Field of the invention

[0001] The present invention relates to methods for dealiasing data such as encountered when acquiring and separating

contributions from two or more different simultaneously emitting sources in a common set of measured signals

representing a wavefield, particularly of seismic sources and of sets of aliased recorded and/or aliased processed seismic signals .

Background

[0002] Seismic data can be acquired in land, marine, seabed, transition zone and boreholes for instance. Depending on in what environment the seismic survey is taking place the survey equipment and acquisition practices will vary.

[0003] In towed marine seismic data acquisition, a vessel tows streamers that contain seismic sensors (hydrophones and sometimes particle motion sensors) . A seismic source usually towed by the same vessel excites acoustic energy in the water that reflects from the sub-surface and is recorded by the sensors in the streamers. The seismic source is typically an array of airguns but can also be a marine vibrator for

instance. In modern marine seismic operations many streamers are towed behind the vessel and the vessel sails, e.g., many parallel closely spaced sail-lines (3D seismic data

acquisition) . It is also common that several source and/or receiver vessels are involved in the same seismic survey in order to acquire data that is rich in offsets and azimuths between source and receiver locations. [0004] In seabed seismic data acquisition, nodes or cables containing sensors (hydrophones and/or particle motion

sensors) are deployed on the seafloor. These sensors can also record the waves on and below the sea bottom and in particular shear waves which are not transmitted into the water. Similar sources as in towed marine seismic data acquisition are used. The sources are towed by one or several source vessels.

[0005] In land seismic data acquisition, the sensors on the ground are typically geophones and the sources are vibroseis trucks or dynamite. Vibroseis trucks are usually operated in arrays with two or three vibroseis trucks emitting energy close to each other roughly corresponding to the same shot location .

[0006] Traditionally seismic data have been acquired

sequentially: a source is excited over a limited period of time and data are recorded until the energy that comes back has diminished to an acceptable level and all reflections of interest have been captured after which a new shot at a different shot location is excited. Being able to acquire data from several sources at the same time is clearly highly desirable. Not only would it allow to cut expensive

acquisition time drastically or to better sample the wavefield on the source side which typically is much sparser sampled than the distribution of receiver positions. It would also allow for better illumination of the target from a wide range of azimuths as well as to better sample the wavefield in areas with surface obstructions. In addition, for some applications such as 3D VSP acquisition, or marine seismic surveying in environmentally sensitive areas, reducing the duration of the survey is critical to save cost external to the seismic acquisition itself (e.g., down-time of a producing well) or minimize the impact on marine life (e.g., avoiding mating or spawning seasons of fish species) .

[0007] Simultaneously emitting sources, such that their signals overlap in the (seismic) record, is also known in the industry as "blending". Conversely, separating signals from two or more simultaneously emitting sources is also known as "deblending" and the data from such acquisitions as "blended data".

[0008] Simultaneous source acquisition has a long history in land seismic acquisition dating back at least to the early 1980' s. Commonly used seismic sources in land acquisition are vibroseis sources which offer the possibility to design source signal sweeps such that it is possible to illuminate the sub ^¬ surface "sharing" the use of certain frequency bands to avoid simultaneous interference at a given time from different sources. By carefully choosing source sweep functions,

activation times and locations of different vibroseis sources, it is to a large degree possible to mitigate interference between sources. Such approaches are often referred to as slip sweep acquisition techniques. In marine seismic data context the term overlapping shooting times is often used for related practices. Moreover, it is also possible to design sweeps that are mutually orthogonal to each other (in time) such that the response from different sources can be isolated after acquisition through simple cross-correlation procedures with sweep signals from individual sources. We refer to all of these methods and related methods as "time encoded

simultaneous source acquisition" methods and "time encoded simultaneous source separation" methods.

[0009] The use of simultaneous source acquisition in marine seismic applications is more recent as marine seismic sources (i.e., airgun sources) do not appear to yield the same

benefits of providing orthogonal properties as land seismic vibroseis sources, at least not at a first glance. Western Geophysical was among the early proponents of simultaneous source marine seismic acquisition suggesting to carry out the separation in a pre-processing step by assuming that the reflections caused by the interfering sources have different characteristics. Beasley et al . (1998) exploited the fact that, provided that the sub-surface structure is approximately layered, a simple simultaneous source separation scheme can be achieved for instance by having one source vessel behind the spread acquiring data simultaneously with the source towed by the streamer vessel in front of the spread. Simultaneous source data recorded in such a fashion is straightforward to separate after a frequency-wavenumber (ωξ) transform as the source in front of the spread generates data with positive wavenumbers only whereas the source behind the spread

generates data with negative wavenumbers only, to first approximation .

[0010] Another method for enabling or enhancing separability is to make the delay times between interfering sources

incoherent (Lynn et al . , 1987) . Since the shot time is known for each source, they can be lined up coherently for a

specific source in for instance a common receiver gather or a common offset gather. In such a gather all arrivals from all other simultaneously firing sources will appear incoherent. To a first approximation it may be sufficient to just process the data for such a shot gather to final image relying on the processing chain to attenuate the random interference from the simultaneous sources (aka. passive separation) . However, it is of course possible to achieve better results for instance through random noise attenuation or more sophisticated methods to separate the coherent signal from the apparently incoherent signal (Stefani et al . , 2007; Ikelle 2010; Kumar et al . 2015) . In recent years, with elaborate acquisition schemes to for instance acquire wide azimuth data with multiple source and receiver vessels (Moldoveanu et al., 2008), several methods for simultaneous source separation of such data have been described, for example methods that separate "random dithered sources" through inversion exploiting the sparse nature of seismic data in the time-domain (i.e., seismic traces can be thought of as a subset of discrete reflections with "quiet periods" in between; e.g., Akerberg et al . , 2008; Kumar et al . 2015) . A recent state-of-the-art land example of simultaneous source separation applied to reservoir characterization is presented by Shipilova et al . (2016) . Existing simultaneous source acquisition and separation methods based on similar principles include quasi random shooting times, and pseudo random shooting times. We refer to all of these methods and related methods as "random dithered source acquisition" methods and "random dithered source separation" methods.

"Random dithered source acquisition" methods and "random dithered source separation" methods are examples of "space encoded simultaneous source acquisition" methods and "space encoded simultaneous source separation" methods.

[0011] A different approach to simultaneous source separation has been to modify the source signature emitted by airgun sources. Airgun sources comprise multiple (typically three) sub-arrays along which multiple clusters of smaller airguns are located. Whereas in contrast to land vibroseis sources, it is not possible to design arbitrary source signatures for marine airgun sources, one in principle has the ability to choose firing time (and amplitude i.e., volume) of individual airgun elements within the array. In such a fashion it is possible to choose source signatures that are dispersed as opposed to focused in a single peak. Such approaches have been proposed to reduce the environmental impact in the past (Ziolkowski, 1987) but also for simultaneous source shooting.

[0012] Abma et al . (2015) suggested to use a library of

"popcorn" source sequences to encode multiple airgun sources such that the responses can be separated after simultaneous source acquisition by correlation with the corresponding source signatures following a practice that is similar to land simultaneous source acquisition. The principle is based on the fact that the cross-correlation between two (infinite) random sequences is zero whereas the autocorrelation is a spike. It is also possible to choose binary encoding

sequences with better or optimal orthogonality properties such as Kasami sequences to encode marine airgun arrays (Robertsson et al . , 2012) . Mueller et al . (2015) propose to use a

combination of random dithers from shot to shot with

deterministically encoded source sequences at each shot point. Similar to the methods described above for land seismic acquisition we refer to all of these methods and related methods as "time encoded simultaneous source acquisition" methods and "time encoded simultaneous source separation' methods .

[0013] Recently there has been an interest in industry to explore the feasibility of marine vibrator sources as they would, for instance, appear to provide more degrees of freedom to optimize mutually orthogonal source functions beyond just binary orthogonal sequences that would allow for a step change in simultaneous source separation of marine seismic data.

Halliday et al . (2014) suggest to shift energy in ωλ-space using the well-known Fourier shift theorem in space to

separate the response from multiple marine vibrator sources. Such an approach is not possible with most other seismic source technology (e.g., marine airgun sources) which lack the ability to carefully control the phase of the source signature (e.g., flip polarity) .

[0014] A recent development, referred to as "seismic

apparition" (also referred to as signal apparition or

wavefield apparition in this invention) , suggests an

alternative approach to deterministic simultaneous source acquisition that belongs in the family of "space encoded simultaneous source acquisition" methods and "space encoded simultaneous source separation" methods. Robertsson et al . (2016) show that by using periodic modulation functions from shot to shot (e.g., a short time delay or an amplitude

variation from shot to shot) , the recorded data on a common receiver gather or a common offset gather will be

deterministically mapped onto known parts of for instance the ωξ -space outside the conventional "signal cone" where conventional data is strictly located (Figure la) . The signal cone contains all propagating seismic energy with apparent velocities between water velocity (straight lines with

apparent slowness of +-1/1500 s/m in ωξ -space) for the towed marine seismic case and infinite velocity (i.e., vertically arriving events plotting on a vertical line with wavenumber 0) . The shot modulation generates multiple new signal cones that are offset along the wavenumber axis thereby populating the ωξ -space much better and enabling exact simultaneous source separation below a certain frequency (Figure lb) .

Robertsson et al . (2016) referred to the process as "wavefield apparition" or "signal apparition" in the meaning of "the act of becoming visible". In the spectral domain, the wavefield caused by the periodic source sequence is nearly "ghostly apparent" and isolated. A critical observation and insight in the "seismic apparition" approach is that partially injecting energy along the ωξ -axis is sufficient as long as the source variations are known as the injected energy fully predicts the energy that was left behind in the "conventional" signal cone. Following this methodology simultaneously emitting sources can be exactly separated using a modulation scheme where for instance amplitudes and or firing times are varied

deterministically from shot to shot in a periodic pattern.

[0015] In the prior art it has been suggested to combine different methods for simultaneous source acquisition. Miiller et al . (2015) outline a method based on seismic data

acquisition using airgun sources. By letting individual airguns within a source airgun array be actuated at different times a source signature can be designed that is orthogonal to another source signature generated in a similar fashion. By orthogonal, Miiller et al . (2015) refer to the fact that the source signatures have well-behaved spike-like autocorrelation properties as well as low cross-correlation properties with regard to the other source signatures used. On top of the encoding in time using orthogonal source signatures, Miiller et al . (2015) also employ conventional random dithering (Lynn et al . , 1987) . In this way, two different simultaneous source separation approaches are combined to result in an even better simultaneous source separation result.

[0016] Halliday et al . (2014) describe a method for

simultaneous source separation using marine vibrator sources that relies on excellent phase control in marine vibrator sources to fully shift energy along the wavenumber axis in the frequency-wavenumber plane. Halliday et al . (2014) recognize that the method works particularly well at low frequencies where conventional random dithering techniques struggle. They suggest to combine the two methods such that their phase- controlled marine vibrator simultaneous source separation technique is used for the lower frequencies and simultaneous source separation based on random dithers is used at the higher frequencies.

[0017] The method of seismic apparition (Robertsson et al . , 2016) allows for exact simultaneous source separation given sufficient sampling along the direction of spatial encoding (there is always a lowest frequency below which source

separation is exact) . It is the only exact method there exists for conventional marine and land seismic sources such as airgun sources and dynamite sources. However, the method of seismic apparition requires good control of firing times, locations and other parameters. Seismic data are often shot on position such that sources are triggered exactly when they reach a certain position. If a single vessel tows multiple sources acquisition fit for seismic apparition is simply achieved by letting one of the sources be a master source which is shot on position. The other source (s) towed by the same vessel then must fire synchronized in time according to the firing time of the first source. However, as all sources are towed by the same vessel the sources will automatically be located at the desired positions - at least if crab angles are not too extreme. In a recent patent application (van Manen et al . , 2016a) we submitted methods that demonstrate how

perturbations introduced by, e.g., a varying crab angle can be dealt with in an apparition-based simultaneous source

workflow. The same approach can also be used for simultaneous source separation when sources are towed by different vessels or in land seismic acquisition. Robertsson et al . (2016b) suggest approaches to combine signal apparition simultaneous source separation with other simultaneous source separation methods .

[0018] Fig. 1(B) also illustrates a possible limitation of signal apparition. The injected part of the wavefield is separated from the wavefield in the original location centered at wavenumber k=0 within the respective lozenge-shaped regions in Fig. 1(B) . In the triangle-shaped parts they interfere due to aliasing and may no longer be separately predicted without further assumptions. In the example shown in Fig. 1(B), it can therefore be noted that the maximum non-aliased frequency for a certain spatial sampling is reduced by a factor of two after applying signal apparition. Assuming that data are adequately sampled, the method nevertheless enables full separation of data recorded in wavefield experimentation where two source lines are acquired simultaneously.

[0019] It is herein proposed to make use of properties of analytic functions for complex representations, and

corresponding structures for hypercomplex representations, to separate simultaneous source data acquired using seismic apparition into separate source contributions. Moreover, the technique can also be used to reduce the effects of aliasing due to limitations in sampling. Further novel methods to reduce aliasing have been submitted in van Manen et al .

(2016b) .

Brief summary of the invention

[0020] Methods for dealiasing recorded wavefield information making use of a non-aliased representation of a part of the recorded wavefield, and a phase factor derived from a

representation of a non-aliased part of the wavefield, and combining both to a non-aliased function from which the further parts of the recorded wavefield information can be gained, suited particularly for seismic applications and other purposes, substantially as shown in and/or described in connection with at least one of the figures, and as set forth more completely in the claims.

[0021] Advantages, aspects and novel features of the present invention, as well as details of an illustrated embodiment thereof, may be more fully understood from the following description and drawings.

Brief Description of the Drawings [0022] In the following description reference is made to the attached figures, in which:

Figs. 1A, B illustrate how in a conventional marine seismic survey all signal energy of two sources typically sits inside a "signal cone" (horizontally striped) bounded by the propagation velocity of the recording medium and how this energy can be split in a transform domain by applying a modulation to the second source ;

Fig. 2 shows a common-receiver gather from the simultaneous source complex salt data example with all four sources firing simultaneously in the reference frame of the firing time of sources 1 and 2 in the Fourier domain;

Fig. 3 shows filtered blended data in the Fourier domain corresponding to the data set depicted in Fig. 1. The data clearly contains a mixture from two directions ;

Fig. 4 shows phase shifted data in the Fourier domain corresponding to the data set depicted in Fig. 2. Note how the energy from the two parts is moved into two centers ;

Fig. 5 shows the filtered reconstruction of source one in the Fourier domain corresponding to the data set depicted in Figs. 1-3;

Fig. 6 shows a common-receiver gather from the simultaneous source complex salt data example with all four sources firing simultaneously in the reference frame of the firing time of sources 1 and 2 in the quaternion Fourier domain. Note that all energy is present in one quadrant; Fig. 7 shows filtered blended data in the quaternion Fourier domain corresponding to the data set depicted in Fig. 6. The data clearly contains a mixture from two directions ;

Fig. 8 shows phase shifted data in the quaternion

Fourier domain corresponding to the data set depicted in Fig. 6. Note how some of the energy is moved into the center and some of the energy remain unaffected. The unaffected energy corresponds only to the (shifted) information of source two;

Fig. 9 shows the filtered reconstruction of source one in the quaternion Fourier domain for source one

corresponding to the data set depicted in Figs. 5-7;

Fig. 10 shows a common-receiver gather from the

simultaneous source complex salt data example with all four sources firing simultaneously in the reference frame of the firing time of sources 1 and 2 in the time domain;

Fig. 11 shows the contribution from source one only for Fig. 10 in the time domain;

Fig. 12 shows the reconstruction of source one as depicted in Fig. 11 using analytic part dealiasing in the time domain;

Fig. 13 shows the reconstruction error between the wavefield shown in Figs. 11-12 using analytic part dealiasing in the time domain;

Fig. 13 shows the reconstruction of source one as depicted in Fig. 11 using quaternion dealiasing in the time domain; Fig. 15 shows the reconstruction error between the wavefield shown in Fig. 11 and Fig. 14 using quaternion part dealiasing in the time domain;

Detailed Description

[0023] The following examples may be better understood using a theoretical overview and its application to simultaneous source separation as presented below.

[0024] It should be understood that the same methods can be applied to the dealiasing of any wavefield in which a non- aliased content can be identified.

[0025] We will use the notation for the Fourier transform.

[0026] Let C denote the cone C = {(ω, ξ): \ω\ > \ξ\}, and let T>

denote the non-aliased (diamond shaped) set > = (?\({(ω, ξ): \ω\ > \ξ— 1/2 |} υ {(ω, : Μ > + 1/2 |}).

[0027] Suppose that d(t,j)=f ₁ (t )+f ₂ (t- A _t (-i ,j) (1) is a known discrete sampling in x = j recorded at a first sampling interval. Note that due to this type of apparition sampling, the data d will always have aliasing effects present if the data is band unlimited in the temporal frequency direction .

[0028] If f and f ₂ represent seismic data recorded at a certain depth, it will hold that supp( i) c C and supp( ₂ ) ^c C. We ill assume that the source locations of f and f ₂ are elatively close to each other. Let and

[0029] It is shown in Andersson 2016 that

[0030] For each pair of values (ω, () E D, most of the terms over k in (2) vanish (and similarly for D ₂ ) , which implies that i (ω, ξ) and ₂ (ω, can be recovered through

£ ₍ _ Pi (&>,f ) - cos (2πΔ _t cj)D ₂ (ω,ξ)

Λ^'^- ₅ ίη ₂ (2πΛ, _ω)

D ₂ (ω,Ο - cos (2πΔ^)D (ω,ξ) '

given that sin(2 A _t ω)≠ 0 .

[0031] By including an amplitude variation in (1), the last condition can be removed. For values of (ω, C D it is not possible to determine the values of Λ(ω, and / ₂ (ω, without imposing further conditions on the data.

[0032] Given a real valued function / with zero average, let (t') _e 2Jri(t-t/) _{W dt} > _do)

[0033] The quantity is often referred to as the analytic part of /, a description that is natural when considering Fourier series expansions in the same fashion and comparing these to power series expansions of holomorphic functions. It is readily verified that Re(f ^a )=f.

[0034] As an illustrative example, consider the case where

/(t) = COS(27Tt) for which it holds that

[0035] Now, whereas \f(t)\ is oscillating, \f ^a (t)\ = 1 , i.e., it has constant amplitude. In terms of aliasing, it can often be the case that a sampled version of \f ^a \ exhibits no aliasing even if / and \f\ do so.

[0036] Let us now turn our focus back to the problem of recovering i (ω, ξ) and A (ω, ξ) for (ω,ξ)£Ί). We note that due to linearity, it holds that d

[0037] A natural condition to impose is that the local directionality is preserved through the frequency range. The simplest such case is when f and f ₂ are plane waves (with the same direction), i.e, when fi(t> ^x ) = ¾i(t + bx), and f2{t,x) = h ₂ (t + bx).

[0038] Without loss of generality, we assume that b > 0 . We note that

CO CO

h _x (5) ₆ -2πί(5 _ω + ^χ 0ί-& _ω ) _{dsdx =} ^ ( _ω )δ(ξ - boS).

—CO —CO A similar formula holds for f ₂ . i

[0039] Let us now assume that ω < - . Inspecting (2) we see that i ²

if, e.g., — -<ξ<0 then all but three terms disappear and therefore the blended data satisfies d ^a ( _ω , ξ) = {h* (ω) + COS(2TT A _t ώ) h% (ω))δ(ξ - ήω) - isin(27r A _t ώ) h% (ω)δ(ξ - (ήω - 1 /2))

= ¾ _! (ω)δ(ξ - bo)) + ¾2 (ω)5(? - (ήω - 1/2)).

[0040] Let w _h and Wj be two filters (acting on the time

variable) such that w _h has unit L ² norm, and such that w _h has a central frequency of ω ₀ and Wj has a central frequency of ω ₀ /2 . For the sake of transparency let Suppose that we have knowledge of Λ(ω, and ₂ ( ^ω Ό f° ^r ω<ω ₀ , and that the

bandwidth of W _j is smaller than ω ₀ /2 .

[0041] Let * ^w i- ^Note that, e.g., g-i t,x) = hi * w _t t + bx),

[0042] so that g is a plane wave with the same direction as f . Moreover, \g \ will typically be mildly oscillating even when f and are oscillating rapidly.

[0043] Let be the phase function associated with g _lr and define p ₂ in a likewise manner as phase function for g ₂ . If j is

narrowbanded, g^ will essentially only contain oscillations of the form i.e., |5Ί(ί, )| is more or less constant. [0044] Under the narrowband assumption on W _j (and the relation ^w h ^{= w} i ) I ^we consider d _h t,x = (d ^a *W _ft )(t,x) «¾ _! (ω ₀ )β ^2πίω ο( ^ί+δ ) +¾ ₂ ( _ωο ) _β 2πί(ω ₀ Ε+(6ωο-1) ^χ )_

[0045] By multiplication by i ₂ , which may be considered as an example of a conjugated phase factor, we get d _h {t,x)p ₁ {t,x)p ₂ {t,x) « (ω ₀ ) + h ₂ (ω ₀ )β ^~πιχ . (4)

[0046] In the Fourier domain, this amounts to two delta- functions; one centered at the origin and one centered at (0, -1/2) . Here, we may identify the contribution that comes from f ₂ by inspecting the coefficient in front of the delta-function centered at (0, -1/2) . By the aid of the low-frequency

reconstructions g and g ₂ , it is thus possible to move the energy that originates from the two sources so that one part is moved to the center, and one part is moved to the Nyquist wavenumber. Note that it is critical to use the analytic part of the data to obtain this result. If the contributions from the two parts can be isolated from each other, it allows for a recovery of the two parts in the same fashion as in (3) .

Moreover, as the data in the isolated central centers is comparatively oversampled, a reconstruction can be obtained at a higher resolution than the original data. Afterwards, the effect of the phase factor can easily be reversed, and hence a reconstruction at a finer sampling, i.e. at a smaller sampling interval than the original can be obtained.

[0047] A similar argument will hold in the case when the filters W _j and w _h have broader bandwidth. By making the approximation that

we get that d _h (t,x) _Pl (t,x)p ₂ (t,x) « ¾! ( _ω ) _β 2πί(ω-ω ₀ )(Ε+6 ^χ ) ₊ ^ ^-nix+e^-^iW^ and since w _h is a bandpass filter, will contain information around the same two energy centers, where the center around (0, -1/2) will contain only information about f ₂ . By suppressing such localized energy, we can therefore extract only the contribution from f ₂ and likewise for f .

[0048] The above procedure can now be repeated with ω ₀

replaced by ω ₁ = βω ₀ for some β>1. In this fashion we can gradually recover (dealias) more of the data by stepping up in frequency. We can also treat more general cases. As a first improvement, we replace the plane wave assumption by a locally plane wave assumption, i.e., let φ _α be a partition of unity (∑α<Ρα ⁼ 1) ' ^an d assume that f _l (t,x)(pa(t,x) « h _la (t +

[0049] In this case the phase functions will also be locally plane waves, and since they are applied multiplicatively on the space-time side, the effect of (4) will still be that energy will be injected in the frequency domain towards the two centers at the origin and the Nyquist wavenumber.

[0050] Now, in places where the locally plane wave assumption does not hold, the above procedure will not work. This is because as the phase function contains contributions from several directions at the same location, the effect of the multiplication in will no longer correspond to injecting the energy of D ₁ (and D ₂ ) towards the two centers around the origin and the Nyquist number. However, some of this directional ambiguity can still be resolved.

[0051] In fact, upon inspection of (2), it is clear that the energy contributions to a region with center (ω ₀ ' ^ο) m st originate from regions with centers at (ω ₀ , ^ ₀ + /ί/2) for some k . Hence, the directional ambiguity is locally limited to

contributions from certain directions. We will now construct a setup with filters that will make use of this fact. [0052] Let us consider the problem where we would like to recover the information from a certain region (ω ₀ ,ξ ₀ ). Due to the assumption that i and f ₂ correspond to measurements that take place close to each other, we assume that f and f ₂ have similar local directionality structure. From (2) we know that energy centered at (ω ₀ ' ^ο) will be visible in measurements D ₁ at locations (ω ₀ , ξ ₀ + k/2) . We therefor construct a (space-time) filter w _h , that satisfies

[0053] We now want to follow a similar construction for the filter Wi . Assuming that there is locally only a contribution from a direction associated with one of the terms over k above, we want the action of multiplying with the square of the local phase to correspond to a filtration using the part of w _h that corresponds to that particular k .

This is accomplished by letting where Ψ = ψ _* ψ .

[0054] Under the assumption that f *w _h and f ₂ * w _h has a local plane wave structure, we may now follow the above procedure to recover these parts of f and f ₂ (by suppressing localized energy as described above) . We may then completely recover f and f ₂ up to the temporal frequency ω ₀ by combining several such reconstructions, and hence we may proceed by making gradual reconstructions in the ω variable as before.

Example

[0055] As an example we have applied one embodiment of the simultaneous source separation methodology presented here to a synthetic data set generated using an acoustic 3D finite- difference solver and a model based on salt-structures in the sub-surface and a free-surface bounding the top of the water layer. A common-receiver gather located in the middle of the model was simulated using this model a vessels acquiring two shotlines with an inline shot spacing of 25 meters. The vessel tows source 1 at 150 m cross-line offset from the receiver location as well as source 2 at 175 m cross-line offset from the receiver location. The source wavelet

comprises a Ricker wavelet with a maximum frequency of 30Hz.

[0056] Sources 1 and 2 towed behind Vessel A are encoded against each other using signal apparition with a modulation periodicity of 2 and a 12 ms time-delay such that Source 1 fires regularly and source 2 has a time delay of 12 ms on all even shots.

[0057] In Figures 2-5 the procedure as well as some results are illustrated in the frequency domain. Figure 2 shows the Fourier transform of the blended data, and the effect of the overlap in C D is clearly visible. Note that the information in the central diamond shaped region can be recovered by using (3) . We now need to recover the remaining part. In Figure 3 the blended data is filtered by multiplication of w _¾ from (5) . Next, using the recovered parts of i and f ₂ we compute the phase functions using the filters from (6), and after that we apply the phase function multiplicatively in the (t,x)-domain using (4) . In Figure 4, the result is illustrated in the (ω, ξ) - domain. Finally, we isolate the two parts and use a similar reconstruction formula as (3) using a phase factor to

approximately recover

(A *w _ft ) li¾.

[0058] This part is expected to be well sampled since much of the oscillating parts are counteracted by the factor p ₁ p ₂ , and it can thus be resampled using a smaller trace distance, if desired. The final reconstruction is obtained by

multiplication with i ₂ , which is displayed in Figure 5. In Figures 10-13 we illustrate the reconstruction in the

temporal-spatial domain. Figure 10 shows the blended data, and the apparition pattern is illustrated in the two smaller inset images. In Figure 11 the original data from source one is shown, and in Figure 12 the reconstruction of source one is shown. Figure 13 finally shows the reconstruction error for source one.

[0059] In an alternative embodiment we will make use of quaternion Fourier transforms instead of standard Fourier transforms, and make use of a similar idea as for the analytic part .

[0060] Let HI be the quaternion algebra (Hamilton, 1844) . An element q E M can be represented as q = q ₀ + iq ₁ + jq ₂ + kq ₃ , where the qj are real numbers and i ² = j ² = k ² = ijk =— 1 . We also recall Euler' s formula, valid for i,j, k :

e ^w = cosO + isinfl, e> ⁹ = cosO + jsmO, e ^ke = cosO + ksmO.

[0061] Note that although i,j, k commute with the reals, quaternions do not commute in general. For example, we

generally have e ^w ≠ e^e ^w which can easily be seen by using

Euler' s formula. Also recall that the conjugate of q = q _Q + iq ₁ + jq2 + kq ₃ is the element q ^* = q ₀ — iq^— jq ₂ — kq ₃ . The norm of q is defined as II q 11= (qq ^* ) ^1/2 = (q ² + q\ + q\ + ql) ^1/2 .

[0062] Given a real valued function / = (t, x) , we define the quaternion Fourier transform (QFT) of / by inverse is given by

[0063] In a similar fashion, it is possible to extend the Fourier transform to other hypercomplex representations, e.g., octanions (van der Blij, 1961), sedenions (Smith, 1995) or other Cayley or Clifford algebras. A similar argument applies to other well-known transform domains (e.g., Radon, Gabor, curvelet , etc . ) .

[0064] Let

4 if<o>0and^>0,

Χ(ω,ξ) = { ⁴

otherwise.

[0065] Using χ we define f ^q : ² →M as f ^q = Q-'xQf-

[0066] We call f ^q the quaternion part of /. This quantity can be seen as a generalization of the concept of analytic part. For the analytic part, half of the spectrum is redundant. For the case of quaternions, three quarters of the data is redundant .

[0067] In a similar fashion, it is possible to extend the analytic part to other hypercomplex representations, e.g., octanions (van der Blij, 1961), sedenions (Smith, 1995) or other Cayley or Clifford algebras.

[0068] The following results will prove to be important: Let f{t, x) = cosii, where u = 27r(at + bx) + c with a>0. If b > 0 then f ^q t,x) = cosii + isinii + y ^' sinii— kcosu, and if b < 0 then f ^q t,x) = cosii + isinii— y ^' sinii + kcosu.

[0069] The result is straightforward to derive using the quaternion counterpart of Euler' s formula. Note that whereas | (t,x)| is oscillating, \\f ^q \\= /2, i.e., it has constant

amplitude. In terms of aliasing, it can often be the case that a sampled version of II f ^q II exhibits no aliasing even if / and \f\ do so. [0070] Assume that f(t, x) = cos(u) , and that g(t,x) = cos(i;) , where u = 2n(a ₁ t + bx) + e and v = 2n(a ₂ t + b ₂ x) + e ₂ with a ₁ ,a ₂ ≥0 . It then holds that

— _ [2(cos(ii— 2v)(l + k) + sin(u— 2v)(i + _/ ^' )) ifb _x , b ₂ ≥ 0,

9 ^q f ^q 9 ^q = [2(cos( )(-l - k) + sin(u)(i + ; ^' )) ifb _x > 0 and b ₂ < 0, with similar expressions if b ₁ < 0 .

[0071] Let us describe how to recover Α (ω, ^) and / ₂ (ω, us the quaternion part. We will follow the same procedure as before, and hence it suffices to consider the case where j = h ₁ (t + bx) , f ₂ ⁼ h ₂ (t + bx), with b > 0, and ω<1/2. Let w _h and W _j be two (real- valued) narrowband filters with central frequencies of ω ₀ and ω ₀ /2, respectively, as before. From (2), it now follows that d * w _h (t, x) « ο ₁ €θ3(2πω ₀ {ί + bx) + e- + c ₂ cos(2noo ₀ (t + b x) + e ₂ ),

[0072] for some coefficients c ₁ , c ₂ , and phases e ₁ , e ₂ , and with b < 0.

[0073] Since i and f ₂ are known for ω = ω ₀ /2, we let 5Ί(ί, x) = f _x * Wi(t, x) « c ₃ cos( it) ₀ (t + bx) + e ₃ ).

We compute the quaternion part g^ of g _lr and construct the phase function associated with it as

and define p ₂ in a likewise manner as the phase function for

92-

[0074] Let d ^q be the quaternion part of d * w _h . It then holds that after a left and right multiplication of conjugate of the phase factors _x and p ₂ p ₁ d ^q p ₁ « c 2 cos e _x — 2e ₃ )(l + k) + sin^— 2e ₃ )(i + _/))

+c ₂ 2(cos(2 io ₀ (t + b x) + e-, —1— fc) + sin(2 io ₀ (t + b x) + e^i + _/)). [0075] This result is remarkable, since the unaliased part of d is moved to the center, while the aliased part remains intact. Hence, it allows for a distinct separation between the two contributing parts.

Example

[0076] We now use the same example data set as in the previous example. In Figures 6-9 the procedure as well as some results are illustrated in the quaternion frequency domain. Figure 6 shows the quaternion Fourier transform of the blended data. Note that there is only information in one of the quadrants. In Figure 7 the blended data (outside the central diamond V ) is filtered by multiplication of w _h from (5) . Next, using the recovered parts of i and f ₂ we compute the quaternion phase functions using the filters from (6), and after that we apply the phase function multiplicatively in the (t,x)-domain using (4) . In Figure 8, the result is illustrated in the (ω, ξ) - domain. Finally, we isolate the two parts by multiplication from the left by and multiplication from the right by j . The final reconstruction is displayed in Figure 9. In Figures 14-15 we illustrate the reconstruction in the temporal-spatial domain. Figure 14 shows the reconstruction of source one by using the quaternion approach and Figure 15 shows the

corresponding reconstruction error.

[0077] The methods described herein have mainly been

illustrated using so-called common receiver gathers, i.e., all seismograms recorded at a single receiver. Note however, that these methods can be applied straightforwardly over one or more receiver coordinates, to individual or multiple receiver- side wavenumbers . Processing in such multi-dimensional or higher-dimensional spaces can be utilized to reduce data ambiguity due to sampling limitations of the seismic signals.

[0078] We note that further advantages may derive from

applying the current invention to three-dimensional shot grids instead of two-dimensional shot grids, where beyond the x- and y-locations of the simultaneous sources, the shot grids also extend in the vertical (z or depth) direction. Furthermore, the methods described herein could be applied to different two-dimensional shot grids, such as shot grids in the x-z plane or y-z plane. The vertical wavenumber is limited by the dispersion relation and hence the encoding and decoding can be applied similarly to 2D or 3D shotgrids which involve the z (depth) dimension, including by making typical assumptions in the dispersion relation.

[0079] The above discussion on separation over one more receiver coordinates also makes it clear that seismic

apparition principles can be applied in conjunction with and/or during the imaging process: using one-way or two-way wavefield extrapolation methods one can extrapolate the recorded receiver wavefields back into the subsurface and separation using the apparition principles described herein can be applied after the receiver extrapolation.

Alternatively, one could directly migrate the simultaneous source data (e.g., common receiver gathers) and the apparated part of the simultaneous sources will be radiated, and

subsequently extrapolated, along aliased directions, which can be exploited for separation (e.g. by recording the wavefield not in a cone beneath the sources, but along the edges of the model) .

[0080] As should be clear to one possessing ordinary skill in the art, the methods described herein apply to different types of wavefield signals recorded (simultaneously or non- simultaneously) using different types of sensors, including but not limited to; pressure and/or one or more components of the particle motion vector (where the motion can be:

displacement, velocity, or acceleration) associated with compressional waves propagating in acoustic media and/or shear waves in elastic media. When multiple types of wavefield signals are recorded simultaneously and are or can be assumed (or processed) to be substantially co-located, we speak of so- called "multi-component" measurements and we may refer to the measurements corresponding to each of the different types as a "component". Examples of multi-component measurements are the pressure and vertical component of particle velocity recorded by an ocean bottom cable or node-based seabed seismic sensor, the crossline and vertical component of particle acceleration recorded in a multi-sensor towed-marine seismic streamer, or the three component acceleration recorded by a

microelectromechanical system (MEMS) sensor deployed e.g. in a land seismic survey.

[0081] The methods described herein can be applied to each of the measured components independently, or to two or more of the measured components jointly. Joint processing may involve processing vectorial or tensorial quantities representing or derived from the multi-component data and may be advantageous as additional features of the signals can be used in the separation. For example, it is well known in the art that particular combinations of types of measurements enable, by exploiting the physics of wave propagation, processing steps whereby e.g. the multi-component signal is separated into contributions propagating in different directions (e.g., wavefield separation) , certain spurious reflected waves are eliminated (e.g., deghosting) , or waves with a particular (non-linear) polarization are suppressed (e.g., polarization filtering) . Thus, the methods described herein may be applied in conjunction with, simultaneously with, or after such processing of two or more of the multiple components.

[0082] Furthermore, in case the obtained wavefield signals consist of / comprise one or more components, then it is possible to derive local directional information (e.g. phase factors) from one or more of the components and to use this directional information in the reduction of aliasing effects in the separation as described herein in detail.

[0083] Further, it should be understood that the various features, aspects and functionality described in one or more of the individual embodiments are not limited in their

applicability to the particular embodiment with which they are described, but instead can be applied, alone or in various combinations, to one or more of the other embodiments of the invention . [0084] For example, it is understood that the techniques, methods and systems that are disclosed herein may be applied to all marine, seabed, borehole, land and transition zone seismic surveys, that includes planning, acquisition and processing. This includes for instance time-lapse seismic, permanent reservoir monitoring, VSP and reverse VSP, and instrumented borehole surveys (e.g. distributed acoustic sensing) . Moreover, the techniques, methods and systems disclosed herein may also apply to non-seismic surveys that are based on wavefield data to obtain an image of the subsurface .

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