\chapter{Correlation processing experiments}\label{chapter:TimeSynth}
\section{Introduction}
Cross-correlation of traces from transmission wavefields, followed by summing
the individual results, yields a data volume with the same kinematics as 
active seismic experiments if sufficient coverage of subsurface sources 
was captured in the transmission wavefields.  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
To analyze the efficacy of 
correlation processing under various source configurations, a suite of 
2D synthetic transmission wavefields were generated.  The data were generated
with a finite-difference two-way operator.  All surface-related and
internal multiples are included, and the transmission 
wavefields have been flux-normalized \cite{draganov:90}.  The model
contains two synclines in an otherwise flat three layer velocity 
model.  Subsurface impulsive 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, spaced 20~m apart.  For comparison, 
Figure~\ref{fig:actshots} shows two modeled surface seismic shot gathers 
at locations in the model from which the forthcoming synthesized 
gathers are extracted.  For clarity, I will use the term ``source'' to
describe the subsurface sources, and ``shot'' or ``virtual shot'' in
reference to the virtual source axis of the synthesized shot gathers.
\plot{actshots}{width=\textwidth}{Two active reflection seismic shot
  gathers, corresponding to the locations of the synthesized gathers
  in the next figures.}
\par
Figure~\ref{fig:data} shows two representative transmission wavefields
and the velocity model with which they were generated.  The left panel
shows data from a source at the left edge of the model domain, while
the center panel is a  transmission wavefield due to a source directly
beneath the center of the model domain.  The first hyperbolas are
direct arrivals and the rest are due to the structure within the
velocity model.  
\plot{data}{width=\textwidth}{Representative transmission wavefields from 
impulsive subsurface sources through a three layer velocity model with 
two synclines (right).  All orders of surface-related and internal multiples are
included.}
\par
Figure~\ref{fig:shot1200} shows two slices through a 3D volume of
synthesized gathers produced by correlating all traces using a single
transmission wavefield (left panel of Figure~\ref{fig:data}).  Thus, 
this volume represents a partial processing of the source summation 
required to synthesize active data from transmission data.  The 
front face of the left cube shows the first trace in the
transmission data correlated with all other traces.  The correlation
does not dramatically change the kinematics of the events seen in
Figure~\ref{fig:data}.  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) crosses the right edge of
the gather at about $t=2$~s.  The same event in
Figure~\ref{fig:actshots} reaches the right edge of the gather at
about $t=3$~s.

\par
The first event, that intersects the $x,t$ axes at
$(1200\mbox{m},0\mbox{s})$, is the manifestation of the delta
function, $delta({\bf x}_r,{\bf x}_s)$, in equation~\ref{rigor}.  The
next two events are the synthesized primaries.  The right face of the
panels have axes time and source position.  Roughly hyperbolic acausal
movement of the reflectors is apparent.  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 causes the two primary events are acausal after virtual shot
position $x=3300$~m.
\par
The trace under the vertical black line in the correlated shot gathers
is the zero-offset trace in the synthesized gather shown and the
location of the trace from the transmission wavefield used as a source
function, or master trace, used to correlate all of the other
traces in the wavefield.  The front face of the right panel of 
Figure~\ref{fig:shot1200} 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.
\plot{shot1200}{width=\textwidth}{Two correlated shot gathers from the
volume generated with a single source below the left edge of the model.}
\par
%%Figure~\ref{fig:shot1200.shallow} are correlated gathers produced with
%%a preprocessed transmission wavefield.  In this case, a hyperbolic
%%mute was applied below 1.3~s in Figure~\ref{fig:data}.  The direct
%%arrival wait (0.5~s) plus the zero-offset traveltime to the second
%%reflector is 1~s.  Therefore most of the multiples have been removed
%%before correlation.  This case represents the 
%%
%%The late-lag events in this gather have been
%%removed as compared to Figure~\ref{fig:shot1200}, but it is unclear as
%%to the effect on the quantity of subsurface information that has been
%%lost since these gathers are only a partial reconstruction of the
%%surface seismic experiment.
%%\plot{shot1200shallow}{width=\textwidth}{Two correlated shot gathers
%%  from the volume generated with a single source below the left edge
%%  of the model that does not contain high-order multiples.}
Figure~\ref{fig:allshots} shows synthesized gathers at locations
corresponding to Figure~\ref{fig:allshots} produced by summing the
correlations from subsurface sources across the entire bottom of the
model.  The left front face is not substantially different from the
correlations of a single transmission wavefield in
Figure~\ref{fig:shot1200}.  This gather suffers significantly from the
limited distribution of subsurface sources used to synthesize the
surface seismic volume of data.  The quality of the result is
predicated on upcoming 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 the 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 that in the left panel of
Figure~\ref{fig:shot1200}.  Notice that on the right face of the right
panel (time-shot axes), the events develop a slight concave-up moveout
toward the edge of the dowain rather than an asymptotic approach to
linearity. 
\par
The right panel of Figure~\ref{fig:allshots} shows a gather from
the center of the model domain, which should be the most complete
gather in the synthesized volume.  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.  However, the receiver
and shot axes are much more symmetric than previously, which is
appropriate for this model with synclines in otherwise flat layers.
\plot{allshots}{width=\textwidth}{Two correlated shot gathers from the
volume generated by summing correlations from 225 sources across the
bottom of the model.}
\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 probably be ignored, as this work is focused on 
recovering 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 above 
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}, migrate images produced
with these data volumes are presented to explore these effects in the
image space. 

\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 
when attempting to process long passive records as a single wavefield.  
\par
Consider field data, $T_f$, composed of
transmission wavefields $a({\bf x},t)$ and $b({\bf x},t)$ from
individual sources.  These individual transmission wavefields
are placed on the field record at unknown times $\tau_a$ and
$\tau_b$ such that
\begin{equation}
T_f(t)=a*\delta(t-\tau_a)+b*\delta(t-\tau_b).
\end{equation}
Correlation in the Fourier domain yields
\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 the desired result.  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.  
\par
Redefine $A$ and $B$ as the impulse response of the Earth, $I_e$,
convolved with source functions, $F$, which now contain their phase
delays.  As such, the cross terms of equation \ref{tt} 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}
Like the first two terms in equation \ref{tt}, the cross terms contain
the desired information about the Earth.  However, the source function 
$f_c$ included is not zero phase.  If the
source functions are random series, terms with residual phase
($F_cI_e^2$) within the gathers will decorrelate and diminish in 
strength as the length of $f$ and the number of cross terms 
increases.  Designing uncorrelable functions $F_i$ is the chief concern
to effectively process sums of phase-encoded gathers.  For passive 
recordings however, the summation of transmission wavefields can
be a natural phenomenon.  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.  This is not probable if the 
noise sources have similar source mechanisms. 
\par
Figure~\ref{fig:shot1200.3500.shift} represents the case where the
energy from two subsurface sources sources ($x=1200, 3500$~m) is processed 
together.  In contrast, Figure~\ref{fig:shot1200csum3500} shows the 
sum of correlations of the two transmission wavefields.  Thus, 
Figure~\ref{fig:shot1200csum3500} is a partial result of the 
summation of correlations presented in Figure~\ref{fig:allshots}.
\plot{shot1200.3500.shift}{width=\textwidth}{Synthesized shot gathers
  produced by correlating the sum of transmission wavefields due to
  sources beneath the left edge and middle of the model.}
\plot{shot1200csum3500}{width=\textwidth}{Synthesized shot gathers
  produced by summing the correlations from sources beneath the left
  edge and middle of the model.}
\par
Both the time sections and time slices in Figures~\ref{fig:shot1200.3500.shift}
and~\ref{fig:shot1200csum3500} show the
differences associated with the improper processing of 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:shot1200.3500.shift} that
are not present in Figure~\ref{fig:shot1200csum3500}, as well as a
strong concave-right hyperbola at shot position $x=3000$~m.  There are
also strong half hyperbolas on the left edge of the front face of the
right panel in Figure~\ref{fig:shot1200.3500.shift} that are not
present in Figure~\ref{fig:shot1200csum3500}.  Correlating
the sum of the transmission wavefields produces a result that contains
all the events of both transmission wavefields spread throughout the 
volume.  Both shot positions in Figure~\ref{fig:shot1200.3500.shift}
look very similar, and the noise is symmetric across the zero-offset 
diagonal in the time-slices.
\par
As more sources are processed together, such as a storm of
small subsurface sources during natural or induced fracturing, this
problem could overwhelm the correlations.  Figure~\ref{fig:shotsmall}
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 cross-talk noise that the gathers are 
uninterpretable.  Figure~\ref{fig:shotsmallsmall} was produced with 
30 shots every 25~m clustered around the center of the model.  There
is little to no energy in the first gather.  The gather in the midst
of the swarm has events, but they produce a ringing time section that
looks nothing like the desired result.
\plot{shotsmall}{width=\textwidth}{Synthesized shot gathers with 25
  subsurface sources spaced evenly across the bottom of the model.}
\plot{shotsmallsmall}{width=\textwidth}{Synthesized shot gathers with
  30 subsurface sources clustered under the center of the model.}
\par
The total correlation volume produced by a single source, 
Figure~\ref{fig:shot1200}, could be confusing as it represents
an incomplete data set.  However, the synthesized gather immediately
above the subsurface source is fairly representative of the complete
summation at the same location.  Processing summed transmission
wavefields copies each shot panel throughout the volume.  This fact
can be observed by noting the similarities in Figures~\ref{fig:shot1200}
and~\ref{fig:shot1200.3500.shift}.  When more than two sources are
captured in the passive data the resulting correlations,
Figures~\ref{fig:shotsmall} and~\ref{fig:shotsmallsmall}, 
become completely uninterpretable.
\par
The cross-talk introduced by processing the reverberations from multiple
sources simultaneously takes the form
of the convolution of the Earth's impulse response (all reflections 
and multiples) with the correlation function of the two sources,
$F_cI_e$.  The cross-talk events due to
$n_s$ sources will likely be more problematic than conventional
multiples as every multiple and reflector can be repeated $n_s!/(n_s-2)!$ 
times.  The ratio of desirable zero phase terms, $I_e$, to cross
terms, $F_jI_e^2$, decreases as $1/(n_s-1)$ if the source terms
are maximally correlable.

\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 correlable.  While local sources probably share similar wavelet
characteristics, it is also informative to explore the effects of
varying the source wavelets used to model the transmission
wavefields.  For this exercise, I convolve the individual transmission
wavefields with a random number sequence before correlation.  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
Figures~\ref{fig:shot1200.3500.rand} -~\ref{fig:shotsmallsmallrand}
recapitulate Figures~\ref{fig:shot1200.3500.shift}
-~\ref{fig:shotsmallsmall} with the addition of convolving the
modeled data with a random source wavelet before further 
processing.  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.
\plot{shot1200.3500.rand}{width=\textwidth}{Synthesized shot gathers
  produced by correlating the sum of transmission wavefields due to
  randomized sources beneath the left edge and middle of the model.}
\plot{shot1200csum3500rand}{width=\textwidth}{Synthesized shot gathers
  produced by summing the correlations from randomized sources beneath the left
  edge and middle of the model.}
\par
Figure~\ref{fig:shot1200.3500.rand}, reproduces the correlation of two
summed transmission wavefields, Figure~\ref{fig:shot1200.3500.shift}, 
with the inclusion of random wavelets before summation.  Many of the
inappropriate features in the time slices have been eliminated in
comparison to Figure~\ref{fig:shot1200.3500.shift}.  Note however,
that the randomized wavelet associated with the 
transmission wavefield for the source on the left side of the model
also included a nearly 2~s time delay.  Therefore, the cross-talk 
visible in the front face of the right panel is still present, 
though shifted to later time.  Interpreting this gather without
such apriori knowledge, one might assume that there is a diffractor 
(or a series of diffractors if not aware of the many orders of multiples
included in the data) on the left edge of subsurface domain.
\par
Figure~\ref{fig:shot1200csum3500rand} is the sum of correlations of
two transmission wavefields to be compared with
Figure~\ref{fig:shot1200.3500.rand} as an example of the appropriate
processing strategy of the two wavefields.  The figures are very
comparable.  The main reason for this is that the two events
were well seperated in time and most of the cross-talk is either
acausal or at late time lags.
\par
Figure~\ref{fig:shot1200csum3500rand} also bears comparison with
Figure~\ref{fig:shot1200csum3500} to see how the randomized 
wavelet has effected the quality of the summed 
correlations.  Figure~\ref{fig:shot1200csum3500rand}, now shows an
extended coda attached to each event and some amplitude variation due to the
modeling parameters of the individual sources.  The longer randomized
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 created more
events.  If the wavelet codas are truly random and 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.  
\par
Figure~\ref{fig:shotsmallrand} is produced with the sum of 25
randomized subsurface sources spaced evenly across the bottom of the
model.  The sources were at the same locations as those used in
Figure~\ref{fig:shotsmall}, but the quality of these synthesized
gathers has clearly diminished.
\plot{shotsmallrand}{width=\textwidth}{Synthesized shot gathers with 25
  randomized subsurface sources spaced evenly across the bottom of the model.}
\par
Figure~\ref{fig:shotsmallsmallrand} shows an contrary image.  Whereas
Figure~\ref{fig:shotsmallsmall} is full of ringing monochromatic
energy, Figure~\ref{fig:shotsmallsmallrand} has events that are
better differentiated and appear more appropriate.  Whether these
more clear events are appropriately interpreted at subsurface information
will be explored in the next chapter in which the correlation 
volumes above are migrated assuming that the synthesized shot gathers
contain the kinematics of the active seismic reflection survey.
\plot{shotsmallsmallrand}{width=\textwidth}{Synthesized shot gathers
  with 30 randomized subsurface sources clustered under the center of
  the model.} 

\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 nose events are included
in the result due to incomplete destructive interference that would be 
enjoyed by a complete distribution of source locations surrounding 
the subsurface volume of interest.  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
If it is not possible to correlate individual transmission wavefields 
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 subsurface structure one is attempting to synthesize.  If the source
functions within the transmission wavefields are identical and impulsive,
the resulting synthesized shot gathers are sometimes clear or ringing 
depending on the subsurface distribution of source locations.  However
believable the gathers are in either case, interpretation of these 
volumes as if they were conventional shot gathers would likely
lead to misinterpreting the subsurface.
\par
Modeling the potential difference 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 are not good encoding 
functions.  However, I do think that they represent a more reasonable
likelihood in the real Earth than has been thus far reported in the
literature.  Depending on the distribution of the 
subsurface sources, the correlation volume can seem more or less 
realistic.  Despite the appearance of realism however, the inclusion
of multiples sources with or without random wavelets does not 
visibly approach the kinematics of the desired active data.
