Correlation

Following the theory from Wapenaar et al. (2004), the basic principle of time processing passively recorded seismic records to yield the kinematics of actively collected data dictates  

$$2\Re [R({\bf x}_r,{\bf x}_s,\omega)]= \delta({\bf x}_s-{\bf ... ...ox{\boldmath$\xi$},\omega)\; \mbox{d}^2\mbox{\boldmath$\xi$}\;,$$ (1)

where ^* represents conjugation. The vector ${\bf x}$ corresponds to horizontal coordinates, where subscripts r and s indicate different station locations from a transmission wavefield T. After correlation, r and s acquire the meaning of receiver and source locations, respectively, associated with an active survey. The RHS represents summing correlations of windows of passive data around the occurrence of individual sources from locations $\mbox{\boldmath$\xi$}$. The symbol $\partial \! D_m$ represents the domain boundary that surrounds the subsurface region of interest on which the sources are located. The transmission wavefields $T({\bf x}_r,\mbox{\boldmath$\xi$},\omega)$ contain the arrival and reverberations due to a single subsurface source. To synthesize the reflection experiment exactly, impulsive sources should completely surround the volume of the subsurface one is trying to image. Alternatively, many impulses can be substituted with a full suite of planewaves emerging from all angles and azimuths.

Because windowing individual arrivals from the 79 Gbytes of data associated with the hydrophone measurement is impractical, it is impossible to honor equation 1 precisely by correlating traces within time windows when only a single event is active. However, this effort makes 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 second records from all the stations were correlated in accordance with equation 1 and the results stacked over the duration of the (hours long) section of available data. Thus  

$$\hat{R}({\bf x}_r,{\bf x}_s,\omega)= \sum_\xi T({\bf x}_r,\omega,\xi) T^*({\bf x}_s,\omega,\xi)$$ (2)

was calculated where $\xi$ is the set of sequential 12 second chunks of data.

Figure  shows correlations from one of the 9 hour sections of data. A trace from the NW corner 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 energy 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  plots the power spectrum of the sum of 100 traces from the gather in Figure . Versions of the data bandpassed around and between the peaks made no appreciable increase in interpretability.

 

cor
Figure 7 Correlated shot-gather with source location at the left side of the panel. 0-60 Hz energy was used for the correlation performed in the frequency domain.

 

spec Figure 8 Power spectrum of the gathers in Figure .

Spectral whitening by division of a trace in the frequency domain with its power produces remarkable results when compared to simple correlation. Gathers were also produced using the relation  

$$\hat{R}({\bf x}_r,{\bf x}_s,\omega)= \sum_\xi \frac{T({\bf x... ...({\bf x}_r,\omega,\xi)\;T^*({\bf x}_s,\omega,\xi)\vert\vert}\;.$$ (3)

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. Smoothing across the frequency axis mixes early- and late-time energy, and therefore is not appropriate.

Figure  shows the same gather as in Figure  using the spectral whitening of equation 3. 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 km² of the array and has reasonable bandwidth. Versions of data were bandpassed across the intervals between and surrounding the energy maximums in power spectrum, Figure , 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. After 42 Hz, the signal begins to diminish markedly, and no coherent energy is present at all after 52 Hz.

 

dec
Figure 9 Trace-by-trace spectral whitening applied to shot-gather in Figure . Line connecting minimum travel-times almost perfectly straight.

Events are hyperbolic. This indicates the event could be caused by any solution to the class of hyperbolic PDE's. 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. This means that 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. This indicates that the source must be to the North and West. Moveout is curve-linear in the inline direction, but very close to linear in the cross-line direction. The line connecting the minimum travel-times of the event on each cable deviates from straight only at the far right side. Notice from the map in Figure  that crossline offset there is twice the normal spacing. This indicates that the array is close to the asymptotic limit of the hyperbolic event in the cross-line direction, and nearer the hyperbola top in the inline direction. These two observations indicate that 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 , the source of energy should be somewhere West of the location toward the English 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 1 has not been honored. If only a single shot is captured by a passively recorded array, correlation simply shifts the event up such that it arrives at t=0 at the source-trace used to create the gather. The moveout of the event will not be altered at all. I believe the energy is localized in space, but seemingly ubiquitous in time. Figure  shows four lines from the array from the same correlated shot-gather from the four time periods into which the data volume was divided. The lines are the first and last two complete receiver lines from Figure . The third panel is a subset of the gather shown in Figure . 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 obvious energy is the same. The last panel was computed with data recorded almost one year after the data used in the first three panels.

 

event
Figure 10 Ubiquitous strong event in correlated and deconvolved shot-gathers from data recorded 9 hours, 25 hours, and 11 months after that used to produce the first panel. The third panel is a subset of the gather shown in Figure .

Figure  shows four synthesized shot-gathers using data shown in the previous section concerning the character of the raw recordings. The source trace used for correlation in all panels was the first trace on North end of line 2 (trace 119). All panels use the deconvolutional variant of correlation described by equation 3 and show the Eastern side of the array. The first panel is the first three seconds of the causal lags using the data shown in Figure  which showed the ringing noise-train. No strong events are present, but there are some faint ringing hyperbolas that show water velocity.

Correlation subtracts the time to the source trace from all the other traces in the gather. The beginning of the noise-train in Figure  on the first trace (number 119), 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. The acausal lags in the second panel of Figure  show exactly what was expected. Overlain on the correlated event are time picks modeled with the same 1450 m/s arrival used to delineate the noise-train in Figure . The hyperbolas ring to the bottom (t=0) of the panel. Given the long (7 s) coda of the noise train, these are probably correlation side-lobes that are also responsible for the events seen in the panel to its left.

The bottom row of Figure  are the deconvolved correlated gathers using trace 119 as a source and the data from Figures  & Figure  respectively. In each case only 12 s of data were correlated. Faint hyperbolas are discernible in the early time of the right side of the right panel. The most obvious energy however is in the events similar to the strong arrivals in Figure  and Figure .

 

evtest
Figure 11 Correlated gathers from the Eastern side of the array. Top row: Early causal and acausal lags of a correlated gather using data in Figure . Picks on right panel have same (1450 m/s) velocity and origin as those delineating the top and bottom boundaries of the raw data. Bottom row: Early causal lags of correlated gathers using data in Figure  & Figure .

Correlation masks some of the characteristics of the source energy. The source could be impulsive or a long sweep containing many 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 , I favor the latter explanation.

Figure  shows the same correlated gathers as Figure . Superimposed on the data are time picks that were auto-picked by amplitude. The picks are not very accurate due to the effort required to maintain stability. 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.

 

picks
Figure 12 Auto-picked time values along event in correlated gathers. Picks are used as data when inverting for the source location.

The equation describing the travel-time of a planar reflection in shot-receiver coordinates is  

$$v^2t^2={\bf x}^2+4z^2+4z{\bf x}\cos{\alpha}\sin{\phi}$$ (4)

where ${\bf x}$ is the map distance from the source to the receiver, z is the depth of the plane under the source location, $\alpha$ is the azimuth of the line ${\bf x}$ minus 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 the data with a single source trace subtracts the time to the source trace from each trace in a gather. Therefore, the forward modeling operator used in the inversion solves for the time, t_s, to the source-trace location using equation 4 and subtracts this value from each trace in the shot-gather to produce 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 result in the first term, ${\bf x}^2$, to dominate the others. The inversion technique used was a micro-genetic algorithm. The model space is: Average velocity through which the rays have passed, areal location of the source, and orientation angles and depth of the reflection plane. The fitness function for the genetic algorithm was the l² norm.

The panels in Figure  show the effectiveness and quality of the inversion results. The genetic algorithm uses a random seed number that determines the characteristics of the first population as well as how the future generations change. Fifteen seeds were used to find the source location and medium velocity for all four data sections. Figure  is a representative sample of the four inversion results. The panels in the figure show the convergence of the algorithm in total (panel 1), and the convergence of the model parameters: velocity, distance East, and distance North respectively. 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 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 similar plots for the model parameters associated with the planar reflection: depth, dip, and azimuth. These parameters were completely in the null space of the inversion.

 

inv Figure 13 Residual energy of the inversion to locate the source of the energy causing the event in Figure . As the inversion iterates, overall residual energy decreases, and the standard deviation of the model parameters diminishes for well constrained members.

Figure  shows the forward modeled time picks from the inversions. The best model parameters from all 15 inversions were 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 . The quality difference between the picks that went into the inversion and the result account for the residual energy in the first panel of Figure . The inversion was not able to match the bad data because it was required to honor the laws of physics. Therefore, the result is not able to converge to zero.

 

mods
Figure 14 The forward modeled time picks from 15 inversions are plotted over each panel in Figure . 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 .

Using graphs such as those shown in Figure , values for map coordinates and velocity were selected at the location of the minimum from all 15 inversions. The best velocity for the three data sets from 2004 was 1460 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 . 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.

Using the values listed above, time picks at every receiver station were forward modeled for use as data in an exhaustive search inversion. The gray scale in Figure  shows the inverse of the data misfit normalized by the maximum value. The velocities used were those stated above. The same line is plotted on the panels for reference (8^o from the horizontal axis). All data volumes resolve the source location at very similar locations with almost identical 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.

 

xy
Figure 15 Sensitivity to horizontal location for the inversions using the four periods of passive data.

The geometric construct used to calculate the kinematics of a reflection from a flat plane is to double the time from the source to the midpoint on the plane. It is also possible that the source location was twice the distance away and the event is a multiple along the same azimuth. Another possible ray path which shares the kinematics of the forward modeling operator is one with three ray paths through the water column. This event begins at the surface, is reflected up from the sea floor, and then once more from the surface to be recorded by the OBC sensors. This travel path is at most 2/3 of the distance from the array along the same azimuth. Similarly, any odd number of raypaths, k, through the water column will move the source toward the array by 2/k. Therefore, it is plausible to move the source location from the inversion almost anywhere along the azimuth of the lines plotted in Figure .

Figure  is a detailed map of the Norwegian and British oil infrastructure around the Valhall reservoir. Also plotted are the footprint of the receiver array, and the location of the trace used as source function (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 provided by the inversions described above. The location of the source for the events in Figure  is exactly over the Ardmore field in British controlled section of the North Sea.

 

mapa Figure 16 A small section of the British infrastructure map with the Norwegian infrastructure and the location of the energy source inverted in Figure . 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.