Tu SRS3 02 Data Reconstruction and Denoising of Different Wavefield Components Using Green s Theorem

Transcription

1 Tu SRS3 02 Data Reconstruction and Denoising of Different Wavefield Components Using Green s Theorem N. Kazemi* (University of Alberta) & A.C. Ramirez (Statoil) SUMMARY Multicomponent technology is likely to become the standard in future towed streamer surveys. Among reasons to acquire multicomponent data is to increase efficiency (deep tow) in acquisition, increase resolution (deghost) of pressure data, interpolate to a fine grid (using crossline directional measurements). One of the challenges of working with towed streamer multicomponent data is the fact that the measured low frequencies in the directional components are rather noisy mostly due to practical aspects/limitations. We present a formalism based on Green s theorem that provides tools to reconstruct, redatum, and/or denoise these components, and/or for deghosting.

2 Introduction and Motivation In marine streamer data, for each unghosted event (primary, multiple, etc) there are three ghosts (source ghost, receiver ghost, and source-receiver ghost). Ghosts interfere with the desired unghosted events. In frequency domain, the interference can be destructive (generating notches at specific frequencies); in time domain, it might visually extend the wavelet s length. Thus, the final image resolution is degraded by the presence of ghosts. Deghosting a single component dataset (i.e., pressure field) is challenging, mostly due to singularities and the need for assumptions about the sea surface and the water column. In deghosting of multicomponent datasets the scalar component, pressure field, is the complementary part of a directional component, normal/vertical pressure field derivative (Zhang and Weglein, 2005). In practice, the measured streamer directional components are noisy. The measurements of geophones or accelerometers (which are scaled and used as directional derivatives) have low signal to noise ratio and suffer from the errors enforced by moving devices (i.e., vessel, currents). In this abstract we provide a formalism based on Green s theorem to predict, redatum, deghost, denoise the pressure wavefield and its spatial gradients. Theory Green s second identity can be derived from divergence theorem and divergence theorem itself can be considered as the generalization of the Fundamental Theorem of Calculus to higher dimensions (Green, 1828; Ramírez and Weglein, 2009). Green s second identity is: ψ(r,r s,ω) 2 φ(r,r,ω) φ(r,r,ω) 2 ψ(r,r s,ω) dr = V [ψ(r,r s,ω) φ(r,r,ω) φ(r,r,ω) ψ(r,r s,ω)].ˆn ds, (1) S where ˆn is a unit vector normal to the surface S enclosing the volume V, ψ(r,r s,ω) and φ(r,r,ω) are arbitrary functions that are defined in a way that their second derivatives and the integral in equation (1) exist. In this work, V is the volume between sea surface and measurement surface (i.e., water in marine acquisition). Two arbitrary functions ψ(r,r s,ω) and φ(r,r,ω) should be consistent with the medium properties inside the volume V. For our proposes, ψ(r,r s,ω) is chosen to be a measured pressure wavefield P(r,r s,ω) that satisfies the acoustic wave equation 2 P(r,r s,ω) + c 2 (r ) P(r,r s,ω) = A(ω)δ(r r s ), (2) where A(ω) is the source signature and satisfies the convolutional theorem. The reference medium satisfies the actual Earth properties in V (half-space of water and half-space of air). By adding a perturbation α(r ) 1 to this reference medium one can change water to Earth. By substituting with c 2 (r ) ω 2 (1 c 2 0 (r ) α(r )) equation (2) becomes ( 2 + c 2 0 (r ) )P(r,r s,ω) = A(ω)δ(r r s ) + c 2 0 (r ) )(1 α(r ))P(r,r s,ω), (3) where c 0 is water speed inside the predefined volume (V ) with a sea surface on top. By choosing different volume (V ), a different position of source (inside or outside of the volume) and replacing φ(r,r,ω) in equation (1) by a different Green s function in the water medium that satisfies the wave equation for the reference medium, one can derive several applications such as deghosting, redatuming, wavefield separation, interpolation, and imaging (Ramírez and Weglein, 2009; Amundsen et al., 2013). In the next section, we derive wavefield prediction equations with two different approaches based on Green s theorem. The first formalism uses both, the measured pressure wavefield and its measured 1 For true Earth α(r ) is a more complicated function that perturbs an acoustic medium into a full elastic medium.

3 normal gradient (normal with respect to the measurement surface) to predict and/or redatum the pressure wavefield, its spatial gradients, or the upgoing field. The second formalism uses a double Dirichlet Green s function to predict/redatum the abovementioned fields using only the pressure component as input. Derivation of field prediction formula By replacing φ(r,r,ω) in equation (1) with the Green s function G 0 in reference medium which satisfies the wave equation 2 G 0 (r,r,ω) + c 2 0 (r ) G 0(r,r,ω) = δ(r r); (4) substituting equations (4) and (3) into equation (1); choosing the space between measurement surface and sea surface as the volume V ; choosing r anywhere inside that volume; considering the input data source location outside the volume V, equation (1) becomes P(r,r s,ω) = [P(r,r s,ω) G 0 (r,r,ω) G 0 (r,r,ω) P(r,r s,ω)].ˆn ds, (5) where is the measurement surface. This formula can predict and redatum the pressure wavefield anywhere within the volume V, using measurements of pressure wavefield and its gradient. By applying a first derivative with respect to any space coordinates in r to both sides of equation (5), the corresponding pressure derivative within the volume V is obtained, χ P(r,r s,ω) = [P(r,r s,ω) χ G 0(r,r,ω) P(r,r s,ω) χ G 0(r,r,ω)].ˆn ds, (6) where χ can be x,y or z directions and (x, y, z) r. Moreover, changing the Green s function G 0 to a causal one G + 0 that satisfies a medium without a sea surface (half space of water) 2 G + 0 (r,r,ω) + c 2 0 (r ) G+ 0 (r,r,ω) = δ(r r), (7) one can show that this new Green s function will allow us to predict and redatum the receiver deghosted wavefields. 2 In other words, equation (5) becomes P + (r,r s,ω) = [P(r,r s,ω) G + 0 (r,r,ω) G + 0 (r,r,ω) P(r,r s,ω)].ˆn ds, (8) and, by extension, equation (6) becomes χ P+ (r,r s,ω) = [P(r,r s,ω) χ G+ 0 (r,r,ω) P(r,r s,ω) χ G+ 0 (r,r,ω)].ˆn ds, (9) which provide deghosted pressure field and its spatial derivatives, respectively. Derivation of field prediction formula via Double Dirichlet Green s function The boundary conditions do not need to be the same for the two functions used to satisfy the Green s theorem. The only requirement is that the medium parameters are equal inside the volume V. Choosing a Green s function satisfying different boundary conditions than the pressure field, one can predict different wavefield components using only measured pressure. Consider again equation (1), and replace φ(r,r,ω) with the double Dirichlet Green s function G DD 0 in reference medium which satisfies 2 G DD 0 (r,r,ω) + c 2 0 (r ) GDD 0 (r,r,ω) = δ(r r), (10) 2 P + (r,r s,ω) is receiver deghosted if the calculations are done on P(r,r s,ω) and P(r,r s,ω) corresponding to shot gathers. Source deghosting and source-receiver deghosting can be applied if we were working in a different data domain (Yarman and Ramírez, 2013).

4 Figure 1 2.5D synthetic example: Left is modeled data. Middle is the prediction using equation (6). Right is the prediction using equation (12). where G 0 DD satisfies the same medium parameters, as the pressure field, within the volume (i.e., water properties) but with different boundary conditions. The boundary conditions are set so that G 0 DD vanishes at the boundaries (i.e. measurement and sea surface). Substituting equations (3) and (10) into equation (1); choosing the space between the sea surface and the measurement surface as V ; selecting r anywhere inside V ; using the proper boundary conditions on G DD 0 ; considering the source position outside the predefined volume; and ignoring the effects of vertical walls 3, equation (1) becomes P(r,r s,ω) = [P(r,r s,ω) G DD 0 (r,r,ω)].ˆn ds. (11) Equation (11) shows that one can predict and redatum the pressure wavefield by just using the measured pressure wavefield and a double Dirichlet Green s function G DD 0. As in the previous section, one can predict and redatum the spatial derivatives of pressure wavefield by applying the corresponding derivative operators on both sides of equatio (11) χ P(r,r s,ω) = [P(r,r s,ω) χ G DD 0 (r,r,ω)].ˆn ds. (12) Synthetic and real data examples We evaluated the performance of our formulations on synthetic and real data. The results shown here dp correspond only to the prediction of the vertical derivative component (i.e., dz ). The synthetic data are 2.5D FD modeled based on a model from the Norwegian Continental Shelf (NCS). Figure 1 shows that the predictions with the two methods are comparable to the modeled synthetic benchmark. Main differences are in the direct arrival (consistent with theory) and at aliased frequencies (especially strong amplitude, aliased events). Our observations indicate that if the amplitude of aliased data have been attenuated, the aliasing-related errors are minimized. There are also differences in the overall strength of the scattered events (i.e., excluding the direct arrival). This can be compensated for. Next we applied equation (6) to field data from the NCS. What we expect is better fidelity in the low frequencies (where noise affects the measurements). The comparison of FK spectra for a sample shot in Figure 2, shows that the reconstruction is good at all frequencies and stronger low frequency signal is observed below 40Hz. 3 This only has a higher order effect, the main contribution comes from the measurement surface.

5 Figure 2 NCS data: Left is measured data and Figure 3 FK spectrum corresponding to the right is the predicted data. leftmost shot gather in Figure 2. Their effect is observed in the reconstructed and 2.4s. dp dz (see Figure 3), especially at mid-far offsets between 1.7s Conclusions The resolution of towed streamer data can be drastically decreased due to the presence of ghosts. New multicomponent streamer measurements provide the means to deghost and interpolate pressure data in a data-driven fashion. However, the measurements of the directional (vertical/horizontal) components are degraded by noise, especially at low frequencies. Using integral equation approaches, the low frequencies of different wavefield components can be recalculated either by using more than one measured component, or pressure data only. We presented two integral formulations that can be used in isolation or in combination to reconstruct, redatum, deghost, or simply denoise measured data. Acknowledgements Nasser would like to thank Statoil, ASA for the summer research internship opportunity at Rotvoll Research Center. References Amundsen, L., Weglein, A.B. and Reitan, A. [2013] On seismic deghosting using integral representation for the wave equation: Use of Green s functions with Neumann or Dirichlet boundary conditions. Geophysics, 78, T89 T98. Green, G. [1828] An essay on the application of mathematical analysis to the theories of electricity and magnetism. Printed for the author by T. Wheel house, 4. Ramírez, A. and Weglein, A. [2009] Green s theorem as a comprehensive framework for data reconstruction, regularization, wavefield separation, seismic interferometry, and wavelet estimation: A tutorial. Geophysics, 74(6), W35 W62. Yarman, E. and Ramírez, A. [2013] Directional wavefield decomposition. Geophysics, 78(2), WA71 WA76. Zhang, J. and Weglein, A.B. [2005] Extinction theorem deghosting method using towed streamer pressure data: analysis of the receiver array effect on deghosting and subsequent free surface multiple removal. 75th Annual International Meeting, SEG, Expanded Abstracts, 24, th EAGE Conference & Exhibition 2016