WO2015155597A2

WO2015155597A2 - Attenuating pseudo s-waves in acoustic anisotropic wave propagation

Info

Publication number: WO2015155597A2
Application number: PCT/IB2015/000620
Authority: WO
Inventors: Botao Qin
Original assignee: Cgg Services Sa
Priority date: 2014-04-07
Filing date: 2015-04-02
Publication date: 2015-10-15
Also published as: WO2015155597A3

Abstract

A method for attenuating pseudo S-waves in acoustic anisotropic wave propagation obtains a plurality of anisotropic media parameters associated with a given subsurface area from which acquisition seismic data are obtained. A shot position of a seismic source used to generate simulated seismic data is identified. The seismic source has a source wavelet that is a function of time. Anisotropic media parameters are modified at the shot position to define an ellipsoidally anisotropic region at the shot position of the seismic source. The function of time of the source wavelet is used to define a source vector used together with the modified anisotropic media parameters in an anisotropic wave equation to obtain a stress vector wavefield. The stress vector wavefield is used in a wavenumber domain representation of the anisotropic wave equation to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation.

Description

ATTENUATING PSEUDO S-WAVES IN ACOUSTIC ANISOTROPIC WAVE

PROPAGATION

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority and benefit from U.S. Provisional Patent Application No. 61/975,931 , filed April 7, 2014, for "Attenuating Pseudo S-waves in Acoustic Anisotropic Wave Propagation", the entire content of which is incorporated herein by reference. TECHNICAL FIELD

[0002] Embodiments of the subject matter disclosed herein generally relate to methods and systems for seismic data processing and, more particularly, to mechanisms and techniques for noise attenuation in seismic signals. BACKGROUND

[0003] The importance of anisotropy in seismic imaging has been recognized for several decades. In recent years, a growing number of anisotropic applications of reverse time migration (RTM) and full wave inversion (FWI) have attempted to account for anisotropic effects of the real-world subsurface physics. In anisotropic media the P-wave, SV-wave and SH-wave are intrinsically coupled; therefore, the media are elastic in nature. But the elastic anisotropic wave equation is

computationally very demanding and requires S-wave velocity models, often for which little information is available. Consequently, the acoustic assumption is widely- used in seeking anisotropic wave equations.

[0004] As first introduced, the acoustic approximation sets the S-wave velocities to zero along the anisotropy axis of symmetry. The simplified dispersion relation and a corresponding fourth-order scalar wave equation were derived. The S-wave velocities, although assumed to be zero in principal planes, still can be non-zero in other directions. This results in strong pseudo S-wave noise in wave propagation simulations. From the simplified dispersion relation, different variants of coupled second-order acoustic anisotropic wave equations have been developed. These equations are kinematically equivalent, but amplitude behavior may differ. More significantly, these equations suffer from instability in media of general inhomogeneity.

[0005] Attempts to overcome the problems of stability and pseudo S-waves derived the pure acoustic wave equation based on the P-wave dispersion relation. Like the one-way wave equation, the decoupled P-wave equation contains a pseudo- differential operator, which is difficult to tackle. For this reason, auxiliary elliptic wave equations are solved together, or some sophisticated numerical method is used to avoid alteration of wave propagation kinematics. By definition, there is no pseudo S- wave in this approach.

[0006] All equations derived from dispersion relations imply the assumption of locally constant media. Another way of obtaining pseudo acoustic wave equations starts from Hooke's law and the equations of motion. These anisotropic wave equations have the advantage of being physically clear and straightforward to implement. Furthermore, their stability has been demonstrated. Therefore, these acoustic anisotropic wave equations are preferred. However undesired S-waves are still present in the modeling.

SUMMARY

[0007] Exemplary embodiments are directed to systems and methods that utilize improved equations for removing pseudo S-waves to cancel the S-wave component by operations on accurate simulated stresses. The pure P-wave is calculated from saved wavefields for each imaging time step for RTM or FWI.

[0008] In one exemplary embodiment, a method for attenuating pseudo S- waves in acoustic anisotropic wave propagation is provided in which a plurality of anisotropic media parameters is obtained. These anisotropic media parameters are associated with a given subsurface area from which acquisition seismic data are obtained. The given subsurface area is associated with a type of acoustic anisotropic media. Suitable types of acoustic anisotropic media include, but are not limited to, vertical orthorhombic media, tilted orthorhombic media, vertical transversely isotropic media or tilted transversely isotropic media. In one embodiment, anisotropic media parameters are obtained based on the type of acoustic anisotropic media. Suitable anisotropic media parameters include, but are not limited to, velocity, Thomsen parameters, anisotropic angles, density, temperature, pressure, mass and

combinations thereof.

[0009] A shot position of a seismic source used to generate simulated seismic data is identified. The seismic source has an associated source wavelet that is a function of time. One or more of the plurality of anisotropic media parameters at the shot position are modified to define an ellipsoidally anisotropic region at the shot position of the seismic source. In one embodiment, the plurality of anisotropic media parameters include a first Thomsen epsilon parameter, a second Thomsen epsilon parameter, a first Thomsen delta parameter, a second Thomsen delta parameter and a third Thomsen delta parameter. Modifying one or more of the plurality of anisotropic media parameters further includes equating the first Thomsen epsilon parameter and the first Thomsen delta parameter, equating the second Thomsen epsilon parameter and the second Thomsen delta parameter and defining the third Thomsen delta parameter as a function of the first Thomsen delta parameter and the second

Thomsen delta parameter.

[0010] In one embodiment, the function of the first Thomsen delta parameter and the second Thomsen delta parameter is a difference between the first Thomsen delta parameter and the second Thomsen delta parameter divided by one plus twice the second Thomsen delta parameter. In addition to modifying anisotropic media parameters at the shot position, one or more of the plurality of anisotropic media parameters at additional positions within a given distance of the shot position can be modified to create a tapered transition from the obtained plurality of anisotropic media parameters and the one or more of the plurality of anisotropic media parameters modified at the shot position to define the ellipsoidally anisotropic region. In one embodiment, this given distance is about 150 meters.

[0011] The function of time of the source wavelet is used to define a source vector. The modified anisotropic media parameters and this source vector are used in an anisotropic wave equation to obtain a stress vector wavefield. The stress vector wavefield is used in a wavenumber domain representation of the anisotropic wave equation to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation. In one embodiment, the function of time of the source wavelet is included in each dimension of the source vector, and the function of time of the source wavelet is modified in at least one dimension of the source vector using one of the plurality of anisotropic media parameters. In one embodiment, modifying the function of time of the source wavelet in at least one dimension of the source vector further includes multiplying the function of time of the source wavelet by a square root of one plus twice a Thomsen delta parameter.

[0012] In one embodiment of using the stress vector to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation, a wavefield is defined in each one of a plurality of dimensions as a product of one of a plurality of linearly

independent eigenvectors and a component of the stress vector in a direction corresponding to that one of the plurality of linearly independent eigenvectors. A weighted arithmetic average of the plurality of wavefields is then calculated to obtain the pure P-wave wavefield. In one embodiment, calculating the weighted arithmetic average includes defining a mathematical equation and coefficients in that

mathematical equation based on a desired dimensionality and a type of acoustic anisotropic media.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate one or more embodiments and, together with the description, explain these embodiments. In the drawings:

[0014] Figure 1 is a flowchart of an embodiment of a method for attenuating pseudo S-waves in acoustic anisotropic wave propagation in accordance with the present invention;

[0015] Figure 2 is an illustration of vertical orthorhombic wavefield snapshots in homogeneous media;

[0016] Figure 3 is an illustration of an embodiment of using the stress vector in a wavenumber domain representation of the anisotropic wave equation to obtain a pure P-wave wavefield;

[0017] Figure 4 is an illustration of two-dimensional vertical transversely isotropic wavefield snapshots for a two-layer model; and

[0018] Figure 5 illustrates an exemplary data processing device or system which can be used to implement the methods. DETAILED DESCRIPTION

[0019] The following description of the embodiments refers to the accompanying drawings. The same reference numbers in different drawings identify the same or similar elements. The following detailed description does not limit the invention.

Instead, the scope of the invention is defined by the appended claims. Some of the following embodiments are discussed, for simplicity, with regard to local activity taking place within the area of a seismic survey. However, the embodiments to be discussed next are not limited to this configuration, but may be extended to other arrangements that include regional activity, conventional seismic surveys, etc.

[0020] Reference throughout the specification to "one embodiment" or "an embodiment" means that a particular feature, structure or characteristic described in connection with an embodiment is included in at least one embodiment of the subject matter disclosed. Thus, the appearance of the phrases "in one embodiment" or "in an embodiment" in various places throughout the specification is not necessarily referring to the same embodiment. Further, the particular features, structures or characteristics may be combined in any suitable manner in one or more embodiments.

[0021] Exemplary embodiments of systems and methods utilize acoustic wave equations and S-wave removal. The focus is on second order acoustic wave equations for the principal stress vector„ = (_σχχ , _{σ yy} , a₂₂Y , although all discussions below can be extended to acoustic wave equations for particle velocities. For simplicity, density is assumed to be a constant. In vertical media, the symmetry planes align with the coordinate axes. In vertical orthorhombic (VORT) media, the anisotropic wave equation is given by:

¾ = cz⁾ _{C + s} (1 ) dt²

where D = diag{d¹ _xx, d¹

s is the source vector, and c = (c_iJ) denotes the following symmetric 3x3 elastic matrix with velocity and six Thomsen parameters:

[0022] A three-dimensional (3D) vertical transversely isotropic (VTI) media is a special case of VORT media with ε₁ =ε₂(=ε), δ_ι =δ₂(=δ), and s₃ = 0. In two-dimensional (2D) VTI media, there is no term _σ and the elastic matrix degenerates to a 2x2 matrix.

In isotropic media the stress vector degenerates to the pressure wavefield, and equation (1 ) becomes the isotropic acoustic wave equation.

[0023] For transversely isotropic (Tl) wave propagation, the source is surrounded with a smooth tapered ellipsoidally anisotropic region, .e., _s =s , to suppress the source- generated S-wave. Although internal conversions between P- and S-waves always exist, this is a plausible and easy way to remove part of the S-wavefield while minimally influencing the kinematics. This method is generalized to orthorhombic media. An ellipsoidal region is defined by:

¾ =<¾, δ₃ = (δ₁ -δ₂)/(ΐ+2δ₂), (3) and three zero angles. The medium is tapered around the source to be ellipsoidal to get rid of source-generated pseudo S-waves. The components of the source vector are not identical.

[0024] The source vector is written as the following form:

where / is a source signature. Marine surface acquisition automatically places the source in the isotropic water layer, that limits the application of this method except in shallow water survey. This approach is useful for vertical seismic profile (VSP), ocean bottom cable (OBC) and land data, where the sources or receivers may be very near to or in strongly anisotropic media.

[0025] In order to attenuate propagation S-waves, the pure P-wave is expressed by eigenvalue analysis. Assuming local homogeneity, equation (1 ) is rewritten in the wavenumber domain as:

The transpose of the matrix G is given

[0026] This transpose matrix has three non-negative eigenvalues^ , x_sr and x_SH corresponding to the squared phase velocities of three wave modes. Supposing that the three corresponding linear independent eigenvectors are x_p , x_sv and x_SH , the matrix G can be diagonalized, and equation (5) becomes:

with

[0027] The three new wavefieldsy , Y_SV and Y_SH are exactly the P-wave, SV-wave and SH-wave. Equation (7) implies that knowing the eigenvector of matrix G^T corresponding to the eigenvalue x_p produces the pure P-wave component.

[0028] Under the acoustic assumption, the S-wave velocities are very small in all directions ( _sv∞ _SH « 0). Employing this approximation, all three columns in matrix (6) can be considered as the eigenvectors to x_p . Taking a weighted arithmetic average of the three columns yields:

₌ (.cX )a _II + {c_yk_y ¹ )a_w + £,²σ„ (8) k + k_y ² + k]

with _{c =} c_u/c_l3 +c_u/c₂₃ +c_l3/c_{33 c =} c_u/c_l3 + c₂₂/c₂₃ +c₂₃/c₃₃ For 2D VTI media, the

3 ' ^v 3

coefficients in the equation are slightly different and are given by:

[0029] In order to consider a more precise but costly formulae, the value of λ_ρ is given by the root of an algebraic equation of degree 3 and its Taylor expansion with respect to anisotropic parameters is considered. Equations (8) and (9) are essentially the zero order approximation with reasonable compensation for higher order effects. To obtain better accuracy, the correct first order approximation is used as given by:

[0030] The eigenvector is computed from the value in (10). [0031] The formula corresponding to (8) is of great complexity with many derivatives in VORT media, and is not very practical. But for VTI media, the formula becomes less complicated:

P {k +k_y ² +k f '

With ₊e_y)k ₊kl For 2D VTI media:

γ (12) with w ^^^ ρ₂₀ = +_ε +δ^ ₊^⁴.

v +25

[0032] To demonstrate the higher order P-wave construction formulae, 2D two- layer model is considered. The first layer is isotropic with velocity equal to 1500 m/s, and the second layer is VTI with _s =0.35, δ =-o.i , and velocity 2000 m/s. A point source is placed in the isotropic layer 25 m above the second layer. Finite-difference (FD) modeling is performed using grid spacings dx = dy = dz = n.5m . Without S-wave attenuation, spurious SV-wavefronts appear in the wavefield. Using equation (9), the residual S-wave is highly attenuated; however, weak pseudo S-waves remain. Applying the more precise equation (12), achieves complete elimination of S-wave energy in the wavefield.

[0033] The acoustic wave equations in transversely isotropic (TTI) media and tilted orthorhombic (TORT) media are computationally demanding. To obtain more efficient equations, the derivatives in equation (1 ) are replaced by rotated differential operators, i.e. (D_X, D

e ¾)^Γ , where R is the transformation from local system to global system with 3 angles, and the adjoints of these operators to yield simplified stable tilted media acoustic wave equations. Those wave equations give accurate amplitudes compared with the exact equations. All the above P-wave construction formulae (8), (9), (1 1 ), and (12) become valid for TTI and TORT media by rotating the wavenumbers.

[0034] The above P-wave construction formulae (8), (9), (1 1 ), and (12) can be implemented by the finite-difference (FD) method, the pseudospectral method or a combined method in a mixed space-wavenumber domain. In order to solve the elliptic equations in the denominators, a FD implementation is preferred, which allows variable- length grids especially in the vertical direction. An approach based on a high-order FD scheme is used. Using equation (8) and a high-order FD scheme, the P-wave is almost unchanged, and S-wave crosstalk is successfully removed.

[0035] Since the stable acoustic anisotropic wave equations have a major drawback in producing unwanted pseudo S-waves, exemplary embodiments utilize new formulae based on eigenvalue analysis for attenuating those pseudo S-waves at each output step. In addition, the method of modifying anisotropic parameters around the source is extended from transversely isotropic (Tl) to orthorhombic (ORT) media, further attenuating source-generated S-waves. These equations provide almost pure P- wave propagation.

[0036] Referring initially to Figure 1 , one exemplary embodiment is directed to a method for attenuating pseudo S-waves in acoustic anisotropic wave propagation 100. The method takes into account the type of acoustic anisotropic media associated with the seismic data. Therefore, the type of acoustic anisotropic media in a given area from which seismic data are obtained is identified 102. In general, there are two types of acoustic anisotropic media, orthorhombic media and transversely isotropic (Tl) media. In fact, Tl is a special and simpler case of orthorhombic. Orthorhombic media is further divided into vertical orthorhombic and tilted orthorhombic. Similarly, Tl media is divided into vertical Tl (VTI) and tilted Tl (TTI). Tilted media are more complicated than vertical media, because tilted media contain anisotropic angle parameters.

[0037] Therefore in one embodiment, the identified types of acoustic anisotropic media include vertical orthorhombic media, tilted orthorhombic media, vertical transversely isotropic media and tilted transversely isotropic media. Although one type may be a special case of another, e.g., VTI can be seen as a special case of TTI, the different types of media have different groups of associated anisotropic parameters, i.e. different groups of anisotropic models of the media. For example, TTI media has more anisotropic models than VTI media. The identified or selected type of anisotropic media can be based on the information available for the given geological area.

[0038] Having identified the type of acoustic anisotropic media in the subsurface geological area, a plurality of anisotropic media parameters associated with that given subsurface area are obtained 104.

[0039] Suitable geological parameters or geological models of the subsurface include, but are not limited to, wave velocity, density, Thomsen parameters (epsilon, delta and gamma), anisotropic angles, temperature, pressure, mass and combinations thereof. As the groups of geological parameters vary based on the type of acoustic anisotropic media, the plurality of anisotropic media parameters are obtained based on the type of acoustic anisotropic media. One of the most important models or parameters is the velocity of the wave in subsurface. For anisotropic media, Thomsen parameters and anisotropic angles are also important. In general, many parameters exist to describe a given geological area. As exemplary embodiments are directed to wave modeling, the wavefields in the geological area are simulated. Therefore, all the terms or parameters or models in the acoustic wave equation that are used to simulate the wavefields are known except the wavefield. Since the geological media is anisotropic, an anisotropic wave equation is used, which includes more parameters than an isotropic wave equation. Those parameters are anisotropic parameters, which can also be referred to as anisotropic models.

[0040] A shot position of a seismic source used to generate the simulated seismic data is identified 106. This seismic source has an associated source wavelet, which is a function of time. The shot position of the seismic source is also referred to as the source position, and the source wavelet can be referred to as the source signature. Suitable sources include the recorded seismic data in wave propagation in RTM and the data residual in wave propagation in FWI.

[0041] In a marine seismic data acquisition system, an air gun provides a wave, which is sometimes called source, and is also referred to as a source wavelet or source signature that varies on time. The wave goes into the subsurface, and the geophones receive the returned wave, called seismic data. Similar systems are used in land seismic data acquisition systems.

[0042] Mathematically, the source signature or source wave is expressed as a function "f(x,y,z,t)" where t is the time. The air gun, for example, is controlled during acquisition; therefore, the source, its position, e.g., its position at the time of a given shot and the variation of the source wavelet over time is known. In numerical modeling, the source is the function that is to be propagated. Before a numerical modeling, the shot position and source wavelet/signature, i.e., the source function is known. For a special numerical method to solve the wave equation, e.g., finite difference (FD), if the source locations are not exact on the grids, they may be spread to FD grids. This is also done before a numerical modeling. [0043] One or more of the plurality of anisotropic media parameters at the shot position are modified to define an ellipsoidally anisotropic region at the shot position of the seismic source 108. In one embodiment, the modified media parameters include one or more of the Thomsen parameters, e.g., epsilon and delta. For example, the plurality of anisotropic media parameters includes a first Thomsen epsilon parameter, a second Thomsen epsilon parameter, a first Thomsen delta parameter, a second Thomsen delta parameter and a third Thomsen delta parameter. The first Thomsen epsilon parameter and the first Thomsen delta parameter are equated, and the second Thomsen epsilon parameter and the second Thomsen delta parameter are equated. In addition, the third Thomsen delta parameter is defined as a function of the first Thomsen delta parameter and the second Thomsen delta parameter. In one embodiment, the function of the first Thomsen delta parameter and the second Thomsen delta parameter is a difference between the first Thomsen delta parameter and the second Thomsen delta parameter divided by one plus twice the second Thomsen delta parameter. In addition, for tilted media, anisotropic angles are zero.

[0044] Mathematically, a first Thomsen epsilon parameter, ε_ι , is changed such that^ ^, i.e. , the first Thomsen delta parameter. In addition, a second Thomsen epsilon parameter, ε₂ is changed such thatg₂ =<5₂ , the second Thomsen delta parameter.

The third Thomsen delta parameter, <¾, is defined or changed such that s₃ = (δ₁ -δ₂)/(ΐ+2δ₂) , i.e., a difference between the first Thomsen delta parameter and the second Thomsen delta parameter divided by one plus twice the second Thomsen delta parameter. This embodiment is suitable for wave propagations that focus more attention on small incident angle information, for example in RTM.

[0045] Alternatively, δ_ι is changed such

and <5₂ is changed such that <¾ =¾ . Again, <5₃ is defined such thats₃ = (δ₁ -δ₂)/(ΐ+2δ₂) . This is a suitable strategy for wave propagations that focus more attention on large incident angle information, for example in refractions/transmissions FWI. For the purpose of S-wave attenuation, any modification of Thomsen parameters which satisfies equation (3) can be used.

[0046] The illustrated examples are for orthorhombic media. For Tl media, which is a special case and much simpler, only the second Thomsen epsilon parameter and the second Thomsen delta parameter are used. In addition, a single source position can be used or multiple source positions are defined, i.e., the source can be one point or an area.

[0047] After modifying the anisotropic media parameters at the shot position, the anisotropic media parameters around the shot position are modified 1 10 to produce a gentle modification of the anisotropic media parameters around the shot position. This yields a model taper around the source positions. In one embodiment, one or more of the plurality of anisotropic media parameters are modified at additional positions within a given distance of the shot position to create a tapered transition from the obtained plurality of anisotropic media parameters and the one or more of the plurality of anisotropic media parameters modified at the shot position to define an ellipsoidally anisotropic region. In one embodiment, the given distance is about 150 meters.

[0048] In one embodiment, A denotes the area of all source positions. Suppose that all models on A have been modified by equation (3). The models aroundv4 are gradually modified. A point x is "around" A if the distance between x andA is less then / meters. In practice, / depends on the level of anisotropy. In one embodiment, / =150m.

on is defined. In one embodiment, the taper function is

For any Thomsen parameter, e.g. , e_v on the point x, the

[0, ifd>l.

letter y denotes the point in A (a source position) which is closest to x. In addition, d(x,y) is the distance between x and . Then ε^χ) is modified to (s_l(y)-s_l(x))W(d(x,y))+s_l(x). Therefore, if x is also a point in A (a source position), y=x, and the parameter will not be changed as this value is already modified by (3). If the distance between x and A is larger than l, W(d(x,y))=0, and the parameter is not changed.

[0050] The function of time of the source wavelet is used to define a source vector 1 12. This is the source vector that is used in the anisotropic acoustic wave equation illustrated in equation (1 ). In one embodiment, the function of time of the source wavelet is included in each dimension of the source vector. For example, a source vector, s can contain 2 or 3 functions, corresponding to 2 or 3 dimensions. In practice, these functions are just the source wavelet function, /. Therefore, the source vector containing three functions is s = (/, /, /). If the source positions are in isotropic media, i.e. , all anisotropic parameters are 0, the source vector is still s = (/, /, /).

[0051] In one embodiment, the function of time of the source wavelet in at least one dimension of the source vector is modified using one of the plurality of anisotropic media parameters. For example, the function of time of the source wavelet is multiplied by a square root of one plus twice a Thomsen delta parameter as illustrated, for example, in equation (4). Therefore, the source wavelet function and source vector are modified in accordance with equation (4).

[0052] Referring to Figure 2, VORT wavefield snapshots in homogeneous media 200 are illustrated (a) without changes of models, models are modified 150 m around the source in the three other cases, (b) source in isotropic region, (c) source in ellipsoidal region with the source term (4) and (d) source also in ellipsoidal region but with identical components in the source vector. The anisotropic media parameters for VORT medium are given by. v_p0 = 2000m / s , ε_ι = 0.3 , ε₂ = 0.25 , ^ = 0.15 , <5₂ = 0.2 , <5₃ = -0.1 . Strong S-wave noise 202 is present in (a), where the source is in the VORT medium. In (b)-(d), the models are modified in a region of 150 m around the source. If the region is isotropic, quick changes in the tapers result in the artifact 204 in (b). In addition, a near 15 m horizontal wavefront delay can be identified that leads to simulation travel time error. This travel time error reduces the accuracy of wave simulation, especially for FWI. If the models are tapered in an ellipsoidal region, the resulting snapshot (c) is nearly perfect, and the horizontal travel time error is negligible. In (d), the model is tapered as in (c), but the source vector components are identical, and the artifact on the source location is very strong.

[0053] Returning to Figure 1 , using the modified anisotropic media parameters and the source vector in an anisotropic wave equation to obtain the stress vector wavefield 1 14. In one embodiment, numerical wave propagation is performed using wave equation (1 ). The obtained wavefields are stress vectors. The stress vector is used in a wavenumber domain representation of the anisotropic wave equation to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation 1 16.

[0054] Referring to Figure 3, an embodiment of using the stress vector in a wavenumber domain representation of the anisotropic wave equation to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation 300 is illustrated. A wavefield in each one of a plurality of dimensions is defined as a product of one of a plurality of linearly independent eigenvectors and a component of the stress vector in a direction corresponding to that one of the plurality of linearly independent eigenvectors 302. In addition, a weighted arithmetic average of the plurality of wavefields is calculated to obtain the pure P-wave wavefield 304.

[0055] In one embodiment, calculating the weighted arithmetic average further includes defining a mathematical equation and coefficients in that mathematical equation based on a desired dimensionality and a type of acoustic anisotropic media. For example, in a three dimensional world, the dimension is 3 and three coordinates, x,y,z, are used. In some embodiments, only two dimensional part of the media is studied, e.g. with unchanged "y". Thus, the wave equation is solved in dimension 2. If the media is ORT media, which includes Tl media, then either equation (8), three dimensions, or equation (9), two dimensions is used. If the media is Tl (transversely isotropic) and has strong anisotropic parameters, equation (1 1 ), three dimensions, or equation (12), two dimensions is used instead of equations (8) or (9) to produce the pure P-wavefield.

[0056] Exemplary embodiments change the anisotropic media parameters in orthorhombic media around the source position in accordance with equation (3). In addition, the source vector in the anisotropic wave equation is changed using the source wavelet in accordance with equation (4). P-wave construction formulae (8), (9), (1 1 ) and (12) are utilized with the modified anisotropic media parameters and modified source vector equation to generate a pure P-wavefield based on the desired dimensionality and the type of acoustic geological media. These equations could also be used with non-modified anisotropic media parameters and a source vector not in accordance with equation (4).

[0057] Referring to Figure 4, two dimensional VTI wavefield snapshots for a two- layer model 400 are illustrated. Section (a) is without S-wave attenuation, and section (b) is with S-wave attenuation using equation (9). Section (c) is with S-wave attenuation using higher order equation (12). Without S-wave attenuation, a diamond-shaped spurious SV-wavefront 402 appears in (a). In (b), the residual S-wave is highly attenuated by equation (9), but weak pseudo S-waves 404 remain. Applying the more precise equation (12), a complete elimination of S-wave energy is obtained in (c).

[0058] Methods and systems in accordance with exemplary embodiments can be hardware embodiments, software embodiments or a combination of hardware and software embodiments. In one embodiment, the methods described herein are implemented as software. Suitable software embodiments include, but are not limited to, firmware, resident software and microcode. In addition, exemplary methods and systems can take the form of a computer program product accessible from a computer-usable or computer-readable medium providing program code for use by or in connection with a computer, logical processing unit or any instruction execution system. In one embodiment, a machine-readable or computer-readable medium contains a machine-executable or computer-executable code that when read by a machine or computer causes the machine or computer to perform a method for attenuating pseudo S-waves in acoustic anisotropic wave propagation in accordance with exemplary embodiments and to the computer-executable code itself. The machine-readable or computer-readable code can be any type of code or language capable of being read and executed by the machine or computer and can be expressed in any suitable language or syntax known and available in the art including machine languages, assembler languages, higher level languages, object oriented languages and scripting languages.

[0059] As used herein, a computer-usable or computer-readable medium can be any apparatus that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. Suitable computer-usable or computer readable mediums include, but are not limited to, electronic, magnetic, optical, electromagnetic, infrared, or

semiconductor systems (or apparatuses or devices) or propagation mediums and include non-transitory computer-readable mediums. Suitable computer-readable mediums include, but are not limited to, a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk and an optical disk. Suitable optical disks include, but are not limited to, a compact disk - read only memory (CD-ROM), a compact disk - read/write (CD-R/W) and DVD.

[0060] In one embodiment, a computing device for performing the calculations as set forth in the above-described embodiments may be any type of computing device capable of processing and communicating seismic data associated with a seismic survey. An example of a representative computing system capable of carrying out operations in accordance with these embodiments is illustrated in Figure 5. The computing system 500 includes a computer or server 502 having one or more central processing units 504 in communication with a communication module 506, one or more input/output devices 510 and at least one storage device 508. All of these components are known to those of ordinary skill in the art, and this description includes all known and future variants of these types of devices. The communication module provides for communication with other computing systems, databases and data acquisition systems across one or more local or wide area networks 512. This includes both wired and wireless communication. Suitable input-output devices include keyboards, point and click type devices, audio devices, optical media devices and visual displays.

[0061] Suitable storage devices include magnetic media such as a hard disk drive (HDD), solid state memory devices including flash drives, ROM and RAM and optical media. The storage device can contain data as well as software code for executing the functions of the computing system and the functions in accordance with the methods described herein. Therefore, the computing system 500 can be used to implement the methods described above associated with the calculation of the induced source shot gather. Hardware, firmware, software or a combination thereof may be used to perform the various steps and operations described herein.

[0062] The disclosed exemplary embodiments provide a computing device, software and method for calculating the induced source shot gather. It should be understood that this description is not intended to limit the invention. On the contrary, the exemplary embodiments are intended to cover alternatives, modifications and equivalents, which are included in the spirit and scope of the invention. Further, in the detailed description of the exemplary embodiments, numerous specific details are set forth in order to provide a comprehensive understanding of the invention. However, one skilled in the art would understand that various embodiments may be practiced without such specific details.

[0063] Although the features and elements of the present exemplary

embodiments are described in the embodiments in particular combinations, each feature or element can be used alone without the other features and elements of the embodiments or in various combinations with or without other features and elements disclosed herein. The methods or flowcharts provided in the present application may be implemented in a computer program, software, or firmware tangibly embodied in a computer-readable storage medium for execution by a geo-physics dedicated computer or a processor.

[0064] This written description uses examples of the subject matter disclosed to enable any person skilled in the art to practice the same, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the subject matter is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims.

Claims

WHAT IS CLAIMED IS:

1 . A method for attenuating pseudo S-waves in acoustic anisotropic wave propagation (100), the method comprising:

obtaining a plurality of anisotropic media parameters associated with a given subsurface area from which acquisition seismic data are obtained (104);

identifying a shot position of a seismic source used to generate simulated seismic data, the seismic source having an associated source wavelet comprising a function of time (106);

modifying one or more of the plurality of anisotropic media parameters at the shot position to define an ellipsoidally anisotropic region at the shot position of the seismic source (108);

using the function of time of the source wavelet to define a source vector

(1 12);

using the modified anisotropic media parameters and the source vector in an anisotropic wave equation to obtain stress vector wavefield (1 14); and

using the stress vector wavefield in a wavenumber domain representation of the anisotropic wave equation to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation (1 16).

2. The method of claim 1 , wherein:

the given subsurface area is associated with a type of acoustic anisotropic media; and

obtaining the plurality of anisotropic media parameters comprises obtaining anisotropic media parameters based on the type of acoustic anisotropic media.

3. The method of claim 2, wherein the type of acoustic anisotropic media comprises vertical orthorhombic media, tilted orthorhombic media, vertical transversely isotropic media or tilted transversely isotropic media.

4. The method of claim 1 , wherein the plurality of anisotropic media parameters comprise velocity, Thomsen parameters, anisotropic angles, density, temperature, pressure, mass or combinations thereof.

5. The method of claim 1 , wherein:

the plurality of anisotropic media parameters comprise a first Thomsen epsilon parameter, a second Thomsen epsilon parameter, a first Thomsen delta parameter, a second Thomsen delta parameter and a third Thomsen delta parameter; and

modifying one or more of the plurality of anisotropic media parameters further comprises:

equating the first Thomsen epsilon parameter and the first Thomsen delta parameter;

equating the second Thomsen epsilon parameter and the second Thomsen delta parameter; and

defining the third Thomsen delta parameter as a function of the first Thomsen delta parameter and the second Thomsen delta parameter.

6. The method of claim 5, wherein the function of the first Thomsen delta parameter and the second Thomsen delta parameter comprises a difference between the first Thomsen delta parameter and the second Thomsen delta parameter divided by one plus twice the second Thomsen delta parameter.

7. The method of claim 1 , further comprising modifying one or more of the plurality of anisotropic media parameters at additional positions within a given distance of the shot position to create a tapered transition from the obtained plurality of anisotropic media parameters and the one or more of the plurality of anisotropic media parameters modified at the shot position to define the ellipsoidally anisotropic region.

8. The method of claim 7, wherein the given distance is about 150 meters.

9. The method of claim 1 , wherein using the function of time of the source wavelet to define the source vector further comprises:

including the function of time of the source wavelet in each dimension of the source vector; and

modifying the function of time of the source wavelet in at least one dimension of the source vector using one of the plurality of anisotropic media parameters.

10. The method of claim 9, wherein modifying the function of time of the source wavelet in at least one dimension of the source vector further comprises multiplying the function of time of the source wavelet by a square root of one plus twice a Thomsen delta parameter.

1 1 . The method of claim 1 , wherein using the stress vector to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation further comprises: defining a wavefield in each one of a plurality of dimensions as a product of one of a plurality of linearly independent eigenvectors and a component of the stress vector in a direction corresponding to that one of the plurality of linearly independent eigenvectors; and

calculating a weighted arithmetic average of the plurality of wavefields to obtain the pure P-wave wavefield.

12. The method of claim 1 1 , wherein calculating the weighted arithmetic average further comprises defining a mathematical equation and coefficients in that mathematical equation based on a desired dimensionality and a type of acoustic anisotropic media.

13. A method for attenuating pseudo S-waves in acoustic anisotropic wave propagation, the method comprising:

identifying a shot position of a seismic source used to generate the simulated seismic data, the seismic source having an associated source wavelet comprising a function of time (106);

using the anisotropic media parameters and the source wavelet comprising the function of time in an anisotropic wave equation to obtain stress vector wavefield (1 14);

using a wavenumber domain representation of the anisotropic wave equation to define a wavefield in each one of a plurality of dimensions as a product of one of a plurality of linearly independent eigenvectors and a component of the stress vector in a direction corresponding to that one of the plurality of linearly independent eigenvectors (302); and

calculating a weighted arithmetic average of the plurality of wavefields to obtain a pure P-wave wavefield (304).

14. The method of claim 13, wherein calculating the weighted arithmetic average further comprises defining a mathematical equation and coefficients in that mathematical equation based on a desired dimensionality and a type of acoustic anisotropic media.

15. The method of claim 14, wherein:

the desired dimensionality comprises two-dimensions or three-dimensions; and

the type of acoustic anisotropic media comprises vertical orthorhombic media, tilted orthorhombic media, vertical transversely isotropic media or tilted transversely isotropic media.

16. The method of claim 13, wherein using the anisotropic media parameters and the source wavelet comprising the function of time in the anisotropic wave equation further comprises:

using the function of time of the source wavelet to define a source vector comprising the function of time of the source wavelet in each dimension of the source vector;

modifying the function of time of the source wavelet in at least one dimension of the source vector using one of the plurality of anisotropic media parameters; and using the anisotropic media parameters and the source vector in an

anisotropic wave equation to obtain stress wavefield vector.

17. The method of claim 16, wherein modifying the function of time of the source wavelet in at least one dimension of the source vector further comprises multiplying the function of time of the source wavelet by a square root of one plus twice a Thomsen delta parameter.

18. A computer-readable storage medium containing a computer-readable code that when read by a computer causes the computer to perform a method for attenuating pseudo S-waves in acoustic anisotropic wave propagation (100), the method comprising:

using the function of time of the source wavelet to define a source vector

(1 12);

19. The computer readable storage medium of claim 18, wherein using the stress vector to obtain a pure P-wave wavefield from acoustic anisotropic wave propagation further comprises:

defining a wavefield in each one of a plurality of dimensions as a product of one of a plurality of linearly independent eigenvectors and a component of the stress vector in a direction corresponding to that one of the plurality of linearly independent eigenvectors; and

20. The computer readable storage medium of claim 19, wherein calculating the weighted arithmetic average further comprises defining a mathematical equation and coefficients in that mathematical equation based on a desired dimensionality and a type of acoustic anisotropic media.