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\chapter{Synthesized shot gathers from Valhall} \label{chapter:TimeShot}
\section{Summary}
BP made available passive recordings from an array of 2300 four component 
receiver stations at the Valhall reservoir in the Norwegian North Sea 
from February 2004 and January 2005.  I used the hydrophone records
for this analysis.  The bulk of the raw records do
not contain obvious events.  Hyperbolic events with water velocity
are sometimes seen which are centered at the location of the
platforms within the array.  Also, there are occasionally noise trains 
in the raw data with the appearance of ringing, noisy shot gathers 
approximately 7~s long.  These are also sourced from the production 
facilities at Valhall.
\par
Simple correlation does not yield interpretable synthesized shot
gathers from the raw data.  Spectral whitening of the correlations 
reveals several interesting features.  Dominating the gathers 
is a single event from a source probably 40~km to the S. West of the 
array traveling at roughly 1450~m/s.  The same event is present in both 
the 2004 and 2005 data.  The inverted source location for the event is 
exactly over the Ardmore field in British waters, operated by Tuscan 
Energy (Scotland) Limited.  

\section{Introduction to the {V}alhall array}
%%\section{North Sea reservoir Valhall}
BP, with partners Hess, Shell, and Total, provided approximately 43
hours of continuous passive seismic records from the permanent sensor
installation above the Valhall reservoir in the Norwegian North 
Sea.  Figure~\ref{fig:areamap} is a map provided by the Norwegian Oil 
Directorate which shows the production infrastructure between the coasts 
of Britain and Norway with the many developments in the region.  It is 
630~km from the Southern tip of the Norway shown and the city of Teeside 
in Britain.  The Valhall development is roughly in the center of the line
connecting those two geographic points.
\par
\activesideplot{areamap}{width=3.in,height=3.in}{NR}{British, Norwegian, and
  Danish coasts surrounding the North Sea oil development area.
  The Valhall reservoir is in the South-central area of the map and
  only 20 km from U.K.-controlled waters to the West, and Danish
  waters to the South.}
There are five surface structures (roughly over the center of the
reservoir), and two wellhead platforms at the North and South ends of
the NW-SE elongate elliptical reservoir (both roughly 6~km from the
central platforms).  The main fluid export pipeline runs NNW to the
Ekofisk reservoir, about 30~km distant.  Figure~\ref{fig:actg} shows
the location of the 13 lines of OBC 4C geophones installed in the
summer of 2003 to facilitate time-lapse imaging of the subsurface for
production monitoring.  The $\sim 9,200$~sensors are permanently installed 1~m
below the sea floor, in roughly 70~m of water.  The array forms a
roughly 4x11~km rectangle.  Nominal inline spacing of receiver
stations is 50~m, while the crossline separation is 300~m.
\activesideplot{actg}{width=2.5in,height=3.5in}{NR}{Valhall permanent subsea
  geophone array.  Each station measures 4 components (4C) of the seismic
  wavefield: Pressure with a piezoelectric sensor, and three
  orthogonal velocities of ground displacement with coil-spring
  geophones.}

\par
Approximately half-way down the North axis of Figure~\ref{fig:actg},
the cables on the Western side of the array are truncated.  This is
where the production facilities are located to which the instruments
are telemetered.  Also noticeable is the deviation of the ends of
cables on the North side of lines 5-7 (E=1000~m) to the NNW.  This is
the corridor for the export lines to Ekofisk.  Production facilities are 
continuously manned and operated, to include: Over 150 total well-bores
into the reservoir section, current production of approximately
80,000~b/d oil and gas from 43 wells, water injection through (at
least) 3 wells, generation of almost 80~MW of electric power with gas
turbines, flow through the export lines, and the continual presence of
multiple modes of transportation\footnote{www.npd.no/engelsk/}.  Valhall 
is located in a heavily developed area, within kilometers of
developments in the U.K. and Norwegian controlled waters and is
about half-way between the two countries.  Also important to the chalk
oil trend of this part of the North Sea is the rate of subsidence
associated with pressure withdrawal during production.  In its first 15
years of production (to 1997), the sea-floor subsided 3.5~m at the
crest of the reservoir structure \cite{subside}. 
\section{Passive seismic recording}
The passive recordings donated by BP are from 29 hours February 15,
2004 and 14 hours January 19, 2005.  Of the former, some 7 hours was
lost due to a failed disk-drive.  The most likely teleseismic arrival
recordable at the array, as determined by query of the National
Earthquake Information Center event
catalog\footnote{http://earthquake.usgs.gov/regional/neic/}, arrived
approximately 5 hours after the last records available.  Of the 2300 4C
stations, this work uses the hydrophone measurement from each
location.  The data are sampled at 0.004~s, and written as
contiguous files between 12-20~s long.  Figure~\ref{fig:wiggle}
shows 12~s of data from a single receiver.  The ten traces are
contiguous 1.2~s sections from the total 12~s record.  Therefore,
both the vertical and horizontal axes are time in seconds.  Figure~\ref{fig:raw1}
shows an example of the character of most of the data.  The traces
are balanced against each other, but are otherwise raw 
recordings.  Scanning through much of the raw data, I was not able to
identify clear events.  Most of the raw data look like incoherent noise,
but I found several interesting features that will be explored below.
\par
\activeplot{wiggle}{width=\textwidth}{ER}{Ten contiguous 1.2~s sections of data from a
  single raw record from the array.}
\activeplot{raw1}{width=6in,height=6.5in}{ER}{Twelve seconds of representative passive
  hydrophone data from February 16, 2004.  Trace number increases from
  the top left to bottom right corners of the array shown in
  Figure~\ref{fig:actg}.} 
Figure~\ref{fig:raw2} shows a conspicuous noise-train in a section of
raw data.  The figure shows all of the traces within the array, progressing
(roughly) North to South down each cable, and West to East across the 
cables.  An inline section from a single cable contains roughly 220 traces, 
and a crossline section contains only 11 traces on average.  While
there are no coherent events in the figure, a noise-train about 7~s
long within the background chatter is reminiscent of a noisy 
shot-gather.  The minimum travel time of the noise-train is at trace
650.  This trace is located at the South end of the top half of the fourth
receiver line.  The various jogs and cable terminations of this region
are associated with the surface facilities on location.  Plotted over
the data are circles corresponding to a direct arrival from the location
of the platforms in the plane of the receivers.  These were shifted to
align with the top and bottom of the noise train.  The velocity used
to model this event is 1450~m/s.  The fact that there is no velocity
increase for the bottom of the envelope suggests that this feature
does not contain reflections from the subsurface.  Similar
examples of this type of noise-train are periodically recognizable
throughout data recordings.  The source of the energy has a complex
coda and a finite duration. 
\activeplot{raw2}{width=6in,height=6.5in}{ER}{Raw data with a powerful noise train
  envelope.  Ten lines of the array are apparent as the hyperbolas are 
  repeated and shifted over crossline offset.  Circles are co-planar direct
  arrivals from the center of the array traveling at 1450 m/s.}
\par
Figure~\ref{fig:raw3ev} shows a time window when a crisp series
of hyperbolic events is captured by the array.  The top panel is the
Eastern half of the array and the bottom is the Western half.  The
minimum traveltimes of the hyperbolas in the data decrease to the West 
in a similar fashion to the onset of the noise section in the
previous figure.  This suggests that the production facilities are
again the source of the energy.  The circles overlaying the data in
Figure~\ref{fig:raw3ev} show the kinematics of a a co-planar direct arrival
from the large 'O' in Figure~\ref{fig:minloc} (explained below) traveling at 
1450 m/s.  The modeled arrival times are plotted on every third trace to avoid 
clutter.  However, even when plotting
all picks in high resolution, the forward modeling clearly shows that
the data are aliased at this slowness value over the center of the
array where the event is not visible.  Similar events are recognizable
throughout the records, though this is a particularly clear example.  
\activeplot{raw3ev}{width=\textwidth,height=3.6in}{ER}{Raw data with a clear
  hyperbolic event annotated with the time calculated for a coplanar direct
  arrival from the array center traveling at 1450 m/s are.} 
\par
Figure~\ref{fig:minloc} is a map of the array rotated to convenient
field coordinates.  The 'X' symbols mark the locations
of the traces with minimum hyperbolic traveltimes that are obvious
in the previous figures.  A line connecting the X's has no inline 
offset variation and includes the area around
$y=6250$ m, which is the location of the platforms (presumably near the
large 'O' symbol).  This figure provided the inline coordinate for the 
overlaid event
plotted in Figures~\ref{fig:raw2}~\&~\ref{fig:raw3ev}.  The crossline
coordinate was deduced by matching the picks to the data 
windows.  The four cables with circled receiver stations indicate 
a subset of the array used for later figures.

\activesideplot{minloc}{width=2.5in,height=3.5in}{ER}{Receiver array in field
  coordinates.  Marked with 'X' are the locations of traces with
  minimum traveltimes picked from the hyperbolas in
  Figure~\ref{fig:raw3ev}.  Surface facilities are roughly at the large 
  $O$.  Trace 118 is the virtual source location for the synthesized
  gathers in Figures~\ref{fig:cor}-~\ref{fig:picks},~\ref{fig:mods} 
  and the origin in the black line in Figure~\ref{fig:mapa}.  Cables 
  marked with circles indicate receiver locations used in 
  Figures~\ref{fig:event}-\ref{fig:picks},\ref{fig:mods}.}
\par
\section{Correlation}
Following the theory from \longcite{kees04}, the basic principle of
time processing passively recorded seismic records to synthesize the
kinematics of conventional shot gathers dictates that individual
transmission wavefields should be correlated before summing the
results.  Because windowing individual arrivals from the +36~hours of
passive data is impractical (because of the large volume and the lack
of identifiable events), it is impossible to honor
equation~\ref{rigor} precisely by correlating traces within time
windows when only a single event is active.  I make the assumption
that processing data in 12 second sections is a reasonable
approximation to the requirement.  The available data were divided
into three sections for processing: one 13.5~hours, two 9~hours, and
one 4.5~hours in duration.  Sequential 12~s records within these
sections were correlated and the results stacked over the duration of
the (hours long) section of the total data.  Thus 
\begin{equation}\label{corr}
\tilde{R}({\bf x}_r,{\bf x}_s,\omega)= \sum_\xi
T({\bf x}_r,\omega,\xi)      T^*({\bf x}_s,\omega,\xi)
\end{equation}
was calculated where $\xi$ is the index of sequential 12 second windows
of data.
\par
Figure~\ref{fig:cor} 
shows correlations from one of the 9 hour sections of data.  A trace from 
the NW corner (receiver 118 in Figure~\ref{fig:minloc}) was used to correlate 
all the other
traces in the array.  If the processing was perfect, the figure would
show something equivalent to an off-end areal shot-gather.  Energy to 60
Hz was used in the correlations, with cosine tapers beginning
at 3 and 57 Hz.  Unfortunately, the image, and the other similar ones,
do not reveal anything of interest.  Figure~\ref{fig:spec} plots the
power spectrum of the sum of 100 traces from the gather in
Figure~\ref{fig:cor}.  The power spectrum of the gathers produced from
all four subsections of the raw data were very similar.  Versions of the 
gathers (produced with all subsections of the raw data) bandpassed around and 
between the peaks in the spectrum (Figure~\ref{fig:spec}) showed no increase in 
interpretability.
\activeplot{cor}{width=6in,height=6.5in}{CR}{Correlated shot-gather with master trace
  from the left side of the panel.  0-60 Hz energy was used for
  the correlation performed in the frequency domain.}
\activesideplot{spec}{width=2in,height=2in}{CR}{Power spectrum of the gathers in
  Figure~\ref{fig:cor}.}
\par
Spectral whitening by division of a trace in the frequency domain with
its power produces remarkable results when compared to simple
correlation.  I produced gathers using the relation
\begin{equation}\label{dec}
\tilde{R}({\bf x}_r,{\bf x}_s,\omega)= \sum_\xi
\frac{T({\bf x}_r,\omega,\xi)      T^*({\bf x}_s,\omega,\xi)}
   {|T({\bf x}_r,\omega,\xi)\;T^*({\bf x}_s,\omega,\xi)|}\;,
\end{equation}
where the denominator is simply the complex absolute value of the 
numerator.  The algorithm was very stable and showed no need to smooth 
the denominator as is common practice in similar deconvolutional 
efforts.  In fact, smoothing the denominator across the trace axis produced
large amounts of hard zero results.  This can only be explained by
large amplitude, uncorrelated energy on neighboring traces.  Dividing
a weak sample by a very strong neighbor will produce very small
values.  
\par
Figure~\ref{fig:dec} shows the same gather as in Figure~\ref{fig:cor}
using the spectral whitening of equation~\ref{dec}.  A very clear
event is now apparent.  Similar features can be seen upon close
inspection of the correlated result after viewing the deconvolved
gather.  The various features of the obvious event in the gather reveal
important features of the presumed source energy.  The event is
coherent across the entire 50 km$^2$ of the array and has reasonable
bandwidth.  The deconvolved gathers, produced with all subsections of the 
raw data, were bandpassed across the intervals
between and surrounding the energy maximums in power spectrum,
Figure~\ref{fig:spec}, of the correlated gathers.  No coherent energy was
present at all through 8 Hz.  Five versions from 9-40 Hz showed only
minor differences, with the strong event equally well represented.  Above
42 Hz, the signal begins to diminish markedly, and no coherent energy
is present at all above 52 Hz.  
\activeplot{dec}{width=6in,height=7.0in}{CR}{Trace-by-trace spectral whitening
  applied to shot-gather in Figure~\ref{fig:cor}.  Line connecting
  minimum traveltimes almost perfectly straight.}
\par
Events revealed by deconvolution processing, equation~\ref{dec}, in 
Figure~\ref{fig:dec} are hyperbolic.  This indicates the event could be
caused by any solution to the Eikonal equation.  Possible solutions
include the direct arrival of a buried source, or the reflection of a
point source from a plane.  No line contains the top of the 
hyperbola.  Therefore, the source of the energy is not contained
within the area of the array.  The moveout of the event is monotonic
to the South and East (in field coordinates).  This indicates that the
source must be to the North and West.  Moveout is curvi-linear in the
inline direction, but very close to linear in the crossline direction
(white line over data in Figure~\ref{fig:dec}).  The line connecting 
the minimum traveltimes of the event on each cable deviates from
straight only at the far right side.  Notice from the map in
Figure~\ref{fig:minloc} that crossline offset there is twice the
normal spacing.  These observations indicate that
the array is close to the asymptote of the hyperbolic event in
the crossline direction, and nearer the hyperbola top in the inline
direction.  Therefore, the source is much
closer to the array in the inline direction than the perpendicular
direction.  Given the true geographical orientation of the array,
Figure~\ref{fig:actg}, the source of energy should be somewhere
S. West of the array toward the British coast.  The fact that the
event does not contain the center of the hyperbola, proves that
processing by correlation to produce conventional shot-gathers has
failed.  Specifically the requirement to sum the correlations from
many shots in equation~\ref{rigor} has not been honored. 
\par
Figure~\ref{fig:conoff} shows two constant crossline offset sections
extracted from a volume of deconvolved synthesized shot 
gathers.  The crossline offset is $2400$~m between the two
OBC lines with bold receiver locations in panel \pan{a}.  Panel \pan{b}
was produced by correlating traces from West to East (bottom to top of
panel \pan{a}), while panel \pan{c} shows reciprocal correlations.  Active
seismic data are symmetric for these two sections.  Therefore,
events in the synthesized sections must be present in both panels to
be geologic reflections.  Alternatively, panels \pan{b} and \pan{c}
are mathematically equivalent to the causal and acausal parts of the
correlations performed in a single direction.  The causal and acausal
parts of the correlations for a successfully synthesized volume of
shot gathers are also symmetric.  The lack of identifiable events in
panel \pan{b}, and the concomitant lack of symmetry between the
panels, indicates that the raw data processed by correlation do not
fulfill the source distribution requirements of the theory of
interferometry, equation~\ref{rigor}.  The strong event below $1$~s in
panel \pan{b} is event discussed in Figure~\ref{fig:dec} on which
the bold white line is overlain.  These sections, and similar ones
exploring a full range of crossline offset, do not reveal geologically
feasible events. 
\activeplot{conoff}{width=\textwidth}{NR}{Reciprocal constant offset sections
  produced by correlating traces in the two bold OBC lines in panel
  \pan{a}.}
\par
If only a single impulsive subsurface source is captured by a 
passively recording
array, correlation simply shifts the event up such that it arrives at
$t=0$ at the master trace used to create the gather.  The panels in 
Figure~\ref{fig:event} all show four lines (receiver locations
marked with circles in Figure~\ref{fig:minloc}) from 
the same virtual shot location as Figure~\ref{fig:dec}.  The panels were 
produced with data from the four time intervals into which the raw data 
volume was divided.  Panel \pan{c} is a subset of the gather
shown in Figure~\ref{fig:dec}.  Each trace was 
correlated with the first trace in the panel.  Thus the 
event on the autocorrelated trace is at time zero.  Some of the
character of the panels changes, but the presence and kinematics of
the strong event is the same.  The last panel was computed with data
recorded almost one year after the data used in the first three panels.
\activeplot{event}{width=6in,height=4.in}{CR}{Ubiquitous strong event in
  deconvolved shot-gathers from data recorded 9 hours (\pan{b}),
  25 hours (\pan{c}), and 11 months (\pan{d}) after that used to produce 
  the panel \pan{a}.  The third panel is a subset of the gather shown in 
  Figure~\ref{fig:dec}.}
\par
Figure~\ref{fig:evtest} shows four synthesized shot-gathers using data
shown in the previous section investigating the character of the raw
recordings.  The master trace used for correlation in all panels was
the first trace on North end of line 2 (trace 118).  All panels use
the deconvolutional variant of correlation described by
equation~\ref{dec} and show the Eastern half of the array.  Only 12~s
of passive data was used to produce all the results in the figure.  Panel
\pan{a} shows the first three seconds of the causal correlation lags using the data
shown in Figure~\ref{fig:raw2}, which showed the ringing noise-train.  No
strong events are present, but there are some faint ringing hyperbolas
described by water velocity.
\par
Correlation, to first order, subtracts the initial time to the direct 
arrival in the master trace from all the other
traces in the gather.  The beginning of the noise-train in
Figure~\ref{fig:raw2} on the first trace (number 118), is at about 3.5
s.  The top of the noise train on the last receiver line (trace number
2150) is at about 2~s.  The complicated coda evident in the raw data
should collapse to a simple wavelet during correlation.  Therefore,
we expect the top of the energy on the last receiver line to lie at
about -1.5~s after correlation.  Figure~\ref{fig:evtest}\pan{b} contains
the acausal lags associated with the correlations presented in panel
\pan{a}.  The water velocity event overlain on the traces has a clear
minimum on the last hyperbola at -1.5~s, which is exactly what is to
be expected for the single-source scenario just described.  The
hyperbola under the overlays rings to the bottom ($t=0$) of the 
panel.  Given the long (7 s) coda of the noise train in the input,
these are probably correlation side lobes.  The side lobes continue 
past $t=0$ and on to panel \pan{a}. 
\par
The bottom row of Figure~\ref{fig:evtest} are the deconvolved
correlated gathers using trace 118 as the master trace, and the raw
data from 
Figures~\ref{fig:raw1}~\&~Figure~\ref{fig:raw3ev} respectively.  In
each case only 12 s of data was correlated.  Faint hyperbolas are
discernible in the early time of the right side of panel 
\pan{d}.  The faint hyperbolas have the same kinematic description as the
water velocity events in panel \pan{b} and Figure~\ref{fig:raw3ev}.  The 
most obvious energy however is in the events similar to the strong 
arrivals in Figure~\ref{fig:dec} and Figure~\ref{fig:event}. 
\activeplot{evtest}{width=6in,height=4in}{CR}{Correlated gathers from the
  Eastern side of the array.  Top row: Early causal (\pan{a}) and acausal
  lags (\pan{b}) of a correlated gather using the data in 
  Figure~\ref{fig:raw2}.  Overlays on panel \pan{b} have the same velocity 
  (1450 m/s) and origin as
  those delineating the top and bottom boundaries of the noise train in 
  the raw data.  Bottom row: Early causal lags of correlated gathers using 
  12~s of raw data.  Panel \pan{c} used data in Figure~\ref{fig:raw1} 
  and panel \pan{d} used data in Figure~\ref{fig:raw3ev}.} 
\par
Correlation masks some of the characteristics of the source energy.  The 
source could be impulsive or a long sweep of frequency
components like a vibrator source.  The latter seems less likely.  The
source could be a single strong event, or many similar repeated events
with low amplitude.  Given the presence of the event during all the
time intervals in Figure~\ref{fig:event} (spanning 1 year), I favor the latter
explanation.  I believe the energy is localized in space, and
ubiquitous in time.  The fact that the event is clear after correlating 
12~s or 9 hours of raw data means that the source must be fairly 
powerful.  The extended noise-train in the data shown in 
Figure~\ref{fig:raw2} used to synthesize the gathers shown in 
Figure~\ref{fig:evtest}\pan{a},\pan{b} indicates that over short time 
intervals, local cultural noise can overpower the temporally stable event
revealed in Figures~\ref{fig:event}\pan{a}-\pan{d}.  However, the local
production noise generated only water velocity in the correlated 
gathers.  Therefore, energy from the production facility did not provide 
subsurface information.
\par
Figure~\ref{fig:fkpow} is the frequency-wavenumber power spectrum of the 
raw data from a line of receivers.  The frequency axis has been windowed 
to 23~Hz to avoid Frequency domain wrap-around of steep linear events 
characterized by water
velocity slope.  Two energy trains are observable on either side of 
zero on the wavenumber axis that could be subsurface signal.  Panel \pan{b}
has been muted to preserve the energy near zero wavenumber and discard
the strong planar water velocity energy.  This Fourier domain mute
was applied to 100~minutes of raw passive data before correlation to 
investigate the steeply dipping (fast) energy near the center of the 
panels.
\activesideplot{fkpow}{width=3in}{ER}{Frequency-wavenumber power spectrum of on
  OBC cable calculated from raw passive data.  Panel \pan{b} shows mute
  applied to focus on energy near zero wavenumber.}
\par
Figure~\ref{fig:fkgath} shows a shot gather produced from the data
preprocessed with the Fourier domain mute in
Figure~\ref{fig:fkpow}\pan{b}.  A single strong event is apparent.  It
is aliased and has much lower frequency content that the event in 
Figure~\ref{fig:dec}.  It is however the same event.  This proves that
the kinematics of the distant event have sufficient curvature in the
inline direction that a FK-mute of slow linear events does not remove
the energy. 
\activeplot{fkgath}{width=\textwidth}{ER}{Correlated shot gather synthesized
  from passive data after preprocessing with the Fourier domain mute
  shown in Figure~\ref{fig:fkpow}.}
\section{Source identification}
Figure~\ref{fig:picks} shows the same correlated gathers as
Figure~\ref{fig:event}.  Superimposed on the data are time picks 
auto-picked by amplitude.  The picks are not very accurate due to the
effort required to maintain continuity.  These picks, and the receiver
locations, are the data used to invert for the source location that
caused these events.  The forward model used for the inversion was
the kinematics of a reflection from an arbitrary plane.  Inversions
for a direct arrival from depth were not successful.  
\activeplot{picks}{width=6in,height=4.in}{CR}{Auto-picked time values along event
in correlated gathers.  Picks are used as data to invert for the
source location.}
\par
The equation describing the traveltime of a planar reflection in 
shot-receiver coordinates is 
\begin{equation}\label{fwd}
v^2t^2={\bf x}^2+4z^2+4z{\bf x}\cos{\alpha}\sin{\phi}\;,
\end{equation}
where ${\bf x}$ is the horizontal distance from the source to the receiver,
$z$ is the depth to the reflector under the source location, $\alpha$ is
the dip direction of the plane, and $\phi$ is the dip of the plane.  At the 
limit of
$z=0$, the operator solves for a co-planar direct arrival.  Correlating 
data with a single master trace subtracts the time from the source to the master
trace from each trace in a gather.  Therefore, the forward modeling
operator used in the inversion solves for the time to every
receiver, and then subtracts the time, $t_s$, to the master trace used
for correlation.  This calculates the time differences produced in a
correlated panel.  Subtracting $t_s$ in the forward modeling operator
greatly diminishes the operator's sensitivity to the terms on the RHS
involving depth.  Also, large horizontal distance will cause the
first term, ${\bf x}^2$, to dominate the others.  The inversion
technique used was a micro-genetic algorithm \cite{Krishnakumar89}.  The 
model space
was defined as the ensemble: Average velocity through which the rays 
have passed, areal location of the source, and orientation angles and 
depth of the reflection plane.  The best results were obtained using 
the $l^2$ norm to evaluate the power of the data fitness function.
\par
The panels in Figure~\ref{fig:inv} show the effectiveness and quality
of the inversion results.  The genetic algorithm uses a seed number
that determines the characteristics of the first population as well as
how the future generations change.  I used fifteen seed numbers to 
begin inversions with the synthesized shot gathers from each of
the four subdivisions of the passive data.  Figure~\ref{fig:inv} is a 
representative sample of the inversion results from the four volumes
of synthesized shot gathers.  The panels in the figure show the
convergence of the inversion.  The total convergence of the inversion 
as a function of generation number is displayed 
in panel \pan{a}.  The convergence of the model parameters velocity, 
distance East, and distance North (respectively) to specific values is shown 
in panels \pan{b}-\pan{d}. In each case the vertical axis is the 
fitness value that the inversion is minimizing.  The relative width of the 
data cloud at the minimum fitness value for the 3 model parameters shown 
reflects the model precision of the inversion for this data.  Velocity and
distance North are very well constrained, but distance East has a wide
region of minimum energy.  Not shown are the plots for the model
parameters associated with the planar reflector: depth, dip, and
azimuth.  These parameters were completely within the null space of the
inversion.

\activeplot{inv}{width=\textwidth}{CR}{Residual energy of the inversion to
  locate the source of the energy causing the event in
  Figure~\ref{fig:picks}.  As the inversion iterates, overall residual
  energy decreases, and the standard deviation of the model parameters
  diminishes for well constrained members.}
\par
Figure~\ref{fig:mods} shows the forward modeled time picks from the
inversions over a synthesized shot gather.  The best model parameters from all 15 inversions are
forward modeled and plotted.  The points are very precise in picking
even minor deviations from regularity in the array geometry and are
much better at describing the event than the auto-picks in
Figure~\ref{fig:picks}.  The difference in quality between the picks that
went into the inversion and the result accounts for the residual energy
in Figure~\ref{fig:inv}\pan{a}.  The inversion was not
able to match the bad data (imprecise picks) because it was required to honor
equation~\ref{fwd}.  Therefore, the result is not able to converge to
zero.
\activeplot{mods}{width=6in,height=4.in}{CR}{The forward modeled time picks
  from 15 inversions are plotted over each panel in
  Figure~\ref{fig:event}.  The accuracy of the times is very faithful
  to the irregularities in the array layout in contrast to the
  auto-picked data input to the inversion, Figure~\ref{fig:picks}.}
\par
I selected values for map coordinates and velocity at the location of the
minimum from all 15 inversions using graphs such as those shown in 
Figure~\ref{fig:inv}\pan{b}-\pan{d}.  The best velocity for the three data
sets from 2004 was $\sim 1455$ m/s, and 1440 m/s for the data from 2005 which
are appropriate for compressional waves in the water column.  The
$(x,y)$ locations measured as distance from the NW corner of the array in
kilometers were: (2.5,-44.0), (2.0,-40.0), (2.4,-43.5), (2.75,-43.5)
in the rotated local coordinate system shown in
Figure~\ref{fig:minloc}.  Because the inversion returned very reliable
water velocities in all cases, the lack of sensitivity for the
subsurface parameters is mitigated.  If the event must be in the
water column, the sea surface and floor are the only potential
reflectors.  Since the receivers are on the sea floor, the most simple
explanation of these results is that the source is also at the sea
floor and reflects from the sea surface once before arriving at the
array.  However, at such large horizontal offset, and shallow water
column, the forward model cannot distinguish between this model and
the direct arrival from the surface to the sea floor.  Conversely, the
source could be at the array, travel roughly horizontally, and reflect
back from a nearly vertical object.  The latter situation is not
plausible, nor is the direct arrival associated with this model
recorded by the array.  From these results, I interpret the energy
as arriving in a direct path through the sea to the array.  The data do 
not contain sufficient information to distinguish between sea floor and sea 
surface source locations.
\par
I used the mathematic description of the direct arrival from the distant 
source to design a stacking operator.  Summing along the trajectory of the 
arrival at all times in the raw data returns a measure of the energy in 
the data associated with this event.  I also summed the energy in the 
raw data volumes associated with zero ray parameter.  The horizontal 
plane wave sum was consistently at least two orders of magnitude less than the 
value returned by the stacking operator designed with equation~\ref{fwd}
and the parameters calculated by inversion.  Similar partitions of 
energy were observed in the summations using the synthesized gathers 
instead of raw data.
\par
Using the velocity and location values calculated by the inversion, I 
forward modeled time picks at every receiver station for use as
data in an exhaustive search over location variables.  Figure~\ref{fig:xy} 
shows the results of the search using the four time sections of passive
recordings.  The gray scale
shows the inverse of the total data misfit normalized
by the maximum value.  The same line is plotted on the panels for 
reference ($8^o$ from the horizontal axis).  All data volumes resolve
the source location at similar locations with similar
trends in precision.  A request to the Norwegian Petroleum Directorate
indicates that there were no active seismic vessels acquiring data in
this region on or around February 15, 2004.  The similar location
derived from the data collected in 2005 indicates that surface seismic 
acquisition is likely not the cause of this feature.
\activeplot{xy}{width=6in,height=4.in}{CR}{Sensitivity to horizontal location
  for the inversions using the four periods of passive data.}
\par
Figure~\ref{fig:mapa} is a detailed map of the Norwegian and British
oil infrastructure around the Valhall reservoir.  Also plotted 
are the footprint of the receiver array (gray rectangle), and the location 
of the trace used as the virtual source function (push-pin icon G118) 
for the gathers presented above.  Superimposed
as a flat image (and thereby introducing small projection errors) is a
map of the British production facilities in the area.  The black line
from the receiver array ends in the SW at the locus of the energy
calculated by the inversions described above.  The location of the
source for the dominant events in the synthesized gathers is exactly over the
Ardmore field in British controlled section of the North Sea.
\activesideplot{mapa}{width=3in,height=3in}{NR}{A small section of the British
  infrastructure map, the Norwegian infrastructure, and the
  location of the energy source inverted in Figure~\ref{fig:xy}.  The
  inverted locus of the energy in the gathers (the SW terminus of the
  black line) corresponds exactly to the location of the
  Ardmore development.}
The British Department of Trade and Industry
\footnote{www.og.dti.gov.uk/environment/permits/TuscanArdmore.htm}
describes the Ardmore development.  It differs from the
Valhall operation in several important ways.  The Ardmore facility 
has no permanent structures, nor is it connected to an export
pipeline.  Instead, two vessels are permanently moored over the field:
One contains production facilities, the other is used to store liquids
for periodic transport to shore.  This operation has no way to handle
produced gas and must therefore flare all produced gas.
\section{Conclusions}
\par
The energy responsible for the features in the raw data,
Figures~\ref{fig:raw2}~-~\ref{fig:raw3ev}, are likely related as they
share a locus at the production facilities within the array and are both 
characterized by water velocity.  I suspect that the noise-train in
Figure~\ref{fig:raw2} is a collection of aliased repeating events
similar that found in Figure~\ref{fig:raw3ev}.  Also, the gathers
generated with data showing the diffuse noise-train,
Figure~\ref{fig:raw2}, have almost identical kinematics to the raw
data showing a crisp water-velocity event, Figure~\ref{fig:raw3ev}. It
is possible that some activity on the platforms could sometimes be
impulsive in nature or more drawn out.  In either case, the correlated
gathers do not show convincing evidence that the energy penetrates the
subsurface after the direct arrival is recorded.  This analysis
also shows that the predominance of observable energy
in the raw data shares a single source location.
\par
Correlating the raw data to produce synthesized shot gathers was
always performed on 12 second records.  These individual results were
summed from 1-2500 times to generate some approximation to active
seismic shot gathers in accordance with the theory of interferometric
imaging.  The gathers produced with a single 12 second record,
Figure~\ref{fig:evtest}, or 15 hours of data correlated in 12 second
sections, Figure~\ref{fig:event}, show an identical strong event
with velocity 1450 m/s arriving from the Ardmore field 40 km to WSW.  
\par
It seems counter-factual to attribute the source of this energy to a
distant production operation when production activities within the 
array have not generated 
events that match the power and bandwidth of this most obvious
arrival.  Gas flaring at Ardmore is the most significant 
difference between the two operations.  I believe that the gas flare is 
the energy source for the strong off-array event revealed by whitened
correlation processing.  Synthesized shot gathers produced with short, 
12~s, time windows of raw data show very different results depending on 
the character of the raw data used.  Figure~\ref{fig:evtest} shows that 
when there are high levels of local energy generated by the Valhall 
production facilities, synthesized gathers produced with short time windows 
do not produce the far-field production event.  Otherwise, the far-field
production energy dominates the correlation result using raw data windows
as short as 12~s and as long as 9 hours.
\par
The synthesized gathers produced with 36 hours of passive recordings
do not provided subsurface information.  The gathers are dominated by 
a cultural noise source.  The energy within this noise is several orders
of magnitude stronger than anything else in the data.  The strength and 
continued presence of this energy makes it very difficult to remove
or ignore.  Several subtraction methods, including inversion, were 
not successful in removing the energy.  
\clearpage
\label{.end}
