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\chapter{Correlation processing experiments}\label{chapter:TimeSynth}
\section{Summary}
Experiments with 2D transmission wavefields modeled with sources across
the bottom of a model domain show that correlation processing can synthesize
the important kinematics in active seismic surveys.  These transmission 
wavefields are limited to nearly vertically propagating body waves,
and result in some noise events in the synthesized gathers due to
incomplete destructive interference required for a perfect result.
Correlating transmission wavefields due to multiple subsurface sources
leads to cross talk that interferes with the desired result linearly
with the number of sources contained within the recording.

\section{Introduction}
Cross-correlation of traces from transmission wavefields, followed by summing
the individual results, can synthesize a data volume with the same kinematics as 
active seismic experiments.  The two main assumptions invoked in developing 
the procedure are concerned with the temporal and areal distribution of 
the subsurface sources which enervate the subsurface domain beneath the 
array of recording stations.  The success of the effort to
image the subsurface with truly passive data is predicated on the strength,
number and distribution of the sources captured during the recording
campaign.  Therefore, the most important investigations into how well 
such an experiment can work in practice deal with the distribution 
and character of the subsurface sources within the data.
\par
I use a suite of 2D synthetic transmission wavefields to analyze the 
efficacy of correlation processing under various source 
configurations.  Figure~\ref{fig:data} shows two representative
transmission wavefields and the velocity model with which they were
generated.  Imaging the subsurface structure with body waves is 
the goal of this study of the details of processing transmission 
wavefields.  Therefore, I use only modeled transmission wavefields due to 
sources located directly beneath the model domain.  All 
surface-related and internal multiples are included, and the transmission 
wavefields have been flux-normalized so we can neglect attenuation 
\cite{draganov:90}.  The model contains two synclines in an otherwise flat 
three layer velocity model.  Impulsive subsurface sources were modeled 
every 25~m across the bottom of the model at the same depth level.  The 
data contain 280 receivers at the surface, axis ${\bf x}$, spaced 20~m apart.  
%%\plot{data}{width=\textwidth}{Representative transmission wavefields 
\activeplot{data}{height=2.5in}{ER}{Representative transmission wavefields 
(\pan{a},\pan{b}) from impulsive subsurface sources through a three 
layer velocity model with two synclines (\pan{c}).  All surface-related 
and internal multiples are included.  Asterisks show subsurface source 
locations.}
\par
Figure~\ref{fig:data}\pan{a} shows data from a source at the left edge
of the model domain, whereas panel \pan{b} is a  transmission wavefield
due to a source directly beneath the center of the model domain.  The
asterisks at the bottom of the panels indicate the location of the
subsurface sources.  The first hyperbolas are direct arrivals and the
rest are reflections due to the structure within the velocity model.  The 
third event, with minimum travel time 1~s, is the reflection from the 
second layer in the model.  Subsequent events are surface related and
internal multiples.  Panel 
\pan{c} is the velocity model with which the transmission wavefields
were modeled. 
\activeplot{actshots}{width=\textwidth}{CR}{Two active reflection seismic shot
  gathers.  These same shot locations, out of the total volume, are used 
  in the forthcoming figures showing synthesized gathers.  Events in the 
  right panel dipping down toward the center of the frames are reflections
  from the edges of the model domain.}
For comparison, Figure~\ref{fig:actshots} shows two modeled surface
seismic shot gathers (where the ``shot'' axis is also defined across
${\bf x}$) at locations in the model from which the
forthcoming synthesized active gathers are extracted.  For
clarity, I will use the term ``source'' to describe the subsurface
sources, and ``virtual shot'' in reference to the synthesized shot
axis developed by the correlation processing of transmission
wavefields.  
\section{Synthesized gathers}
Summing the correlations of traces within a transmission wavefield 
synthesizes active surface data through destructive and constructive
interference of the individual volumes.  At the limit of processing
only a single transmission wavefield due to one source, the product
is incomplete and the gathers produced do not represent active surface
data.  As more transmission wavefields are processed, the 
synthesized shot gathers more accurately model active seismic data.
\par
Figure~\ref{fig:allshots} shows synthesized active shot gathers (at locations
corresponding to previous figures) produced by summing the
correlations from subsurface sources across the entire bottom of the
model.  The front face of panel \pan{a} is not substantially 
different from the correlations of a single transmission wavefield in
Figure~\ref{fig:shot1200}.  This gather, from the left edge of the
domain, suffers from the limited distribution of subsurface sources
used to synthesize active surface data.  The quality of a gather is
predicated on up-coming energy reflecting down at a surface receiver
location, and then returning back to the surface after investigating
the subsurface.  With most of the subsurface sources located to the
right of this virtual shot location, only the source directly beneath the shot
location contributes to the synthesis of the gather.  This is why the
gather is of no better quality than Figure~\ref{fig:shot1200}\pan{a}.
\activeplot{allshots}{width=\textwidth}{CR}{Two correlated shot gathers from the
volume generated by summing correlations from 225 sources across the
bottom of the model.  The ``virtual shot'' axis is built by
correlating every trace in the data wavefield with all of the other
traces. }
\par
Figure~\ref{fig:allshots}\pan{b} is a gather from
the center of the model domain, which should be the most complete
gather in the synthesized volume.  Notice that on the right face
(time-virtual shot axes), the events develop a slight concave-up moveout 
toward the edges of the domain rather than approach an asymptote.  The
inflection points of the events around virtual shot locations $x=2700$
and $6700$~m represent the limits of accurately synthesized shot
gathers for this distribution of subsurface sources.  There is
substantial noise at early time.  Despite using transmission
wavefields from 225 sources across the bottom of the model, the
summation has not perfectly canceled this noise through destructive 
interference.  Modeling transmission wavefields only from the bottom 
of the model ignores the energy contributions from locations at the 
sides of the model needed for a perfect result.  
\par
Comparing Figure~\ref{fig:actshots} with Figure~\ref{fig:allshots} 
highlights several key differences between surface seismic data and
the synthesized gathers.  Obvious amplitude differences should be
ignored, as this work is focused on recovering only the kinematics of
the active experiment.  The direct arrival in the surface seismic
gathers interferes with events after about 2000~m offset, and is
largely absent in the synthesized gathers.  Because the transmission
wavefields do not contain much energy with high ray parameters, the
direct arrivals and refractions are not well represented in the
synthesized gathers.  The correlations in Figure~\ref{fig:allshots}
also have events that would be acausal relative to the direct arrival
of the surface seismic.  This energy can easily be removed by muting
before further processing.  The main events in Figures~\ref{fig:allshots}
and~\ref{fig:actshots} look very similar.  However, the synthesized
gathers suffer from substantial coherent noise due to not using
subsurface sources completely surrounding the domain of interest.  In
Chapter~\ref{chapter:DepthSyn}, migrated images produced with these
data volumes are presented to explore these effects in the image space.
\par
Figure~\ref{fig:shot1200} shows two slices through a 3D volume of
synthesized gathers produced by correlating all traces in a single
transmission wavefield (Figure~\ref{fig:data}\pan{a}).  The
trace under the vertical black line in the correlated shot gathers 
is the zero-offset trace in the synthesized gather.  It is also the
location of the source function, or master trace, from the
transmission wavefield for correlation to generate the synthesized
gather.  This volume is a partial processing of the source summation
required to synthesize active data from transmission data.  This
experiment also represents the situation where a powerful or repeated
source from a single subsurface location dominates the ambient
noisefield.  
\activeplot{shot1200}{width=\textwidth}{CR}{Two correlated shot gathers from the
volume generated with a single source below the left edge of the
model.}
\par
The time line at $t=0.5$~s is at the zero-offset two-way traveltime to
the second reflector.  The front face of
Figure~\ref{fig:actshots}\pan{a} shows the first trace in the
transmission data correlated with all other traces.  The 
correlation does not dramatically change the relative kinematics of
the events seen in Figure~\ref{fig:data}\pan{a}.  The first-order
effect is a simple subtraction of the time to the direct arrival in
the trace used as a virtual source function from each subsequent
trace.  Notice that the second primary event (beginning at 0.5~s on
the left edge of Figure~\ref{fig:shot1200}) crosses the right edge of
the gather at about $t=2$~s.  The same event in
Figure~\ref{fig:actshots}\pan{a} reaches the right edge of the gather
at about $t=3$~s.  These observations highlight the inaccuracies
incurred by not completing the integral in equation~\ref{rigor}.  
\par
The first event in Figure~\ref{fig:shot1200}\pan{a} that intersects
the $x,t$ axes at $(1200\;\mbox{m},0\;\mbox{s})$, is the manifestation
of the delta function, $\delta({\bf x},{\bf x}_s)$, in
equation~\ref{rigor}.  It is the time-shifted direct arrival.  The
next two events are the synthesized primaries.  The right face of the
panels have axes time and virtual shot position.  The reflections show
roughly hyperbolic acausal moveout on this section.  As traces further
to the right in the transmission wavefield are used as source
functions for correlation, the hyperbolic increase in the delay time
of the direct arrival moves the two primary events to negative time
after virtual shot position $x=3300$~m.  The front face of
Figure~\ref{fig:shot1200}\pan{b} contains only the late-time multiples
from the two layer model.  In real data, it is likely that these
high-order multiples would be much weaker than in this model or
completely absent.  If only a small number of subsurface sources
are captured in a passive seismic recording campaign, it is likely
that such synthesized gathers would be discarded.
\par

\section{Multiple sources}
If multiple sources are captured in a passive recording, processing
them together violates the rigorous theory of interferometry, and the 
resulting volume of correlations can be more complicated than the 
desired shot gathers.  It may be impossible to separate passive
data into transmission wavefields due to only a single subsurface
source if sources are weak and unidentifiable, fire simultaneously, or 
by processing long passive records as a single wavefield.  
\par
The same wavelet was used to model the transmission wavefields in
every case.  Therefore, events in summed transmission wavefields are
perfectly correlable.  Because the sources were placed at the same 
depth in the model domain, there is perfect constructive and destructive
interference when events overlap.  To model field data, $T_f$, composed of
two single source transmission wavefields, $a$ and $b$, a bulk time
shift $\tau$ can introduce time delays between the activation of the
sources.  Therefore, data modeling multiple sources has the form 
\begin{equation}\label{fieldT}
T_f(t)=a*\delta(t-\tau_a)+b*\delta(t-\tau_b).
\end{equation}
In general, capital variables represent Fourier domain wavefields, and 
lowercase variables represent their time domain equivalents.  The equation
above violates this convention simply to avoid confusion between the 
transmission wavefield and the time variable.  Time correlation of the 
transmission wavefield defined in equation~\ref{fieldT} performed in the 
Fourier domain is
\begin{equation}\label{tt}
T_fT_f^*= AA^* + BB^* +AB^*\mbox{e}^{-i\omega(\tau_a-\tau_b)} 
        +BA^*\mbox{e}^{-i\omega(\tau_b-\tau_a)} \;.
\end{equation}
The sum of the first two terms is a partial result of the summation 
over subsurface sources in equation~\ref{rigor}. The last two 
terms are cross talk.  Though passive recordings are likely very long,
the two-way traveltime to the deepest reflector of interest, $\max(t)$,
is the latest lag that need be maintained after correlation.  If
$|\tau_b-\tau_a|>\max(t)$, one cross talk term is acausal, and the other 
is at late lags that can be windowed away in the time domain.  If 
$|\tau_b-\tau_a|<\max(t)$, the cross terms are included in the
correlated gathers.  For the modeled transmission wavefields due to
impulsive sources at the same depth level, the phasors in the 
cross talk terms simplify to $1$ since $|\tau_b-\tau_a|=0$.
\par
Figure~\ref{fig:vshot1200.3500} represents the case where the energy
from two subsurface sources ($x=1200, 3500$~m) is processed
together.  In contrast, Figure~\ref{fig:vshot1200csum3500} shows the  
sum of correlations of the two transmission wavefields.  Thus, 
Figure~\ref{fig:vshot1200csum3500} is a partial result of the 
summation of correlations presented in Figure~\ref{fig:allshots}.

\activeplot{vshot1200\DOT 3500}{width=\textwidth}{CR}{Synthesized shot gathers
  produced by correlating the sum of transmission wavefields due to
  sources beneath the left edge and middle of the model.  Compare 
  artifacts within the ovals to the more correctly processed result
  in Figure~\ref{fig:vshot1200csum3500}}
\activeplot{vshot1200csum3500}{width=\textwidth}{CR}{Synthesized shot gathers
  produced by summing the correlations from sources beneath the left
  edge and middle of the model.  Ovals indicate locations of artifacts
  in the results shown in Figure~\ref{fig:vshot1200.3500}.}
\par
The differences in the time sections and time slices in 
Figures~\ref{fig:vshot1200.3500} and~\ref{fig:vshot1200csum3500} show the
inaccuracies associated with correlating the
sum of two transmission wavefields.  There are ringing concave-up half
hyperbolas in the top-left quadrant of the time-slices in 
Figure~\ref{fig:vshot1200.3500} (upper ovals) that are not present in
Figure~\ref{fig:vshot1200csum3500}.  A strong saddle shape
within the central oval overlay mimics the energy in the center of the
time slice.  The lowest oval on the left edge of
Figure~\ref{fig:vshot1200.3500}\pan{b} highlights strong half-hyperbolas 
that are not present in Figure~\ref{fig:vshot1200csum3500}\pan{b}.  Correlating 
the sum of the transmission wavefields causes all of the events due to
each source to be present in the result instead of inappropriate
events canceling due to destructive interference.  Both shot positions
in Figure~\ref{fig:vshot1200.3500} look very similar, and the noise is
symmetric across the zero-offset diagonal in the time-slices.  Note
the similarities in Figure~\ref{fig:vshot1200.3500} and the
correlations of a single subsurface source, Figure~\ref{fig:shot1200}.
\par
As more sources are processed together, such as a storm of
small subsurface sources during natural or induced fracturing, the
introduction of cross talk noise can overwhelm the 
correlations.  Figure~\ref{fig:manysrc}\pan{a}
shows the result of simultaneously processing 25 sources spaced every
125~m across the bottom of the model.  Although the entire model is
illuminated by this distribution, correlating the sum of transmission
wavefields leads to so much coherent noise that the gathers are 
uninterpretable.  
\activeplot{manysrc}{width=\textwidth}{CR}{Left: Synthesized shot gathers with 25
  subsurface sources spaced evenly across the bottom of the model.  Right:
  Synthesized shot gathers with 30 subsurface sources clustered under 
  the center of the model.}
\par
Figure~\ref{fig:manysrc}\pan{b}
was produced with 30 shots every 25~m clustered around the center of 
the model.  Clustered subsurface sources may be a common situation experienced 
with field data when energy from a localized area dominates the energy 
budget of the ambient noisefield.  The gather in the midst of the 
localized swarm of sources has an undesirable ringing 
character.  Concave-up moveout on the right 
face is indicative of the incomplete illumination of the domain by 
sources across only $600$~m of the model.

\section{Introducing randomness}
In the previous examples concerning processing transmission wavefields
containing multiple subsurface sources, the same wavelet was used in
every case.  Therefore, events in summed transmission wavefields are
perfectly correlatable.  While local sources probably share similar wavelet
characteristics, it is probably more realistic to vary the source
wavelets in the transmission wavefields.  Therefore, before
summing multiple wavefields, I convolve the individual transmission
wavefields with a random number sequence.  Random source wavelets will
act to decrease the correlation cross talk when it is not possible or
practical to perfectly separate the passive data into individual
transmission wavefields.
\par
Define two transmission wavefields $A$ and $B$ as the convolution of 
the impulse response of the Earth, $I_e$, and source functions, $F_{a,b}$, 
which contain bulk phase delays and wavelet characteristics.  As such, 
the cross terms associated with correlating their sum are 
\begin{equation}\label{cross}
AB^*=(F_aI_e)(F_bI_e)^*= F_aF_b^* \; I_e^2 = F_c \; I_e^2 \;.
\end{equation}
We see that the cross talk is the convolution of the Earth's impulse 
response (all associated reflections and multiples), with the correlation 
function, $F_c$, of the two source functions.  The correlation of the two 
source functions is not zero phase and could be a complicated function of 
space and time.  The cross talk events due to
$n_s$ sources will likely be more problematic than conventional
multiples as every multiple and reflection is repeated $n_s!/(n_s-2)!$ 
times.  The swarm of events in Figure~\ref{fig:manysrc} is caused by 
the repeated inclusion of every event due to each source in the 
result.  The ratio of desirable zero phase terms, $I_e^2$, to cross
terms, $F_jI_e^2$, decreases as $1/(n_s-1)$ if the source terms
are maximally correlable.
\par
If the source functions are random series, the cross talk terms
($F_cI_e^2$) within the gathers will decorrelate and diminish in
strength as the length of $f_i(t)$ and the number of cross terms
increases.  While we may hope to collect a large number of sources, it
is probably unreasonable to expect many of them to be perfectly random
series of great length, especially if the noise sources
have similar source mechanisms.  Previous authors have convolved
impulsive transmission wavefields with long random time series to
simulate passive data.  Such modeling implies long, continuously
ringing source functions, whose energy is distributed evenly over the
time axis.  I believe it is more appropriate to restrict the duration
of the wavelet (no longer than 1~s), use random amplitude terms, and
apply a bulk phase shift to  the entire record.  This modeling
simulates finite duration, random phase source functions, that can
happen at any moment within the entire recording interval.
\par
Figure~\ref{fig:vshot1200.3500.rand} shows the correlation of two
summed transmission wavefields with randomized wavelets.  The two
sources were located below the left edge and the center of the model
domain.  The figure is directly comparable to 
Figure~\ref{fig:vshot1200.3500} to show the influence of the
randomized source functions on the resulting synthesized 
gathers.  The ovals drawn on the figure are the same as those on the
comparison figure.  The noise identified within the ovals in 
Figure~\ref{fig:vshot1200.3500} is mostly absent.  The half hyperbolas
identified by the lowest oval in Figure~\ref{fig:vshot1200.3500}
has now moved later in time as indicated by the arrow.  These are the
time shifted events from the source at the left edge of the model
domain seen in panel \pan{a}.  Before 2~s, this result is nearly the
same as that produced by processing the transmission wavefield due to
the source located below the center of the domain.  The
cross talk introduced by summing the transmission wavefields is at
comparatively late time.  This could be windowed away since
the zero offset two-way traveltime to the deepest reflector in this
model is only 0.5~s.  Without knowing that two sources were active
during the recording interval, these gathers would be incorrectly
interpreted.
\activeplot{vshot1200\DOT 3500\DOT rand}{width=\textwidth}{CR}{Synthesized shot gathers
  produced by correlating the sum of transmission wavefields due to
  randomized sources beneath the left edge and middle of the model.}
\par
Figure~\ref{fig:vshot1200csum3500rand} is the sum of correlations of
two transmission wavefields to be compared with
Figure~\ref{fig:vshot1200.3500.rand} as an example of the appropriate
processing strategy of the two wavefields.  The cross talk below 2~s on
panel \pan{b} is not present.  The random source wavelet introduces
an extended wavelet coda to each event that leads to the events
overlapping and an overall ringing character that diminishes the
quality of the gathers.  Random source functions result in overall
less energy within their correlation functions, but this energy is
distributed in time as diminishing correlation side-lobes.  Therefore,
though the randomized wavelets decreased the total power of the
cross talk, it has also introduced a ringing coda that can
look like events.  If the source wavelet codas are very long, the ringing in this 
result would be attenuated by the square root of the length of the
wavelet.  Limiting the random wavelet to 1~s in duration  
does not allow for much cancellation of the correlation side-lobes.  
\activeplot{vshot1200csum3500rand}{width=\textwidth}{CR}{Synthesized shot gathers 
  produced by summing the correlations from randomized sources beneath
  the left edge and middle of the model.}
\par
Figure~\ref{fig:manysrcrand}\pan{a} shows the result of 
simultaneously processing 25 sources spaced every
125~m across the bottom of the model.  Although the entire model is
illuminated by this distribution, processing a transmission
wavefield due to so many sources leads to so much cross talk
noise that the gathers are completely uninterpretable.  This result
is worse than that produced without the randomized 
wavelet, Figure~\ref{fig:manysrc}\pan{a}.  The synthesized gathers
including randomized wavelets contain more events and are not interpretable.
\activeplot{manysrcrand}{width=\textwidth}{CR}{Left: Synthesized shot gathers with
  25 randomized subsurface sources spaced evenly across the bottom of
  the model.  Right: Synthesized shot gathers with 30 subsurface sources 
  clustered under the center of the model.} 
\par
Figure~\ref{fig:manysrcrand}\pan{b} was produced with 30 randomized
subsurface sources clustered within 600~m beneath the center of the
model.  This result contains higher frequency, more distinguishable events
than that produced without randomized wavelets,
Figure~\ref{fig:manysrc}\pan{b}.  However, these events are not
subsurface reflections.  They are cross talk artifacts that completely
mask the subsurface information.

\section{Conclusion}
Synthetic transmission wavefields due to impulsive sources across the 
bottom of an arbitrary velocity model show that correlation of 
every trace with every other trace does synthesize a volume close
to that produced by modeling the active surface seismic 
experiment.  However, if only sources directly beneath the array are 
captured during the recording, some coherent noise events are included
in the result due to incomplete destructive interference.  Positively 
however, direct arrivals
and most of the refracted energy is not synthesized, which is often 
problematic in the conventional processing strategies used for active
seismic data.  
\par
Synthesized gathers with virtual shot locations a large
distance from the areal coordinates of the subsurface sources are
very poorly reconstructed.  Upon comparing such correlated gathers to
the active gathers they are meant to synthesize, it is likely that these
results would be discarded when produced with field data.  However, in 
Chapter~\ref{chapter:DepthSyn} it is shown that despite the poor
quality, these gathers contribute positively to the image produced by 
migrating all possible synthesized shot gathers.  The subsurface information
in these synthesized gathers is due to the high-order multiples 
included in these models.  If high-order multiples are not recorded
in field data, these gathers would not contain useful information.
\par
If it is not possible to correlate transmission wavefields from individual 
subsurface sources before summation, the resulting volume of synthesized 
shot gathers will contain coherent artifacts.  These artifacts can not be 
distinguished as such and add coherent noise to the result that can quickly 
overwhelm the events from subsurface reflectors.  For the scenarios of 
clustered or well distributed subsurface sources correlated
simultaneously, the resulting volumes contain so much
cross talk that they can not be reliably interpreted.  
\par
Modeling the potential differences between local source functions  
as finite, random perturbations is not a sure way to alleviate the
cross talk introduced by processing summed gathers.  The model I have
chosen to vary the source parameters does not provide very substantial
damping of the cross talk.  However, I do think that they represent a
more reasonable likelihood in the real Earth than has been thus far
reported in the literature.  
\clearpage
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