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\chapter{Introduction}\label{chapter:Intro}
\section{Introduction}
Passive seismic imaging is a processing methodology which attempts
to extract subsurface information due to unconventional 
sources.  In this thesis, the goal is the production of subsurface
structural images by recording the ambient noisefield of the Earth
with surface arrays of seismometers or geophones.  The images produced
with this technique are directly analogous to those produced with
conventional reflection seismic data.  Manipulating wavefields
with such unconventional sources has been given the general descriptor
in the literature ``interferometry''.  Passive imaging is not
new science.  This statement is true for two reasons.  
\par
First, the idea of imaging the subsurface by correlating the 
recordings from subsurface sources was first  
introduced by \longcite{GEO33-02-02640269}.  That work 
provided a one-dimensional proof that the auto-correlation of time
series collected by a receiver array can produce the
equivalent to a zero-offset time section.  Subsequently, 
\longcite{Zhang.sep.60.85.art}, used plane wave decomposition 
to prove the result in 3D over a homogeneous medium.  \longcite{fink}
developed the Green's function of a heterogeneous medium
with acoustic waves via correlation and validated the observational 
theory with an ultrasonic experiment.  \longcite{kees04}, through 
one-way reciprocity, 
proved that cross-correlating traces of the transmission response of
an arbitrary medium synthesizes the complete reflection response,
in the form of shot-gathers, collected in a conventional active source
experiment.  In fact, in recent years, there has been a sudden
increase in interest in interferometry that has generated a wealth
of literature \cite{wapenaar:SI1}.  
\par
The second reason that passive imaging is not new science, is 
that interferometry is simply a generalization of the seismic 
problem formulated by earthquake and active-source 
exploration seismology.  On the continuum of purposefulness,
active experiments occupy an endpoint where humans 
initiate an experiment over a specific patch of Earth that is 
of interest.  In this case the source function is (hopefully) zero
phase, impulsive, and time zero is precisely known.  Toward the middle of the 
continuum, humans install receivers over an area of interest and 
capitalize on naturally occurring far-field (teleseismic) 
earthquake energy to probe the subsurface \cite{recfun,BRS3,SAW}.  Here, 
the source energy is areal, and time zero is defined by windowing data
beginning at the onset of the earthquake arrival.  At the opposite end
of this spectrum is this work, which asks if the complicated,
low amplitude ambient noisefield in the Earth, about which one has no 
knowledge, can provide sufficient energy to image the subsurface
\cite{artman:SI177,surftom}.  However, the physics and mathematics used to 
manipulate these various data sets are the same.  The more purposeful 
the experiment, many of the details of seismic interferometry can be 
simplified or are implicitly included in the form of the 
acquisition.  The less one knows about the source energy probing the 
subsurface, the more precise the processing needs to be in terms of 
the rigorous formulation of the more general theory.
\par
To date,
the literature concerning subsurface structural imaging at the reservoir 
scale has been largely relegated to theoretic and modeling pursuits due 
mainly to a lack of suitable data acquisition.  This thesis adds to 
the body of work in this field by presenting results from the
processing of a large volume of passive records with both the standard
processing sequence and a new methodology developed herein: direct
migration.  This new algorithm produces a subsurface image directly
from the raw passive data without first synthesizing shot gathers
through correlation.

\section{Development of the correlation theory}
Figure~\ref{fig:sketchP}, simplified for clarity,
shows the basic kinematics exploited in processing transmission
data.  The figure includes two recording stations 
capturing an approximately planar wavefront emerging from a two-layer
subsurface.  Panel \pan{a} shows the raypaths associated with the direct
arrival and one reflected both at the free surface and a subsurface
interface.  The dashed travel path (labeled reflection ray) has the 
familiar kinematics of the reflection seismic experiment if a source
were excited at receiver one.  The transmission
wavefield is depicted in panel \pan{b}.  Wavelet polarity is
appropriate for direct arrivals and reflection.  The three main
features of transmission wavefields can be appreciated here.  First, the
exact timing of the arrival is unknown.  Second, the phase 
and duration of transmitted energy are unknown and likely
complicated.  Third, if the incident wavetrain coda is long,
arrivals in the transmission record can interfere.
\par
%%\sideplot{figure1}{width=3.3in, height=2.4in}{ ({\tt a}) Approximately
\activesideplot{sketchP}{width=3.3in,height=2.2in}{NR}{({\tt a}) Approximately
  planar arrival with rays showing important propagation paths for
  passive imaging.  ({\tt b}) Idealized traces from a transmission
  wavefield. ({\tt c}) Shot-gather (reflection wavefield) synthesized
  using trace \textit{r1} as the source.} 
Choosing trace \textit{r1} as the comparison trace (or source
function), panel \pan{c} 
depicts the correlation spikes associated with the arrivals in the
data (panel \pan{b}), where $\otimes$ is correlation.  A solid line
with linear move-out is super-imposed across the correlated traces
corresponding to the direct arrival recorded at each receiver
location.  The dashed line on panel \pan{c} has hyperbolic moveout.
However, no correlation peak exists on the \textit{r1}$\otimes$
\textit{r1} trace under the hyperbola.  Not drawn, the second arrival
on \textit{r2} will have a counterpart on \textit{r1} from a ray
reaching the free surface farther to the left of the model.  
\par
In fact, the correlations produced from a single plane wave will 
produce another plane wave.  However, each planar reflection is moved 
to the lag-time associated with
a two-way trip from the surface to the reflector.  Correlation 
removes the wait time for the initial arrival and maintains the 
time differences between the direct arrival and reflections.  Summing the 
correlations from a full suite of plane waves builds hyperbolic events 
through constructive and destructive interference.  Analyzing seismic 
data in terms of plane wave constituents is a commonly invoked tool in 
seismic processing.  Summing the correlations from incident plane waves 
is a plane wave superposition process that builds hyperbolas.
\par

Cross-correlation of each trace with every other trace handles the
three main difficulties of transmission data: timing, waveform, and
interference.  First, the output of the correlation is in lag units,
that when multiplied by the time sampling interval, provide the
time delays between like events on different traces.  The zero lag of the
correlation is the zero time for the synthesized 
shot-gathers.  Second, each trace records the character and duration
of the incident energy as it is reflected at the surface.  This
becomes the source wavelet analogous to a recorded vibrator 
sweep.  Third, overlapping wavelets are separated by correlation. 
\par
%%\subsection{Natural phase encoding}
%%\subsection{Direct migration}
To calculate the Fourier transform of the reflection response, 
$R({\bf x}_r,{\bf x}_s, \omega)$, of a loss-less (to neglect amplitude
and attenuation issues) subsurface, \longcite{kees04}, proved
\begin{equation}\label{rigor}
2\Re [R({\bf x}_r,{\bf x}_s,\omega)]=
\delta({\bf x}_s-{\bf x}_r)-\int_{\partial\! D_m} 
T  ({\bf x}_r,\xi,\omega)
T^*({\bf x}_s,\xi,\omega)\;
\mbox{d}^2\xiv \;,
\end{equation}
where the symbol $^*$ represents complex conjugation, and subscripts $r$ and
$s$ indicate different station locations, ${\bf x}$, at which a transmission
wavefield, $T$, is recorded.  Only the real part of the correlation,
$\Re$, is maintained.  After correlation $r$ and $s$ acquire the meaning of receiver
and source locations, respectively, associated with an active survey,
$R$.  The RHS represents summing correlations of windows of passive data around
the occurrence of individual sources, $\xi$.  The
symbol $\partial \! D_m$ represents an arbitrary domain boundary that surrounds
the subsurface region of interest on which the sources are 
located.  The delta function, $\delta$, is a negligible artifact of
correlation processing.  Alternatively, many impulses can be substituted for a 
full suite of plane waves emerging from all angles and azimuths 
as in the kinematic explanation above.
\par
The two main assumptions within the equation are: 1) processing a 
complete set of transmission wavefields (integral), and 2) processing
wavefields due to individual sources, indexed by $\xi$.  If these requirements 
are met, the sum of the many correlation volumes synthesizes active
surface seismic data.  I will use the notation $\tilde{R}$ to
distinguish the synthesized gathers from the active seismic
experiment.
\par
If too few sources are captured (and the integral above not
completed), $\tilde{R}$ may not accurately reproduce the kinematics of 
$R$.  For some applications, completing the integral of 
sources around the volume of interest may not be required.  If one is 
specifically interested in body waves probing subsurface structure, 
only point sources below the domain (or relatively flat plane waves)
are required to image the structure.  However, the synthesized gathers 
will contain some artifacts.  Alternatively, \longcite{louie}, 
described using surface waves generated by cultural noise for 
near-surface refraction velocity analysis, and \longcite{surftom}
extends a similar analysis to crustal scale observations.
\par
\subsection{Direct migration}
The basic premise behind imaging with uncontrolled seismic energy is
simple.  Correlation of data traces calculates the time differences
that energy packets require to travel between two locations.  In a
heterogeneous environment, the various reflections and diffractions
that add travel paths beyond the simple direct ray between the
receivers provides information about the medium.  The entire field of
seismic analysis is built around the interpretation of time
differences between two locations.  Most often, the time differences
that are recorded and analyzed are those between the initiation of
source energy (earthquakes, explosions, or vibrations) and the arrival
of that energy at receiver locations.  Correlating traces removes the
initial waiting time for energy to arrive at a recording station.  The
resulting volume keeps the time delays of all the possible ray paths through
the medium between the receivers.  Simultaneously, complicated wavelet
functions are collapsed to a bandlimited impulse.
\par
In fact, the seismology community has been routinely analyzing complicated
data by correlation algorithms that correspond to various specific cases
of interferometric processing.  Global seismologists correlate teleseismic
arrivals with windows of data after the first arrival.  These
traces, and more recently wavefields, are then back-projected along
the arrival path of the energy in receiver function analysis
\cite{Morozov}.  Results are then summed from many earthquake sources
to investigate the deep structure of the Earth.  The cascade of
projection and summation developed in receiver-function analysis can
be defined as a migration strategy.  Migration is a processing concept
that is not specific to a particular implementation.  It is applied with varying
degrees of complexity and accuracy with many different algorithms to
move energy in data records laterally and vertically (time or depth)
to locations in an Earth model where energy has changed propagation
mode or direction due to subsurface discontinuities.  Producing images with
receiver-function analysis, therefore, can be simplified to a two step
algorithm: Correlate, migrate.   
\par
In the exploration community, vibrator sources sweep across a frequency
spectrum using various source functions that are not impulsive in
time.  To define time-zero, records are correlated with the known
source function to collapse the wavelet coda.  The preprocessed shot
gathers are then migrated to image the subsurface structure.  Imaging
with this acquisition strategy can be simplified to a two step
processing flow: Correlate, migrate.  For marine acquisition, the
source bubble associated with underwater sources reflecting from the
sea surface makes the real source signature a function of time rather 
than a simple impulse.  Deconvolving the effect is an
important step in processing marine data which can be analyzed in
terms of a multi-step correlation process.  Again, producing a final
image of the subsurface can be simplified to the familiar two step
algorithm.
\par
Passive seismic imaging with many subsurface sources was presented in
equation~\ref{rigor} as another two step process: Correlate, sum.  The
summation required is a sum of the correlations produced by individual
unknown sources.  The synthesized shot gathers can then be processed
(migrated) as a volume of conventional data with many surface 
sources.  The total processing flow is therefore a three step process:
Correlate, sum, migrate.  The basic components used to process passive
data are the same as those described above, with the introduction of a
summation. 
\par
I have capitalized on the flexibility of a specific migration strategy in
this thesis: the wave-equation shot-profile migration algorithm
\cite{GEO36-03-04670481}.  The descriptor ``wave equation'' denotes
a method of extrapolating seismic energy through a (accurate)
subsurface velocity model with a propagator derived from the kinematic
solution to the wave equation.  The physics of wave propagation is
the same for impulsive source functions or complicated wavefields that
are a function of space and time.  Seismic data records provide
initial conditions for wavefields extrapolated with with these
propagators.  Extrapolation moves data away from the acquisition
plane to a new datum, by modeling the physics of wave propagation and
maintains appropriate spatial and temporal differences between events
within the wavefield.  After changing the experimental datum, the
traces in the extrapolated wavefields can be still be processed in
accordance with all appropriate techniques developed for seismic data
to produce any desired result.  Extrapolation is the first step of
wave-equation migration to produce a subsurface image. 
\par
Shot profile migration algorithms extrapolate the source wavefields
and data wavefields independently, in opposite temporal directions.  When the
energy within the two wavefields is collocated after propagation, we
have located a subsurface discontinuity that gave rise to a reflection
in the field experiment.  Finding collocated energy is accomplished by
calculating time differences between the two wavefields, and keeping
only energy at $t=0.$  The zero-lag of the correlation of the traces
within the two wavefields provides the desired result.  Images from
many experiments (shot gathers) are summed for a final 
result.  Therefore, this migration algorithm 
can be distilled to components: Extrapolate, correlate, sum.
\par
In this thesis I introduce direct migration as a simplified processing 
algorithm
for passive seismic data.  Instead of the three step processing flow
explained above (correlate, sum, migrate), direct migration uses
the correlation and summation within the migration algorithm to
process transmission wavefields.  If the extrapolation has been
performed accurately, correlation will calculate accurate time
differences between source events and reflections at a new reference 
datum.  The migration algorithm calculates time differences
by correlation at all subsurface depth levels in the model instead of
only at the acquisition surface.  The resulting volumes due to
individual sources are then summed to provide a complete subsurface
wavefield.  This procedure honors the correlate/sum processing 
flow, with the addition of an extrapolation, or depth, axis.  However,
after extrapolating the data, we can discard non-zero lags
in the correlations to remove the now unnecessary time axis.  In so
doing, the final result is a migrated image of the subsurface produced
without creating shot gathers by correlation (that would then be ready for
migration).  This processing flow can be simplified to: Extrapolate,
correlate, sum = migrate.
\par
Direct migration of passive data completely skips the generation of
synthesized gathers in the time domain.  Without producing shot
gathers, a velocity model for migrating the data must be provided from
another source.  Removing the generation of the shot gathers from the
processing flow can substantially reduce the computational cost of
processing data from a large passive seismic recording campaign.
\par
\longcite{SAW} presented images produced with transmission wavefields
due to earthquake arrivals and the resulting local reverberations beneath
a recording array.  Teleseismic arrivals were used as source
functions for migrating data windows.  The authors capitalize on the
relatively simple, planar source functions to separate the up-coming
from down-going energy in the transmission wavefields with a
deconvolution algorithm.  The separated wavefields from each
earthquake event were migrated with a shot profile algorithm to
produce a subsurface image.  Acknowledging the kinematic similarity
between correlation and deconvolution, teleseismic migration
represents a hybrid of direct migration and migrating correlations. 
\par
Imaging with transmission wavefields is the same as imaging
primaries with multiples in a marine survey.  Multiples are due to
strong up-coming events that reflect from the free surface and return
to depth as source energy.  Therefore, the water bottom reflection
fulfills the same role as the direct arrival from a 
subsurface source.  If the water bottom reflection is used as a source
function in shot-profile migration \cite{guitton:1296}, all the
first-order multiples are imaged at the subsurface location of primary
reflections.  That migration is an example of the processing flow:
Extrapolate, correlate.  If all shot gathers are correlated and
summed, the time delay to the water bottom reflection is removed from
the records, and we can migrate the resulting data volume
\cite{Shan.segab.2003}.  That processing flow is an example of:
Correlate, sum, migrate.  Further similarities to passive seismic
imaging within the literature addressing the presence of multiples in
seismic data can be immediately drawn from the ``Inverse data space''
concept \cite{berkhout:2684}. 
\subsection{Chapter description}
Chapter~\ref{chapter:TimeSynth} analyzes the effectiveness of
correlation processing using a two-layer, 2D synthetic model.  Issues
such as source character, distribution, and the inclusion of multiple
source functions within a transmission wavefield are illustrated with
examples.   
\par
Chapter~\ref{chapter:TimeShot} introduces the Valhall field data and
describes and interprets the shot gathers generated by correlation and
spectral whitening.  Chapter~\ref{chapter:TimeCMP} extends the analysis
of the data by further processing the gathers with the normal moveout
correction (NMO) followed by stack.  NMO-stack is another example of a
simple migration algorithm performed in the time domain.
\par
Chapter~\ref{chapter:DepthSyn} presents depth migrations of the correlated
gathers shown in Chapter~\ref{chapter:TimeSynth} to explore the
effects of the artifacts within synthesized shot gathers in the image
space.  I then introduce the concept of passive seismic processing as an
imaging problem within the course of shot profile migration: 
direct migration.  I illustrate the effectiveness of the method with
the same 2D synthetic data used previously.
\par
Chapter~\ref{chapter:DepthVal} presents depth migration results from the
Valhall passive data set produced by both correlation followed by
imaging and direct migration strategies.  An image provided by BP,
produced with data from an active seismic survey is used as ground
truth to interpret the subsurface content of the final results.
\par
Chapter~\ref{chapter:DepthMult} extends the concept of data processing
at the imaging condition of shot profile migration presented in
Chapter~\ref{chapter:DepthSyn} where the transmission wavefield was
used as both the source and data wavefields.  In this case,
shot gathers are convolved instead of correlated, and the resulting image is a 
prediction of the surface-related multiples in the depth domain.  The
prediction volume is equivalent to migrating the multiple prediction
produced by surface-related multiple prediction (SRMP) in the data
domain.   The prediction can be calculated during the course of 
a shot profile migration at minimal extra computational cost.

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