An iterative SVD method for deblending: theory and examples
`Margherita Maraschini*, Richard Dyer, Kent Stevens, Dave Bird, Simon King
`*Fugro Seismic Imaging Ltd.
`
`Summary
`
`Blended acquisition is an important concept because it
`offers the unusual economic prospect of higher quality
`(increased sampling) at a reduced cost (shorter acquisition
`time). The technique consists of activating two or more
`sources almost simultaneously, which allows many shots to
`be recorded in the time normally taken to acquire just one.
`The main challenge is to separate each blended shot into its
`constituent unblended shots, or equivalently remove for
`each of the contributing sources the noise contamination
`due to the firing of the other sources. In this expanded
`abstract we present an iterative method based on rank
`reduction filtering which removes most of the crosstalk
`noise, even in the presence of a low signal to noise ratio. Its
`effectiveness is demonstrated on an artificially blended
`dataset and a real blended wide azimuth dataset.
`
`Introduction
`
`Blended (or simultaneous source) acquisition is a novel
`shooting technique that has the potential to both reduce
`acquisition time and increase spatial sampling (Berkhout
`2008). In conventional acquisition sources are fired such
`that the energy from one source has decreased to a low
`level before the next source is fired. In blended acquisition
`this is not the case, sources are fired almost simultaneously.
`Blended acquisition is widely used in land acquisition
`(Bagaini 2010) where sources can be encoded and thus
`separated more easily. However, marine blending has in
`recent years been the subject of considerable research. In
`order
`to make marine blending
`feasible, efficient
`deblending algorithms are required. Some examples of
`source separation algorithms can be found in the literature:
`Beasley et al. (1998) described a source separation method
`for the case where the sources are positioned at opposite
`ends of the streamer, Stefani et al. (2007) introduced
`randomization of the firing time of the second source that
`makes its energy incoherent in common receiver, offset and
`midpoint domains, making the energy easier to remove;
`Moore et al. (2008, 2010) outlined a signal separation
`technique using sparse inversion of a linear system, and
`Mahdad et al. (2011) proposed an iterative method based
`on coherency filtering and thresholding.
`
`The algorithm presented in this expanded abstract is an
`iterative algorithm
`that gradually
`reconstructs
`the
`individual source contributions. It relies on each source
`firing quasi-randomly in a small time interval about a
`regular shooting sequence; during the firing process the two
`boats operate independently, firing around predetermined
`
`times or spatial positions. The signal reconstruction
`exploits the fact that each source response can be viewed in
`one time configuration as random noise and in a different
`time configuration as coherent signal: in each iteration
`some coherent components are extracted from the original
`dataset translated in the appropriate time configuration.
`The results obtained on a synthetically blended real dataset
`(with both sources on the same side of the streamer) and a
`real blended dataset confirm the effectiveness of our
`algorithm.
`
`Method
`
`The kernel of the algorithm is a noise attenuation technique
`called matrix rank reduction or truncated singular value
`decomposition. Singular value decomposition (SVD) is a
`mathematical process which decomposes an arbitrary,
`complex-valued matrix A into the sum of rank-1 matrices Ii
`(called eigenimages):
`
`
`
`
`
`
`
` A = I1 + ... + In .
`A fundamental property of this decomposition is that
`
`
`
`Ak = I1 + ... + Ik
`(where 1≤k≤n) is the optimal rank-k approximation of A (in
`the least-squares norm sense). Rank reduction is the
`process of reducing a matrix to an optimal lower rank
`approximation by truncating the SVD.
`
`Rank reduction filtering has been successfully applied in
`the seismic processing field for noise suppression, both
`directly in the t-x domain and to constant frequency slices
`of f-x-y cuboids (Trickett 2003, Trickett and Burroughs
`2009, Oropeza and Sacchi 2011, Gao et al. 2011). We
`follow the latter approach, applying rank reduction to
`Hankel matrices generated from constant frequency slices
`of the f-x-y domain, called Cadzow filtering. Coherent
`energy is localized on the first few eigenimages, while
`random noise is spread evenly over all eigenimages.
`Cadzow filtering thus suppresses random noise whilst
`preserving coherent events.
`
`The proposed method is now described. Suppose we have a
`blended acquisition with N shooting vessels. The seismic
`data can be written as D = ∑i Ti(Di) where Di is the
`theoretical response of the ith source (without delays) and Ti
`is a time-delay operator: the time-delay operator Ti shifts
`each trace of Di by the shot variant time-delay (with sign)
`of source i. We consider D to be a single source-cable
`combination from one sail-line of a 3D marine survey,
`conceptually a data cuboid whose dimensions are t-x-y
`(time-offset-shot).
`
`
`© 2012 SEG
`SEG Las Vegas 2012 Annual Meeting
`
`DOI http://dx.doi.org/10.1190/segam2012-0675.1
`Page 1
`
`Downloaded 09/18/15 to 64.124.209.76. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`PGS Exhibit 2024
`WesternGeco v. PGS (IPR2015-00309, 310, 311)
`
`

`
`Iterative SVD for deblending
`
`(Tj)-1
`the
`remove
`(i.e. we
`to D
`If we apply
`Example 1: a shallow water synthetically blended real
`j)
`time-delay
`sequence of
`source
`then we get
`dataset
`(Tj)-1(D)=Dj + ∑i≠j (Tj)-1TiDi. If viewed in the common
`In order to evaluate the effectiveness of the deblending
`offset domain, component Dj will appear coherent, while
`algorithm and to determine the influence of the acquisition
`each other component (Tj)-1TiDi will be random due to the
`parameters, several tests on synthetically blended datasets
`random time shifting operator (Tj)-1Ti (which applies to the
`have been performed. In the following figures (all plotted
`dataset the difference between the time-delays of source i
`with the same scale) we show the results of the separation
`and j). This shows that it is important that the respective
`of a synthetically blended dataset, obtained by
`the
`time-delay sequences (encoded in the T operators) are pair-
`combination of two source cable combinations of a marine
`wise relatively random. Call (Tj)-1(D) the original dataset in
`dataset acquired in a shallow water environment offshore
`the time configuration j. Our algorithm exploits knowledge
`Australia; the two lines are combined applying random
`of the time-delay operators, continually switching between
`time-delays between [-100ms, 100ms] to the second line
`configurations.
`and summing them.
`
`
`The structure of our algorithm has similarities with the
`method presented by Mahdad et al. (2011). Figure 1 shows
`the pseudocode of the algorithm used. It is an iterative
`algorithm
`that gradually
`reconstructs each
`source
`component. The algorithm ends when the maximum
`number of iterations or the maximum rank that we are
`interested in reconstructing is reached. In each iteration the
`algorithm repeats the same procedure for all the sources:
`remove from the original data the previous estimates of the
`other sources, remove part of the data that we can decide a
`priori does not belong to the component we are trying to
`reconstruct by means of a filter function F, move to the
`time configuration of the considered source (calculation of
`Ai in Figure 1) and apply filtering (calculation of Aik) and
`tresholding (calculation of Di). Filtering and tresholding
`parameters are updated on every iteration. At the end we
`can choose either to output the raw estimates (Di) or
`conservative estimates (Ti)-1 (D - ∑i≠j Tj (Dj)).
`
`1 D = Blended data t-x-y cuboid
`2 Di = 0 for all i
`3 for (iter < niter) & while (k < rank_max)
`4 {
`for (i < Nblend)
`5
`6 {
`= (Ti)-1 (F ( D - ∑i≠j Tj (Dj)))
`7
` Ai
`
`= Cadzow_filter_k(D*
`i)
`9
` Aik
`= Threshold(D*
`ik)
`10 Di
`11
` update parameters
`12 }
`13 }
`14 if (conservative estimates)
`15 {
`16 Di_final = (Ti)-1 (D - ∑i≠j Tj (Dj)) for all i
`17 }
`18 else
`19 {
`20 Di_final = Di for all i
`21 }
`
`a) b)
`
`c) d)
`
`e) f)
`
`
`
`
`
`
`
`g) h)
`
`
`
`
`
`Figure 2 Example of common shot gather: a) Signal 1- Original
`shot gather; b) Signal 2 - Original shot gather; c) Signal 1 -
`Deblended shot gather; d) Signal 2 - Deblended shot gather; e)
`Signal 1 - Unblended shot gather; f) Signal 2 - Unblended shot
`gather; g) Signal 1 - Difference between c and e; h) Signal 2 -
`Difference between d and f.
`
`Figure 2 shows the results in term of shot gathers: the
`figures on the left show the results relative to the first
`
`Figure 1: Iterative algorithm pseudocode for source separation.
`
`© 2012 SEG
`SEG Las Vegas 2012 Annual Meeting
`
`DOI http://dx.doi.org/10.1190/segam2012-0675.1
`Page 2
`
`Downloaded 09/18/15 to 64.124.209.76. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`PGS Exhibit 2024
`WesternGeco v. PGS (IPR2015-00309, 310, 311)
`
`

`
`Iterative SVD for deblending
`
`source cable combination, while the figures on the right
`show the results relative to the second source cable
`combination. Figures 2 a-b show an example of the original
`-1(D), respectively); Figures 2 c-
`
`shot record (T1-1(D) and T2
`d show the deblended results (our best estimation of D1 and
`D2), Figures 2 e-f show the original dataset D1 and D2, and
`Figures 2 g-h show the difference between our best
`estimation and the theoretical solution. The analysis of
`these figures demonstrates that the blending noise has been
`almost completely removed from the shot gathers by the
`source separation algorithm.
`
`
`
`a) b)
`
`c) d)
`
`e) f)
`
`
`
`
`
`
`
`
`
`g) h)
`
`Figure 3 Stack of signal 1: a) Original dataset; b) Zoom of (a); c)
`Deblended dataset; d) Zoom of (c); e) Unblended dataset; f) Zoom
`of (e) ; g) Difference between b and c ; h) Zoom of (g).
`
`
`Figure 3 shows the stack of the first line, the images on the
`right are a zoom into a noisy area of the figures on the left.
`No other filter than the source separation algorithm has
`been applied to the dataset before the stacking, in order to
`obtain a fair comparison of the results. Observing these
`figures, we can notice that most of the blending noise has
`been removed (see Figures a-c versus b-d), while no
`coherent signal has been attenuated (see Figures g-h).
`
`Example 2: a real marine blended dataset
`
`The source separation algorithm has then been applied to a
`real blended dataset (random time-delays of the second
`sources between [0ms, 500ms]) acquired in a shallow water
`environment in the Eastern Mediterranean Sea; the second
`source is located about 1km cross-line relative to the first.
`This dataset is a challenging test for the source separation
`algorithm due to the magnitude of the time-delays: due to
`the attenuation of signal 1 with time, the blending noise
`generated by the second source can be 5 times stronger than
`the signal we would like to preserve. In the following the
`reconstruction of the events associated with the second
`source is not shown because the blending noise overlaying
`it is already low due to the contrast in amplitudes.
`An example of the iterative reconstruction of the deblended
`shot record is shown in Figure 4. In these figures only the
`first 4 seconds and 317 channels are shown. From the
`comparison of Figure 4 e-f we can observe that most of the
`blending noise has been removed by the algorithm.
`
`
`
`
`a) b)
`
`c) d)
`
`
`
`
`
`e) f)
`Figure 4 Eastern Mediterranean Sea dataset – Deblended shot
`records - zoom: a) Iteration 1; b) Iteration 2; c) Iteration 5; d)
`Iteration 10; e) Iteration 19 (final); f) Original
`
`
`
`
`
`© 2012 SEG
`SEG Las Vegas 2012 Annual Meeting
`
`DOI http://dx.doi.org/10.1190/segam2012-0675.1
`Page 3
`
`Downloaded 09/18/15 to 64.124.209.76. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`PGS Exhibit 2024
`WesternGeco v. PGS (IPR2015-00309, 310, 311)
`
`

`
`Iterative SVD for deblending
`
`a) b)
`
`c) d)
`
`e) f)
`
`
`
`
`
`
`
`
`
`g) h)
`
`Figure 5 Eastern Mediterranean Sea dataset: a) Stack of the
`original dataset; b) Zoom of (a); c) Stack of the deblended dataset;
`d) Zoom of (c); e) Stack of the unblended dataset; f) Zoom of (e);
`g) Noise removed; h) Zoom of (g).
`
`Figure 5 shows the comparison between the stack of the
`original data, of the deblended data, and of another line
`recorded in about the same position; images on the right are
`zooms of the dark green rectangles in the images on the
`left. Figures 5 a-b show that the blending noise present in
`the dataset is seriously compromising the stack, hiding the
`first two seconds. Most of this noise is removed by the
`deblending process (Figures 5 c-d) with good signal
`preservation. The source separation process allows to
`identify events that in the stack of the original dataset were
`hidden. The results are close to the unblended line used for
`comparison (Figures 5 e-f). Also in this example, no other
`filtering process than the source separation has been
`applied before the stacking.
`
`
`a) b)
`
`
`
`
`
`c) d)
`
`Figure 6 Eastern Mediterranean Sea dataset – zoom in the light
`green rectangle: a) Original dataset; b) Deblended dataset; c) Noise
`removed; d) Unblended dataset.
`
`When we look at the light green rectangle in Figure 5 g, we
`can note a coherent event. Figure 6 shows the zoom on that
`area, and the green circle the coherent event that has been
`removed by the source separation. We can note that this
`event is present in the original stack (a) but not in the
`deblended stack (b). But this event is absent also in the
`stack of the unblended line used for comparison. That
`means that this event belongs to signal 2, and it has been
`recognized by the source separation code even if it is
`coherent in the stack.
`
`Conclusions
`
`This paper describes a new method to perform the
`separation of blended datasets based on the presence of
`varying time-delays between the firing of the sources. The
`method works without any restriction on the position of the
`sources with respect to the receiver array, allowing it to
`deal with situations like wide-azimuth or flip-flop shooting.
`The presented results, for both the synthetically blended
`real data and the real shallow water data, show the ability
`of the algorithm to separate two blended sources preserving
`the signal amplitudes, also when the signal to blending
`noise ratio is very low.
`
`Acknowledgments
`
`We thank Stewart Trickett, Stuart Merrylees, and Thomas
`Hertweck from Fugro Seismic Imaging, Thomas Elboth
`from Fugro-Geoteam and Robert Pfau and Niklas Thiel
`from Karlsruhe Institute of Technology for their help. We
`thank our colleagues of other Fugro companies for their
`help in acquiring and processing the data. We thank Edison
`International S.p.A. for permission to publish the Eastern
`Mediterranean Sea data, and Fugro management for the
`permission to publish these results.
`
`© 2012 SEG
`SEG Las Vegas 2012 Annual Meeting
`
`DOI http://dx.doi.org/10.1190/segam2012-0675.1
`Page 4
`
`Downloaded 09/18/15 to 64.124.209.76. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`PGS Exhibit 2024
`WesternGeco v. PGS (IPR2015-00309, 310, 311)
`
`

`
`http://dx.doi.org/10.1190/segam2012-0675.1
`
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`© 2012 SEG
`SEG Las Vegas 2012 Annual Meeting
`
`DOI http://dx.doi.org/10.1190/segam2012-0675.1
`Page 5
`
`Downloaded 09/18/15 to 64.124.209.76. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`PGS Exhibit 2024
`WesternGeco v. PGS (IPR2015-00309, 310, 311)