GEOPHYSICS, VOL. 76, NO. 3 (MAY-JUNE 2011); P. Q9–Q17, 9 FIGS.
`10.1190/1.3556597
`
`Separation of blended data by iterative estimation and subtraction of
`blending interference noise
`
`Araz Mahdad1, Panagiotis Doulgeris1, and Gerrit Blacquiere1
`
`ABSTRACT
`
`Seismic acquisition is a trade-off between economy and qual-
`ity. In conventional acquisition the time intervals between suc-
`cessive records are large enough to avoid interference in time.
`To obtain an efficient survey, the spatial source sampling is
`therefore often (too) large. However, in blending, or simultane-
`ous acquisition,
`temporal overlap between shot records is
`allowed. This additional degree of freedom in survey design
`significantly improves the quality or the economics or both.
`Deblending is the procedure of recovering the data as if they
`were acquired in the conventional, unblended way. A simple
`least-squares procedure, however, does not remove the interfer-
`ence due to other sources, or blending noise. Fortunately, the
`character of this noise is different in different domains, e.g., it is
`coherent in the common source domain, but incoherent in the
`common receiver domain. This property is used to obtain a con-
`
`siderable improvement. We propose to estimate the blending
`noise and subtract it from the blended data. The estimate does
`not need to be perfect because our procedure is iterative. Start-
`ing with the least-squares deblended data, the estimate of the
`blending noise is obtained via the following steps: sort the data
`to a domain where the blending noise is incoherent; apply a
`noise suppression filter; apply a threshold to remove the remain-
`ing noise, ending up with (part of) the signal; compute an esti-
`mate of the blending noise from this signal. At each iteration,
`the threshold can be lowered and more of the signal is recov-
`ered. Promising results were obtained with a simple implemen-
`tation of this method for both impulsive and vibratory sources.
`Undoubtedly, in the future algorithms will be developed for the
`direct processing of blended data. However, currently a high-
`quality deblending procedure is an important step allowing the
`application of contemporary processing flows.
`
`INTRODUCTION
`
`In current seismic data acquisition, sources are fired with
`large time intervals in order to avoid interference, leading to
`time-consuming and expensive surveys. Furthermore, the source
`side of the acquisition geometry is often coarsely sampled, caus-
`ing spatial aliasing. The concept of simultaneous acquisition has
`been introduced to address these issues by either reducing the
`temporal interval between successive shots, leading to reduced
`acquisition costs, or by increasing the number of sources within
`the same survey time, leading to a higher data quality. Note that
`a combination of the two approaches combines these benefits.
`Several authors have discussed the concept of simultaneous
`and near-simultaneous shooting for impulsive and vibroseis-type
`sources and their particular advantages. The use of simultaneous
`vibrators transmitting the same or different reference signals
`was proposed by Silverman (1979). Beasley et al. (1998) and
`
`Beasley (2008) proposed acquiring seismic data by means of si-
`multaneous impulsive sources with large spacing between illu-
`minating shots. The High Fidelity Vibratory Seismic (HFVS)
`method has been developed by Sallas et al. (1998) in order to
`increase the productivity of land seismic acquisition and to
`reduce acquisition costs. Romero et al. (2000) suggested the use
`of phase encoding in prestack shot record migration such that
`multiple shots can be migrated simultaneously. Although this
`work was focused on the migration process, it can be simply
`generalized to the acquisition phase. Acquiring marine seismic
`data with random, quasirandom, or systematic delay times
`between firing sources was proposed by Vaage (2002). Bagaini
`(2006) discussed various
`simultaneous vibroseis acquisition
`methods including simultaneous sweeps, cascaded sweeps, and
`slip sweeps. The term source blending was introduced by Berkh-
`out
`(2008), and the differences with plane wave synthesis
`
`Manuscript received by the Editor 29 July 2010; revised manuscript received 8 November 2010; published online 28 April 2011.
`1Delft University of Technology, Geotechnology Department, Section of Applied Geophysics and Petrophysics, Delft, The Netherlands. E-mail: a.mah-
`dad@tudelft.nl; p.doulgeris@tudelft.nl; g.blacquiere@tudelft.nl.
`VC 2011 Society of Exploration Geophysicists. All rights reserved.
`
`Q9
`
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`
`Q10
`
`Mahdad et al.
`
`(Rietveld and Berkhout, 1994) were pointed out. A near-simulta-
`neous
`shooting technique with small
`random time delays
`between impulsive sources was presented by Hampson et al.
`(2008). Vibroseis acquisition by means of Simultaneous Pseu-
`dorandom Sweep Technology (SPST) has been suggested by
`Sallas et al. (2008). Berkhout et al. (2009) extended the concept
`of source blending to the detector side by combining incoherent
`shooting with incoherent sensing and introducing the concept of
`double blending.
`The separation process (deblending) was addressed as a blind
`signal separation problem by Ikelle (2007), using independent com-
`ponent analysis as the tool to distinguish between the different
`blended sources. Lin and Herrmann (2009) and Herrmann et al.
`(2009), under the umbrella of compressive sensing, use an inver-
`sion approach constraining the separated data to be sparse in the
`curvelet domain. It is worth mentioning that both these approaches
`use sophisticated source codes (e.g., sweeps or random phase or/
`and amplitude encoding). Furthermore, Neelamani et al. (2008)
`have used simultaneous sources and compressive sensing in a simi-
`lar way to speed up forward modeling.
`By reforming the deblending problem into a denoising one,
`treating the interference due to blending as noise, one can use
`all kinds of signal processing tools available.
`It has been
`reported by various authors — e.g., Moore et al. (2008), Aker-
`berg et al. (2008) — that by sorting the acquired blended data
`into a different domain than the common source domain (e.g.,
`the common offset domain), the interference noise appears as
`random spikes; thus, the separation process turns into a typical
`random noise removal procedure. Based on this property, Huo
`et al. (2009) use a vector median filter after resorting the data
`into common mid-point gathers. This 2D filter acts locally and
`effectively reduces the amplitude of the random spikes. Moore
`(2010) uses an inversion process with sparsity constraint applied
`in the radon domain.
`Spitz et al. (2008) introduced the idea of building a noise
`model based on the subsurface’s velocity model and the wave
`equation. The modeled interference noise is adaptively sub-
`tracted from the data. Moving a step further, Kim et al. (2009)
`built a noise model from the data itself and then adaptively sub-
`
`Figure 1. Schematic representation of data matrix. Every element
`is a complex valued number that represents one frequency
`component.
`
`tracted the modeled noise from the acquired data. This algo-
`rithm was implemented in the common offset domain and was
`applied to OBC (ocean bottom cable) data.
`In the present work, we developed an iterative noise estimation-
`and-subtraction process for the effective separation of blended
`data. A noise model is progressively built from the blended data
`itself and subsequently subtracted. The method, which can also be
`formulated as a steepest descent type of method, was applied to
`field data, where the blending process had been simulated numeri-
`cally for both impulsive and vibrating sources.
`
`The concept of source blending and pseudodeblending
`
`Berkhout (1982) showed that seismic data (2D or 3D) can be
`arranged in the so-called data matrix P. In the temporal fre-
`quency domain, each element of P corresponds to a complex-
`valued frequency component of a recorded trace, each column
`represents a shot record, each row represents a detector gather,
`each diagonal represents a common offset gather, and each anti-
`diagonal represents a common midpoint gather. Figure 1 illus-
`trates the data matrix. An example of the different gathers
`extracted from the data matrix is given in Figure 2 for a numeri-
`cal data set.
`A system representation of seismic data is given by the fol-
`lowing monochromatic expression (Berkhout, 1982):
`
`ð
`P zd; zs
`
`Þ ¼ D zdð
`
`
`
`ÞX zd; zsð
`
`ÞS zsð Þ:
`
`(1)
`
`Here zs and zd correspond to the source and detector depth level
`respectively. S represents the source matrix where each column
`corresponds to the source wavefield at zs due to one source
`(array). D represents the detector matrix where each row repre-
`sents one detector (array). The X matrix is the multidimensional
`transfer function of the earth, which contains the entire subsur-
`face impulse response, including (internal) multiples, wave con-
`version, etc. The importance of the source and detector sampling
`and other acquisition design parameters can be clearly realized
`from the logical combination of the D and S matrices with the
`X matrix. If the source and detector side of the acquisition are
`sparsely sampled or badly designed, X can not be well repre-
`sented by P.
`The concepts of source blending and incoherent shooting
`stand for the continuous recording of sources that are encoded
`with incoherent codes. Source blending is theoretically consist-
`ent with plane wave synthesis and controlled source illumination
`(Rietveld and Berkhout, 1994) in the sense that multiple sources
`are activated within certain time intervals. However, it differs in
`the way that the latter methods generate continuous (coherent)
`wavefronts. In the case of source blending it is important that
`such wavefronts are not generated. This is because of the spatial
`band limitation introduced in this way. Instead, we know that
`the full temporal and spatial bandwidth is preserved in the case
`that a white, random signal is arriving at every subsurface loca-
`tion, i.e., white within the available bandwidth. By incoherent
`shooting of the sources in blending we are aiming at preserving
`the full
`temporal and spatial bandwidth; see also Lin and
`Herrmann (2009). In general, source blending can be formulated
`as follows:
`
`ð
`P0 zd; zs
`
`
`
`Þ ¼ P zd; zsð
`
`ÞC ¼ D zdð
`
`
`
`ÞX zd; zsð
`
`ÞS zsð ÞC;
`
`(2)
`
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`
`Separation of blended data
`
`Q11
`
`where P0 is the blended data matrix. Blending matrix C contains
`the blending parameters. Each column C‘ is related to a blended
`shot record and its elements Ck‘ are the source codes that can
`be phase and/or amplitude terms. For example, in the simple
`case of a marine survey with random firing times, Ck‘ ¼ ejxsk‘
`is a linear phase term that expresses the time delay sk‘ given to
`source k in blended source array ‘. Similarly, in the case of
`vibrating sources transmitting a linear sweep, Ck‘ ¼ ejbk‘x2
`is a
`quadratic phase term describing the source code. An example of
`the different gathers extracted from a blended data matrix and
`
`their f -k spectra is given in Figure 3. Here, five shot records
`from the data set used in Figure 2 were blended. In this example
`linear phase encoding was applied (corresponding to applying
`time delays). Note the incoherent structure of the blended data
`in different domains, and the data compression due to the
`blending.
`To retrieve individual “deblended” shot records from blended
`data, a matrix inversion has to be performed. In general, the
`
`Figure 2. Different sections of the data matrix and their corre-
`sponding f -k spectra for a numerical data set. (a) One column of
`the data matrix (common source gather), (b) one row of the data
`matrix (common detector gather), (c) one diagonal of the data ma-
`trix (common offset gather), and (d) one antidiagonal of the data
`matrix (common midpoint gather).
`
`Figure 3. Different sections of the blended data matrix and their
`corresponding f -k spectra for the same data set as shown in Figure
`2. The number of sources that are blended together is five. (a) One
`column of the blended data matrix, (b) one row of the blended
`data matrix, (c) one diagonal of the blended data matrix, and (d)
`one antidiagonal of the blended data matrix.
`
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`

`
`Q12
`
`Mahdad et al.
`
`blending problem is underdetermined (i.e., P0 has fewer col-
`umns than P), which means that the blending matrix is not in-
`vertible. Instead, a least-squares inverse could be used accord-
`ing to
`
`
`
`ð
`C1 zd; zs
`
`Þ
`
`
`
`
`
`¼ CHC
`
`1CH:
`
`(3)
`
`It can be shown that if the blending matrix only contains phase
`terms (phase encoding), its least-squares inverse corresponds to
`the transpose complex conjugate (Hermitian). This leads to the
`following expression for pseudodeblending:
`
`h
`
`ð
`P zd; zs
`
`Þ
`
`
`
`i ¼ P0 zd; zsð
`
`ÞCH:
`
`(4)
`
`From the physics point of view, the pseudodeblending process
`carries out an expansion corresponding to the number of sources
`that are blended together; for example, if this number is b, each
`blended shot record is copied b times. Then, each of these cop-
`ies is corrected for the corresponding time delays introduced in
`the field or decoded in the case of encoded sources (correlation).
`Due to the fact
`that
`the responses of multiple sources are
`included in a single blended shot record and the source codes
`are not orthogonal, the pseudodeblending process generates cor-
`relation noise. This correlation noise is known as “blending
`noise” or “cross terms.” Figure 4 contains an example of differ-
`ent
`sections of pseudodeblended data recovered from the
`the corresponding f -k spectra are
`blended data of Figure 3;
`shown as well. Note the existence of incoherent blending noise
`in the pseudodeblended common detector and common offset
`and common midpoint gathers, in contrast with the signal that is
`coherent in all gathers. The incoherent nature of the blending
`noise is the discriminating power we will use in the deblending
`process, to be discussed in detail in the next section.
`
`METHOD
`
`The blending information contained in C is known. This
`means that CH is known as well, and therefore, if the unblended
`ð
`Þ were known, the blending noise that is present in
`data P zd; zs
`ð
`Þ
`h
`i could be computed as the
`the pseudodeblended data P zd; zs
`difference between the pseudodeblended and the unblended
`data.
`Using equation 4 we obtain
`
`Þ; (5)
`
`ÞCH P zd; zsð
`
`Þ ¼ P0 zd; zsð
`
`i P zd; zsð
`
`Þ
`N ¼ P zd; zsð
`
`h
`where N represents the blending noise. However,
`the initial
`unblended data are not available and obviously, if they were,
`there would be no need for a deblending method. Suppose
`ð
`Þ could be extracted from the pseu-
`though, that part of P zd; zs
`ð
`Þ
`i. Then, an iterative estimation-and-
`h
`dodeblended data P zd; zs
`subtraction process could be initiated where more of the cross
`terms could be computed and removed at each iteration. In the
`following section, we give an intuitive explanation of our itera-
`tive method for the separation of blended sources in the com-
`mon detector domain. The mathematical formulation in the form
`of a general framework as well as the extension of the method
`to other domains will be discussed later.
`
`Iterative deblending in the common detector domain
`
`In the common detector domain, the signal is arranged in
`coherent events, whereas the cross terms appear as random
`spikes (see Figures 4b and 5b). Hence, any method that can dis-
`tinguish between coherent events and random spikes to some
`degree could be used to suppress the blending interference to
`some degree (Doulgeris et al., 2010).
`A simple example of such a method is a frequency-wavenumber
`(f -k) filter that passes only the part of the pseudodeblended data
`
`Figure 4. Different sections of pseudodeblended data and their
`corresponding f -k spectra, generated by expanding and restoring
`the phase applied to the blended data of Figure 3. (a) One column
`of the pseudodeblended data (common source gather), (b) one
`row of the pseudodeblended data (common detector gather), (c)
`one diagonal of
`the pseudodeblended data (common offset
`gather), and (d) one antidiagonal of the pseudodeblended data
`(common midpoint gather).
`
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`
`Separation of blended data
`
`Q13
`
`that resides in the f-k band of the signal (i.e., in the signal cone).
`We propose to compute the signal bandwidth based on the highest
`velocity observed in the data. The random spikes have a white
`spectrum in the spatial wavenumber direction, extending out of the
`signal cone (Figure 4d). Thus, by applying the f -k filter, these
`spikes are somewhat suppressed and we may assume that the high-
`est amplitudes of the pseudodeblended data now belong to the
`ð
`Þ (Figure 5c). Again, any spike removal tool
`desired signal P zd; zs
`could be used instead of or in combination with the f -k filter.
`In order to select the unblended part of the data from the out-
`put of the filter, we apply a threshold in the x-t domain that
`keeps only the parts of the signal that have an amplitude higher
`than a certain predefined value (e.g., 0.9 times the maximum
`amplitude present). Note that alternatively the application of the
`threshold could take place in the filtering domain (in this exam-
`ple, f -k), being integrated in this way in the filtering process.
`This thresholding process is shown in Figure 5d. Note that,
`ideally, after the thresholding process no blending noise is pres-
`ent anymore. In practice, there may be some leakage of blend-
`ing noise. The consequence of this will be discussed later.
`We can now predict part of the blending noise, based on the out-
`put of the previous step. This output contains part of the unblended
`data, so by applying the blending matrix C to it we can simulate
`the blending that took place in the field. Then, we can apply the
`pseudodeblending process via CH, producing in this way data that
`contain part of the unblended data as well as the cross terms that
`were created by that part. Because that part of the unblended data
`is known, it can be subtracted, resulting in only the cross terms.
`
`
`Using the matrix formulation, this could be summarized as multi-
`plying the known part of the unblended data by a term CCH I
`.
`Hence, an estimate of the blending noise has now been computed
`(Figure 5e).
`At this point, we can subtract this blending-noise estimate
`from the pseudodeblended data. In the case that the blending
`parameters are not known exactly, the subtraction could be car-
`ried out in an adaptive way. The new estimate of the unblended
`data, which has less blending noise, can now serve as the
`updated input to the f -k filter (or in fact any spike removal fil-
`ter). The filter output is expected to contain less blending noise
`than in the previous iteration, so that the threshold can be low-
`ered, leading to an even better estimate of the blending noise.
`Repeating this process leads to the gradual removal of cross
`terms
`from the pseudodeblended gather, until no further
`improvement is achieved. A criterion has been implemented to
`monitor and terminate the iterative procedure (see the next sec-
`tion). Once terminated, the last output of the filter is taken as
`the deblended common detector gather (Figure 5f). Figure 5g
`displays a shot record of the final deblended data.
`As mentioned, after thresholding some blending noise may still
`be present. For example, think of a weak cross term interfering with
`a strong signal event such that it partly passes the threshold. In our
`experience, this type of leaked blending noise appears to spread out
`spatially and decrease its amplitude in the course of the highly non-
`linear iterative process. Note, that a better incoherency filter might
`do a much better job than our simple thresholding procedure. Still, it
`is our belief that this leakage sets a lower bound on the residual
`blending noise in the final result. However, this is a subject of fur-
`ther research.
`As a final remark we mention that a common detector gather
`has a unique property that can prove very useful in the imple-
`
`mentation of the algorithm. Namely, the blending noise present
`in the gather after pseudodeblending is solely produced by the
`signal present in the very same gather. This stems from the fact
`that all the sources are present in each common detector gather.
`It follows that each pseudodeblended common detector gather
`can be treated individually. This means that the algorithm can
`be very easily parallelized when implemented in the common
`detector domain.
`
`Convergence and stopping criterion
`
`The algorithm iterates to the correct solution if no blending
`noise remains in the output of the thresholding process and, for
`convergence,
`the threshold decreases at each iteration. These
`requirements rely on the sequence of thresholds chosen during
`the execution of the algorithm. We can predefine the thresholds
`or set them during execution. As a rule of thumb, a predefined
`sequence of the form ni, where i is the iteration number and n,
`being the
`threshold at
`the first
`iteration,
`is
`chosen as
`n  0:9  max P0 zd; zs
`ð
`Þ
`f
`g, is a safe choice. On the other hand,
`setting the threshold manually after inspection of the results at
`each iteration gives an extra degree of freedom that can be used
`to fine-tune the process and leads to faster convergence.
`The stopping criterion is based on an energy measure that is
`computed as follows. The deblended output of each iteration is
`
`Figure 5. The flow chart of the deblending algorithm (CDG, com-
`mon detector gather; CSG, common source gather). (a) Blended
`common source gather, (b) pseudodeblended common detector
`gather, (c) the output of the f-k filter on the first iteration, (d) the
`output of the thresholding process on the first iteration, (e) the
`blending-noise estimate on the first iteration, (f) the deblended com-
`mon detector gather, and (g) the deblended common source gather.
`
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`

`
`Q14
`
`Mahdad et al.
`
`blended again by applying the blending matrix C and is then
`subtracted from the corresponding blended shot record as it was
`acquired in the field; the root mean square of this difference,
`once integrated over the record, provides the measure. The itera-
`tive procedure is stopped once this measure is no longer
`decreasing or when it is lower than a predefined value.
`
`Iterative deblending in other domains
`
`The iterative deblending method can also be implemented in
`domains other than the common detector domain, for example,
`in the common offset, common midpoint, or common source do-
`main. Some of the properties of the algorithm will now be dis-
`cussed for each of these domains.
`
`Common offset domain
`
`The noise removal filter involved in our method requires a
`sufficiently large number of traces in each gather to be proc-
`essed. In many acquisition configurations, the common offset
`domain offers the largest gathers, making it a good choice for
`this method.
`Another interesting property of the pseudodeblended common
`offset gathers is displayed in Figure 6. The signal-to-blending
`noise ratio varies among different common offset gathers. Near-
`offset gathers tend to contain high-amplitude signal and lower-
`amplitude interference (Figure 6a). On the other hand, far-off-
`sets tend to have lower-amplitudes, thereby suffering more from
`the blending noise from all the other offsets (in particular from
`the strong near-offsets; see Figure 6c). Treating the different
`bands of offsets sequentially and starting with the near-offsets
`allows the method to process data with more blending noise
`faster. By the time the algorithm starts processing the far-off-
`
`Figure 6. Pseudodeblended common offset gathers. (a) Near-off-
`set gather, (b) mid-offset gather, and (c) far-offset gather.
`
`sets, most of the cross terms that were caused by the near- and
`mid-offsets will have been removed. Information on the particu-
`lar blended acquisition geometry used here is given in the sub-
`section “Application to impulsive sources.”
`
`Common midpoint domain
`
`Our method can also be implemented in the common mid-
`point (CMP) domain. Applying normal moveout correction to a
`pseudodeblended common midpoint gather forces the signals’
`f -k spectrum to be concentrated in a narrow-band cone. On the
`other hand, the cross terms still have a white spectrum in the
`spatial wavenumber direction. Hence, an f -k filter can suppress
`the cross terms better
`in the CMP domain than in other
`domains.
`
`Common source domain
`
`Because the cross terms have a coherent structure in the com-
`mon source domain (see Figure 4b), a spatial filter (e.g., an f -k
`filter) is not able to discriminate the desired signal from the
`cross terms. However, the use of sophisticated source codes,
`such as incoherent sweeps or random phase, may result in cross
`terms with a lower amplitude level
`than that of the signal.
`Therefore,
`thresholding can be done directly after pseudode-
`blending without the need for a spatial filter. However, depend-
`ing on the cross-term structure, different types of filters could
`be applied prior to thresholding (e.g., time-frequency filters) to
`suppress the cross-term energy further. An advantage of the
`common source domain is that the deblending process can be
`executed per blended shot record, which may result in an effi-
`cient implementation.
`
`General framework
`
`The iterative deblending method can be generalized to work
`ð
`Þ as a whole rather than on subsets
`on the data matrix P zd; zs
`such as common detector gathers individually. Dropping the
`depth variables zd and zs as well as the brackets from the
`deblended estimate for notational convenience,
`the general
`framework can be formulated as
`
`
`Piþ1 ¼ P0CH Pi CCH I
`;
`(6)
`where Piþ1 is the deblended estimate at iteration i þ 1 and Pi is
`the deblended estimate at iteration i processed in such a way that
`only (part of) the signal is contained. The f -k filtering and thresh-
`olding, as described earlier, is a simple example of such a process-
`ing step. However, the domain-specific filter can now be replaced
`by a multidimensional filter. Such a filter can take even better
`advantage of the lateral consistency of the seismic data versus the
`incoherency of the blending noise. In fact, any (multidimensional)
`denoising technology can be integrated into this step.
`The second term on the right-hand side of equation 6 trans-
`forms the estimated unblended data Pi into blending noise. This
`is achieved by applying both blending and pseudodeblending
`via multiplication with CCH, while making sure that the initial
`signal is removed by subtracting PiI.
`It is interesting to notice that an alternative way of deriving
`equation 6 can be obtained by formulating deblending as an
`optimization problem and solving it with a steepest-descent type
`method. The optimization problem could be written as
`
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`

`
`Separation of blended data
`
`Q15
`
`P0 PC
`k
`
`minimize f Pð Þ ¼ 1
`2
`subject to CPj
`j > b;
`
`k2
`
`2
`
`(7)
`
`where CP denotes a normalized coherency measure of the data,
`(e.g., normalized crosscorrelation or semblance coefficient, inte-
`grated over the whole dataset; see Neidell and Taner, 1971).
`The absolute value of this coherency measure takes values in
`the interval 0,1 with values close to 1 showing high coherency.
`It follows that the parameter b takes values in the same interval.
`Ignoring, for the moment, the inequality constraint in Equation 7,
`a steepest descent iteration in matrix notation would be
`
`
`Piþ1 ¼ Pi þ aiþ1 P0 PiC
`with P0 ¼ 0 and a being the step length. In the absence of noise
`in the forward model, i.e., when the blending parameters are
`known precisely, the blending matrix C can be chosen such that
`the diagonal of the CCH matrix is populated with ones. In this
`case, the parameter a should be equal to 1. Equation 8 can then
`
`
`be written as
`Piþ1 ¼ P0CH Pi CCH I
`
`CH;
`
`(8)
`
`:
`
`(9)
`
`In order to take into account the inequality constraint in Equa-
`tion 6, a projection of the current estimate onto the feasible set
`is required, i.e., the set of coherent signals in the model space.
`This projection can be implemented as a coherency-pass filter.
`If we denote the projected Pi as Pi, then Equation 6 is obtained.
`Note that this means that our method belongs to the class of
`inversion type of methods. Other examples of such methods are
`Abma and Yan (2009) and Abma et al. (2010).
`
`RESULTS
`
`We now demonstrate how this approach performs when
`applied to marine field data with two examples, one for encoded
`sources and one for impulsive sources. We applied numerical
`blending to a data set that was acquired using a traditional acqui-
`sition design by Statoil in the Haltenbanken field, in Norway.
`The temporal and spatial sampling interval for this example are 4
`ms and 25 m, respectively.
`
`Application to encoded sources
`
`As stated before, the deblending process can be performed in the
`common source domain as long as the sources are encoded with so-
`phisticated incoherent codes. The process of deblending has been
`carried out for a blended record containing two marine shot
`records. Each shot record contains 281 traces and 1024 time sam-
`ples. In this example, the shot records, which were acquired with
`an air-gun source, are encoded with up-sweep and down-sweep sig-
`nals of 6 seconds, respectively, and blended numerically. The result
`can be interpreted as the recording of two marine vibrators firing
`simultaneously with mentioned sweeps. The two original shot
`records and the blended shot record are shown in Figure 7. The
`deblending results, obtained after 67 iterations, are depicted in
`Figure 8, and can be compared with the pseudodeblended results.
`
`Figure 7. (a) Marine shot record 1, (b) marine shot record 2, and
`(c) numerically blended shot record (sum of two encoded shot
`records).
`
`Figure 8. (a) Pseudodeblended shot record 1, (b) deblended shot
`record, (c) difference of deblended and unblended shot record 1
`(8b–7a), (d) pseudodeblended shot record 2, (e) deblended shot
`record 2, and (f) difference of deblended and unblended shot re-
`cord 2 (8c–7b).
`
`Downloaded 09/18/15 to 173.226.64.12. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
`
`PGS Exhibit 2023
`WesternGeco v. PGS (IPR2015-00309, 310, 311)
`
`

`
`Q16
`
`Mahdad et al.
`
`Note the cross terms that are still present in the pseudodeblended
`results. Records containing the residual cross-term energy are also
`shown. As illustrated, the two shot records have been deblended
`almost perfectly. The signal-to-blending noise ratio (S/N) after n
`iterations is calculated as follows:
`
`S=N ¼ 20 log10
`
`Prms
`Þrms
`Pn P
`
`ð
`
`;
`
`(10)
`
`where the subscript rms stands for root mean square; the mean
`being computed over all elements of P as well as overall fre-
`quency components. The S=N of the deblended shot records is
`27 dB. In practice, equation 10 cannot be used because the
`unblended data are not available. However, this S=N definition
`provides the most direct measure of separability and could be
`computed in our examples because blending was performed
`numerically.
`
`Figure 9. (a) Unblended shot record, (b) initial estimate (pseudo-
`deblended result), (c) estimate after 26 iterations, (d) estimate af-
`ter 36 iterations, (e) final deblended result after 44 iterations, and
`(f) residual energy.
`
`is often
`in practice the use of downsweeps
`Note that
`avoided because of the presence of harmonics (Abd El-Aal,
`2010). Alternatives are the use of upsweeps with different
`lengths or, in the case of the same sweep length for all sour-
`ces, application of the method in one of the other domains dis-
`cussed earlier.
`
`Application to impulsive sources
`
`We have simulated a 2D blended marine survey based on a
`subset of the unblended data set of the previous example. The
`blended acquisition design consists of one streamer in which
`three sources fire with small random time delays. The detectors
`of the strea