot gathers (left) and 
\activesideplot{disccomp}{width=3in}{CR}{Synthesized shot gathers (left) and 
direct migration images (right) generated at the same computational
cost using transmission wavefields due to a cluster of subsurface 
sources below the center of the domain (top) and a even distribution
of sources below the entire domain (right).}
\par
The synthesized gathers due to multiple subsurface sources in
Figures~\ref{fig:disccomp}\pan{a},\pan{c} are uninterpretable.  For
this scenario, the images in panels \pan{b},\pan{d} (and Figures~\ref{fig:allcomp},~\ref{fig:smallcomp}, 
and~\ref{fig:smallsmallcomp}) show that migration produces very good
results despite the poor quality of the synthesized gathers.  Images
produced by direct migration or migrating the correlated gathers are
very similar in quality.  When immersed in the lower signal-to-noise
regime likely for field data, the subtle differences would likely be hard to 
find.  However, because the synthesized gathers in Figure~\ref{fig:disccomp}
are not useful, creating the synthesized gathers with these transmission 
wavefields is a useless intermediate processing step that still
requires an expensive migration to produce images like those next to
them produced by direct migration.  If the synthesized shot gathers
provide a useful domain in which to address noise or aliasing
problems, this expense could be justified.  However, it may be impossible
to know what subsets of the energy in the gathers contribute to the image
positively or negatively.  Direct migration images of time windows from a
passive recording campaign are cost effective and easily interpretable 
intermediate results that can be produced for quality control purposes 
for transmission wavefields containing either a single subsurface source 
or multiple sources.
\par
The ratio of the cost to migrate a single transmission wavefield to that 
of preparing synthesized gathers from the data
depends, most significantly, on the number of traces in the
transmission wavefield and the number of depth samples in the image
space.  Figure~\ref{fig:cost2} shows the expense ratio as function of
the number of geophones, $g$, and the number of time samples, $t$, in the
transmission wavefield and the number of depth levels, $z$, in the
image domain.  For a single transmission wavefield, the curves show
the ratio of the cost to synthesize gathers, $C_{syn}$, over the
cost of direct migration, $C_{dm}$
\begin{equation}  \label{cost2}
\frac{C_{syn}}{C_{dm}}=\frac{(gt\log t +g^2t\log t)
  +2tg^2}{gt\log t+Mztg\log g}\;.
\end{equation}
The terms in parenthesis are forward and inverse Fourier
transform costs.  The last term in the numerator is the cost of
correlating the traces.  The first term in the denominator is another
Fourier transform over time, and the second term is the migration
cost, which is proportional to a Fourier transform over the trace
variable $g$.  $M$ is a cost multiplier representing migration expense, which
is usually dominated by the accuracy of the extrapolation operator
used and the calculation of common image gathers.  The
migration cost multiplier, $M$, for my implementation of SSF-PI shot
profile depth migration to produce zero subsurface-offset images is
about 5.  
\par
The three groups of curves in Figure~\ref{fig:cost2} are associated with 
the size of the image domain $z=(60,120,700)$.  The cost of migration
with a given number of receivers, $C_{dm}$, is dominated by the size
of the image space.  Therefore, the lowest group of curves used
$z=700$ for the number of depth samples in the image space.  The synthetics
presented above used 120 depth samples for migration.  The images
using the Valhall data I present in the next chapter use about 
700 depth samples.  The three curves within each group correspond to
the number of time samples within the input wavefields,
$t={1024,2048,8192}$.  The dotted line is the ratio when the
transmission wavefield contains 8192 time samples, and the solid line
is due to 1024 time samples.  The cost of creating the gathers with a given
number of receivers, $C_{syn}$, is dominated by the number of time
samples in the wavefield.  Therefore, the highest curve in each group
is associated with the largest number of time samples.  Since
migration is the most expensive operation, the number of depth samples
in the image space has the most drastic effect on the cost ratio.  It
is interesting that for both the synthetic and the Valhall data, the
combination of the number of receivers in the data and the number of
depth samples in the image happen to result in a cost ratio $\sim 1$.
%%\sideplot{cost2}{width=4in}{Cost ratio of processing a single
\activesideplot{cost2}{width=3in}{NR}{Cost ratio of processing a single
  transmission wavefield by synthesizing shot gathers or imaging with
  direct migration.  Cost ratio values greater than $10^0$ indicate
  that it is more expensive to synthesize shot gathers.}
\par
When passive data is collected over a long period of time, the best
way to process the volume is to work with short time windows from the
total data set.  This provides more reasonably sized data volumes to
process, and will minimize the cross talk in the results associated
with processing transmission wavefields due to multiple sources.  When 
strong identifiable events are present, such as teleseismic arrivals
or local explosions, it may be relatively easy to collect a small
number of time windows from the total volume of data.  Otherwise, the
passive seismologist must window the data into arbitrary sections to
minimize the number of subsurface sources captured within an
individual transmission wavefield.  The length and number
of such time windows from a long recording campaign that can be
efficiently processed can be determined by analyzing the relative 
computational costs incurred by processing the data with different
strategies.
\par
Figure~\ref{fig:cost1} shows the computational cost ratio of migrating
synthesized shot gathers compared to direct migration as a function of
the number of time windows processed.  Assuming time windows 15~s long,
$10^5$ windows is 17.36~days.  The two curving lines
represent processing transmission wavefields with 2800 (dashed) and 280
(solid) traces.  These numbers correspond roughly to the
size of the Valhall data and the synthetic used herein.  The ordinate
of the graph is the ratio of the cost to create and migrate
synthesized gathers, $C_{msyn}$ over the cost of direct migration, $C_{dm}$, 
\begin{equation} \label{cost1}
\frac{C_{msyn}}{C_{dm}}=
\frac{w(gt\log t + 2tg^2) + g^2t\log t + Mztg^2\log g}
{w(gt\log t+Mztg\log g)}\;,
\end{equation}
where the independent variable $w$ is the number of time windows
to process.  This cost ratio, as compared to the previous, contains a
migration cost in the numerator.  However, this is a fixed cost that
is not a function of $w$ since the synthesized gathers can be migrated
once after they have been summed.  Likewise, the inverse Fourier
transform of the synthesized gathers need only be performed one 
time.  The entire denominator, the cost of direct migration, is
multiplied by the number of time windows, $w$.  This equation
simplifies to approximately $g/w$ for small numbers of time windows
since the migration dominates the operation count to produce an image
and the variables $t$ and $z$ appear on both sides of the ratio.  The
thin solid lines plot $g/w$ for the two transmission wavefield sizes,
$g={280,2800}$.  These asymptotic lines cross $C_{msyn}/C_{dm}=1$ when
$g=w$.  The departure of the cost curves from the asymptote is due to
the $2tg^2$ cost of correlation in the numerator of
equation~\ref{cost1}.  For an array the size of Valhall 
(upper solid line) it is less expensive to process with direct
migration until the number of time windows processed is approximately
$10^4$.  Processing 15~s time windows, more than 40 hours of data need
be available before it is more cost effective to produce a subsurface
image by first synthesizing shot gathers.  
%%\sideplot{cost1}{width=4in}{Cost ratio of processing many transmission
\activesideplot{cost1}{width=3in}{NR}{Cost ratio of processing many transmission
  wavefields by synthesizing shot gathers or imaging with direct
  migration.  Cost ratio values greater than $10^0$ indicate that it
  is more expensive to create and migrate synthesized shot gathers.} 
\par
The curves in Figure~\ref{fig:cost1} both approach a horizontal
asymptote.  The solid lower curve, calculated with a 280 receiver array,
reaches an asymptote of 0.1.  However, processing data from such a
small array takes so little time with modern computers, that 
either processing selection is tractable.  The dashed upper curve
was calculated assuming a number of receiver stations similar to the
Vahall array.  The horizontal asymptote for the large array is
$C_{msyn}/C_{dm}=0.75$.  This is likely not a severe computational
performance cost if synthesized gathers are never made.  This also
eliminates storage of the $g^2$ traces in a correlation volume and 
produces continuous intermediate images that can be evaluated for 
quality control purposes.  A parallel curve with horizontal asymptote 
of $10^0$ can be computed by assuming 4000 sensors in a hypothetical 
array.
\par
These cost functions assume that all time windows of data
are equally important.  If that is not the case, it would be
beneficial to produce image volumes more often than one time to
determine if the passive data recorded were useful.  Producing such
intermediate volumes quickly increases the computational burden of
migrating synthesized gathers.  Producing a few gathers
with each time window does not have significant impact on the cost 
ratio.  However, a second migration of the gathers approximately doubles 
the number of time windows that can be processed less expensively by
direct migration.  No matter how many intermediate migrations are
produced, the cost ratio approaches the same horizontal asymptote.

\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 input data volume, or transforms it to an 
obvious character.  Transmission wavefields are inherently imperfect 
data volumes compared to those produced with active seismic 
experiments.  Migration greatly simplifies the complexity and volume
of data to interpret, provided a reasonably accurate velocity
model is available.  While this could be a potential disadvantage, 
the expense related with mounting a passive recording campaign will,
for the near future, limit passive acquisition to fairly well understood
areas where a good velocity model is likely available.  If not,
a reasonable initial model can be constructed and then refined through 
iterative methods.  While the discussion above was limited to zero 
subsurface-offset, both migration methods (direct migration and 
conventional migration of synthesized gathers) can produce images
with complete common-image gathers as a function of subsurface offset
\cite{GEO67-03-08830889,SAW} by generalizing the imaging condition,
equation~\ref{ic}.  Doing so introduces the possibility of residual
migration to correct the image for errors in the migration velocity
model \cite{biondi:1283}.
\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.  Even though coherent noise 
events remain in the synthesized gathers, the migrated images do not 
suffer substantially from their presence.  When multiple sources are 
captured in a transmission wavefield during the recording of a passive 
seismic experiment, the cross talk associated with correlation can lead to 
misinterpreting the shot gathers.  The same is true when only a single 
subsurface source in captured in the transmission wavefields.  This 
misinterpretation of the subsurface is minimized, though not mitigated 
completely, by migration.  
\par
Correlation removes a round trip from depth to the surface.  Therefore,
the direct arrival is the appropriate areal/temporal source function
to images the the first-order multiples.  Similarly, the first-order
multiples correctly image the subsurface structure as the source
energy for second-order multiples, and so on.  If high-order multiples
are recorded in the transmission wavefield, these events are important
contributors to imaging reflectors at distant horizontal location from
the subsurface source.  If the sources are weak, and attenuation of
the energy does not allow recording high-order multiple reflections
from the reflectors at depth, synthesized shot gathers and direct
migration images will not image structure at large horizontal distance
from the subsurface source.
\par
Rather than framing the correlation processing of transmission
wavefields as an interferometric calculation of Green's functions, I 
believe that analyzing the use of transmission data as an
imaging problem provides an intuitive and simple framework: direct
migration.  The experiments presented above show that processing
transmission wavefields as an imaging problem within the framework of
shot profile depth migration is a valid strategy akin to phase-encoded
migration or a repeated datuming exercise.  The images created by
correlation followed by migration are very similar to those created by
direct migration.  Correlation and extrapolation are
commutable operators, and we are free to choose a strategy most
convenient for the task, or data, at hand.  If passive data are
available from an area with no knowledge of the subsurface velocity
field, it may be necessary to correlate the records before 
migration for velocity analysis.  Synthesized gathers may also provide a
convenient domain to identify and remove coherent noise.  However, even if 
the synthesized gathers seem to show little or no correct information 
about the subsurface, migrating the data may provide surprisingly good results.
\par
If a velocity model is available, and migrating the data is a foregone
decision, direct migration can save substantial computational 
cost.  Because depth migration is an expensive operation, minimizing
the volume of input data drastically effects the total cost to process
a data set.  The cost comparisons discussed above show that if the raw data 
can be windowed into fewer transmission wavefields than approximately 4 times 
the number of deployed receivers, it is always less expensive to migrate the 
raw data.  Of course these ratios depend on the parameters explained
above and the break-even point will decrease by increasing migration
costs.  These cost comparisons are true however for computing only one
inverse Fourier transform and migration of the $g^2$ traces in the
volume synthesized gathers.  If intermediate volumes of gathers are
viewed in the time domain and/or migrated, the number of time windows
necessary to make direct migration more expensive increases almost
linearly with the number of intermediate results produced.
\par
For a passive experiment of the size of the Valhall array,
the added computational burden of processing more time windows by
direct migration than the calculated break-even point reaches an
asymptotic limit of 1.3 times the cost of migrating synthesized
gathers.  Direct migration of data from an array with 4000 channels 
is always less expensive or equivalent to migrating synthesized
gathers with these parameters.  Direct migration of data from larger
arrays is always the less expensive option.
\par
With current computer technology, a single transmission wavefield collected
with a large array takes approximately one hour to image with direct
migration.  Assuming that transmission wavefields 15~s long are windowed 
to processes independently, a single day of passive records results in
5760 time windows.  Therefore, a cluster of 240 computers can process
passive data in real time.  The images presented above showed that direct migrations
of transmission wavefields due to multiples sources are still interpretable,
with often minor imperfections.  Therefore, the number of windows that can
be summed before direct migration can likely be increased by several times
more to further reduce the computational cost.  Summing four
transmission wavefields before processing reduces the number of
processors to $240/4=60$ to produce images with minimal cross talk
artifacts associated with imaging data due to multiple sources.
\par
The cost ratios presented above implicitly assume that all windowed
transmission wavefields add equal contributions to the final 
image.  For field data this is probably not the case, as much 
of the records may be useless background/instrument noise between 
episodes providing useful data.  Summing data windows containing only
background noise will decrease the signal to noise ratio of the
product.  However, it is unrealistic to view 5760 results per day for
quality control purposes.  Therefore, a summation is required either
before or after imaging.  Therefore, I view decreasing computation costs
by some level of summation prior to processing to be a reasonable
compromise.  Hopefully, balancing the local rate of seismicity and the
available personnel and computer resources can be facilitated with the
analysis presented above.

\clearpage
\label{.end}
