Pressure and shear wave separation of ocean bottom seismic data

Transcription

1 Pressure and shear wave separation of ocean bottom seismic data Ohad Barak ABSTRACT PZ summation is a common method for separating the upgoing wavefield from the downgoing wavefield in data acquired by four-component ocean-bottom seismic surveys. It assumes that the vertical geophone component records mostly pressure waves. If this assumption is not satisfied, non-pressure wave energy (such as shear waves) will be introduced as pressure waves into the receiver wavefield, which may generate artifacts in the migration image. I formulate a medium independent inversion designed to reconstruct the wavefields in the vicinity of the receivers. I then use separation operators which require the wavefield s spatial derivatives to separate P-waves from S-waves. PZ summation is applied to the separated P-wave data to determine propagation direction. INTRODUCTION Ocean-bottom node (OBN) surveys record four-component data: hydrophone pressure data, vertical particle velocity, and two perpendicular horizontal particle velocities. The receivers are located on the sea bed. The source is typically an airgun fired from near the sea surface. One problem of OBN is the acquisition of water surface multiples. These multiples are waves which have reflected off the subsurface, and then bounce back down off the water surface. Another problem is that unlike streamer marine data, shear-wave and Scholte-wave data are recorded by the geophones in addition to compressional waves, since they are coupled to the sea bed. A current implementation for processing of OBN data is to regard the hydrophone component and the vertical particle velocity component as containing P-waves only. PZ summation is applied to these components. It is a weighted summation of the hydrophone and the vertical geophone components, designed to determine the vertical propagation direction of the recorded arrivals (Barr and Sanders, 1989). The basic principle of PZ summation is that the hydrophone and the vertical geophone data will have opposite polarity if the energy they recorded was propagating downward. If that energy was propagating upward, they will have identical polarity. This upgoing/downgoing wavefield direction determination method has been used to create separate images from the primary arrivals and the mirror arrivals (the ones reflecting off the sea-surface) at the nodes on the sea bottom (Ronen et al., 2005; Dash et al.,

2 Barak 2 P/S separation 2009; Wong et al., 2009). This method is known as mirror imaging. Separating the imaging process removes artifacts resulting from cross-talk between the upgoing and downgoing wavefields. Additionally, mirror imaging increases the effective aperture of the survey, and provides more information than the upgoing image alone. However, because shear waves do not propagate in the water column, the inevitable result of PZ summation will be to introduce any shear-wave energy recorded by the vertical geophone into the summation result. This data will be construed as a P- wave energy recording. When these data are then migrated using acoustic wavefield propagation, they may generate artifacts in the resulting image. A simple sketch of this effect is shown in Figure 1 Figure 1: Simplified sketch of PZ summation. Top: For acoustic data, the summation of the hydrophone data and the vertical geophone data will result in the downgoing energy being eliminated, leaving only upgoing pressure data. Bottom: If the data contains shear waves, they will be recorded only by the geophone, and the PZ summation will mistakingly insert them into the result as upgoing pressure data. [NR] Downgoing (P) Upgoing (P) Hydrophone Downgoing (P) Upgoing (P) Hydrophone (P) (P) Geophone Ver2cal (P) (P) (S) Geophone Ver2cal Upgoing Hydrophone Upgoing Hydrophone A particular problem of OBN data (as compared to streamer marine data) is Vz noise (Paffenholz et al., 2006), which results from surface waves propagating along the water-solid boundary. They are considered to be the result of shear waves incident on an irregular sea bed topography, or on near surface scatterers. Additionally, these surface waves can be generated by upgoing shear-waves incident on the sea bed at large angles (beyond a certain critical angle). Since surface waves appear mainly on the geophone components, they will also contribute to an incorrect PZ summation result. Alternate separation methods Dankbaar (1985); Wapenaar et al. (1990); Amundsen (1993) have shown how to do a decomposition of P and S-waves of OBS data, and carry out a better separation between the upgoing and downgoing energy. Their methods require a knowledge of medium parameters in the vicinity of the receivers. Schalkwijk et al. (2003) implement adaptive decomposition, in order to account for the unknown near-surface medium

3 Barak 3 P/S separation parameters on which the separation depends. This decompostion requires the user to manually identify the downgoing waves in each receiver gather, and select a window that contains upgoing waves only. An itterative inversion process is then applied using the windowed upgoing arrivals. From the inversion results, both the medium parameters and the calibration between the different sensors can be deduced. These methods however do not address the additional surface-wave noise which may appear on the geophone component. Any interface waves must be removed before processing. Zhou et al. (2011) propose a novel method of extracting the P-wave data from the geophone components. It involves a complex wavelet transform of the hydrophone and the vertical geophone data. A matching of the geophone data to the predominantly P-wave energy hydrophone data is done in the complex wavelet domain. Afterward, the inverse transform is applied to the result. The end product is a substantially P-only vertical geophone component. This component can then be combined with the hydrophone to separate the upgoing from the downgoing P-wave. Outline of proposed inversion For an isotropic medium there are known P/S separation operators, which utilize the spatial derivatives of the particle displacement values. However, for most acquisition methods the wavefield is spatially sampled horizontally, but not vertically. The inversion attempts to recreate the wavefield at an additional depth level very near the receivers, in order to acquire the vertical derivative needed for P/S separation. The recorded data is a function of the medium parameters and the acquisition geometry. However, a recorded dataset could conceptually have been the result of other medium parameters, and other acquisition geometries than the real ones. The question the proposed inversion attempts to answer is: Given a set of virtual medium parameters and a certain acquisition geometry of virtual sources, what are the virtual sources which, upon injection into the medium, create the observed data at the receivers level. The virtual sources are the model to invert for, but calculating them is not the purpose of the method. It is only necessary that they reproduce the observed data at the receiver level. That means that the wavefield which they generate is equal to the true wavefield observed at the receiver level. If the medium parameters used are a reasonable estimate of the true medium parameters, then the wavefield one depth level below the receivers will likewise be a close representation of the true wavefield. Since now the wavefield has been recreated at additional depths, the spatial P/S separation operators can be applied. The elements of the inversion are: 1. Observed data: The original displacement components recorded by the geo-

4 Barak 4 P/S separation phones. 2. Model: A virtual source gather, injected at some location into a homogeneous elastic medium. 3. Calculated data: The recorded displacements at the receiver level, as a result of the injection of the virtual sources. 4. Desired model: The virtual sources which generate recorded data equal to the observed data. 5. What we actually want: The wavefield displacement values both at the receiver level AND one depth level below them. The inversion is applied to an OBN dataset in order to separate pressure and shear displacements on the geophone components. The standard PZ summation can then be applied to the hydrophone and the P-wave vertical geophone in order to separate upgoing data from downgoing data. These data can then be migrated separately to produce the primary and the mirror image. The expectation is that the resulting image will contain fewer artifacts, as a result of the shear wave removal. PZ summation THEORY PZ summation involves summing the pressure data recorded by the hydrophone data with the vertical particle velocity data recorded by the geophone, with some scaling factor: U(z r ) = 1 2 [P (z r) βv z (z r )], D(z r ) = 1 2 [P (z r) + βv z (z r )], (1) where P is the pressure data, V z is the vertical particle velocity, U is the upgoing data, D is the downgoing data and z r is the receiver depth. β is a scaling factor, which can be defined in several ways. Theoretically, β is the acoustic impedance at the wave s incidence angle. Therefore it is different for the upgoing wavefield and the downgoing wavefield. In practice, as a result of frequency dependent instrument response, and as a result of the two different impedances above and below the receivers, β is frequency dependent and wave-mode dependent.

5 Barak 5 P/S separation Wave mode separation and imaging The Helmholtz separation operator is based on the assumption that any isotropic vector field can be described as a combination of a scalar and vector potential fields: u = Φ + Ψ, (2) where Φ is the scalar potential field and Ψ is the vector potential. u is the elastic displacement vector wavefield. The scalar potential generates pressure waves, and the vector potential generates shear waves. Therefore, the Helmholtz method of separating the P-wave amplitude from the S-wave amplitude is to apply a divergence operator and a curl operator to the displacement field: P = u = 2 Φ; (3) S = u = 2 Ψ. (4) Equations 3 and 4 apply only for an isotropic medium. Dellinger and Etgen (1990) extend these operators for an anisotropic medium. The Helmholtz separation operator is useful for distinguishing between P and S- wave amplitudes, but it is not reversible. The derivation of a reversible P-wave and S-wave displacement decomposition by Zhang and McMechan (2010) is done in the wavenumber domain. For an isotropic medium, the linear equation system they arrive at is: k ũ P = 0, (5) k ũ = k ũ S, (6) k ũ S = 0, (7) and k ũ = k ũ P, (8) where ũ is the 3D spatial fourier transform of the displacement field ũ = (ũ x, ũ y, ũ z ). ũ P and ũ S are the unknown P and S displacements. k = (k x, k y, k z ) is the wavenumber vector that describes the particle displacement direction. k i = ω V i, where V i is the phase velocity in the i direction, and ω is angular frequency. The solutions to equations 5-8 are the decomposed pressure and shear displacements. This decomposition is reversible, since u i = u P i + u S i.

6 Barak 6 P/S separation Inverting for the virtual-source model The isotropic elastic wave equation relates displacements to accelerations via two elastic constants - the Lamé parameters λ and µ ((λ + µ) u) + (µ u) + f = ρü. (9) where u are the particle displacements in each dimension, f is the force function and ρ is medium density. For a homogeneous medium, and using a Green s function to describe the energy propagation between any two locations x = (x, y, z) and y = (x, y, z), the equation takes the form: ( (λ + µ) + µ 2 + ρω 2) G (x, y, ω) = δ (x y) (10) The forward elastic propagation operator injects a virtual-source model into some location in the medium, and records the resulting wavefield at some other location: d (x, ω; x s ) = y m (y, ω; x s ) G (y, x, ω), (11) where m is the model of injected sources at location y in the medium, and d are the recorded displacement fields u at location x in the medium. ω is angular frequency and x s is the shot gather. In vector notation, this is expressed as d = Fm, (12) where F is the forward elastic propagation operator. In a two-dimensional medium it is expressed as: [ 1 ρ F = x (λ + 2µ) x + 1 ρ zµ z t 2 1 ρ xλ z + 1 ρ zµ x 1 ρ zλ x ρ xµ z ρ z (λ + 2µ) z + 1 ρ xµ x t 2 ] (13) The adjoint operator injects the data from the same recording locations, and records the resulting wavefield at the model injection points: m (y, ω; x s ) = x d (x, ω; x s ) G (x, y, ω), (14) which in vector notation is

7 Barak 7 P/S separation where F is the adjoint operator. m = F d, (15) The inversion is done by least-square fitting of modeled geophone data to the recorded geophone data. It is not possible to recreate the entire original wavefield that was recorded by the geophones without accurate knowledge of the acquisition geometry and the medium parameters. However, given a particular acquisition geometry, and some reasonable medium parameters, it is possible to recreate the original wavefield in the vicinity of the geophones. This wavefield will form as a result of the injection of the virtual sources, and will become more similar to the originally recorded wavefield at the receiver locations as it propagates toward them. The inversion starts from a zero-value initial model of displacement sources m 0 = 0. This initial model is injected as a force function, at locations x in the medium, and the energy is propagated with the forward elastic operator: d = Fm 0. (16) The residual that must be minimized is the difference between the calculated data d, and the observed data d obs. The objective function is: The model gradient is: J(m) = d d obs 2 2 = Fm d obs 2 2, (17) m = J ( ) T r m = r = F r, (18) m where F is the adjoint elastic propagation operator. The data gradient is the forward operator applied to the model gradient: r = F m. (19) The model gradient and the data residual are updated using an itterative minimization scheme (Claerbout and Fomel, 2011), with the conjugate direction solver. The end result of the inversion is the model m which, when injected at locations x in a homogeneous medium with the parameters used in the inversion, will generate recorded data at locations y that are equal to the observed data. Even if the medium parameters that were used to generate the wavefield were not a good approximation to the true parameters, this difference will be minor with regard to the wavefield at a close proximity to the receivers. Therefore, if the data residual is zero, we can assume that the wavefield near the receivers has likewise been reliably recreated.

8 Barak 8 P/S separation The displacement field values are calculated for all time steps by running the forward propagation operator using the final virtual-source model as an input: F [ 1 m ] ρ x 1m = ρ z [ ux u z ] (20) The separation operators in equations 3-4 and 5-8 are applied to the displacements fields (u x, u z ) at the receiver locations, in order to separate P-waves from S-waves. CURRENT RESULTS Synthetic data were modeled using a medium that contained a water column, overlying two horizontal layers with different elastic impedances. The upper boundary was free, so that reflections would occur as they do off the sea surface. The source was at the sea surface, and the receivers were on the sea bed. This geometry simulates OBS acquisition. The observed synthetic data is shown in Figures 2(a)-2(d). Note that the direct arrival has been muted. Therefore, the first arrival in each panel of the figures is an upgoing P-wave. The arrival s wave mode and propagation direction are designated in the figures. The shear-wave in Figure 3(d) is not a representation of actual field data, but only serves to assist in the interpretation of the results. The medium parameters which were used to generate the OBS synthetic data were: 1. P-wave velocity: v p1 = 1500 m s, v p2 = 1700 m s, v p3 = 2000 m s 2. S-wave velocity: v s1 = 0 m s, v s2 = 500 m s, v s3 = 800 m s 3. Density: ρ 1 = gr/cm 3, ρ 2 = 1.7 gr/cm 3, ρ 3 = 1.6 gr/cm 3 The inversion was run with virtual sources located below the receivers, in a homogeneous medium with the following parameters: 1. P-wave velocity: v p = 1500 m s 2. S-wave velocity: v s = 600 m s 3. Density: ρ 1 = gr/cm 3 The virtual source array had the same length and number of injection points as the receiver array. Figures 3(a)-3(d) are the results of forward modeling of virtual-source models, which were produced after 200 iterations. Comparing Figures 2(a) and 2(b) to Figures

9 Barak 9 P/S separation (a) (b) (c) (d) Figure 2: Observed synthetic OBS data, generated by elastic modeling. A. Vertical particle velocities. B. Horizontal particle velocities. C. Pressure wave. D. Shear wave. [ER]

10 Barak 10 P/S separation 3(a) and 3(b), the accurate reconstruction of the vertical and horizontal particle velocities is evident. However, some of the phases of the reconstructed hydrophone in Figure 3(c), which represent the P-waves, are opposite to the phases of the observed hydrophone in Figure 2(c). The arrivals with opposite phases correspond to the downgoing waves of the synthetic data. Additionally, the reconstructed shear-waves in Figure 3(d) contain not only the shear arrival of the synthetic data, but also the downgoing P-arrivals. The inversion has correctly reconstructed the displacement values at the receiver locations. Also, the spatial derivatives of the displacements have been correctly reconstructed for the upgoing waves, as evidenced by the fact that inverted upgoing P-wave has the right phase. However, for the downgoing waves, the inversion has yielded a combination of upgoing P-waves and upgoing S-waves. The reasons for this are twofold: 1. There is no constraint on the displacement values over or under the receivers, and therefore no constraint on their spatial derivatives. These spatial derivatives determine the wave-mode. 2. The virtual sources were placed below the receivers, and consequently only upgoing wavenumbers could be modeled. As a result, the inversion has used a combination of upgoing P-waves and upgoing S-waves to explain the downgoing P-waves in the synthetic data. This is a nullspace problem. This effect can be seen in the inverted P and S particle velocities as compared to the observed particle velocities, in Figures 4(a)-4(f). The upgoing arrival s P particle velocities (Figures 4(c) and 4(d) at t = 0.3s) are correct, as evidenced by their similarity to the observed particle velocities. However, notice that for all the other downgoing arrivals, the horizontal P particle velocity (Figure 4(d)) has an opposite phase to the true horizontal particle velocity (Figure 4(b)). Additionally, the downgoing P arrivals have been reconstructed using shearwave particle velocities, as seen in Figures 4(e) and 4(f). Note that the shear-wave itself has been reconstructed correctly, owing to the fact that it is an upgoing wave. Even under these conditions, the inversion result can still be used to separate upgoing P from downgoing P. Since the downgoing waves were reconstructed with an opposite phase, the summation of the inverted hydrophone with the true hydrophone data will result in mostly upgoing P data. The difference will yield the downgoing data. This is shown in Figures 5(a)-5(d). The separation is not perfect, since some of the P-wave energy has leaked into the shear-wave energy, therefore impacting the P-wave amplitude. However, for the lower wavenumbers where leakage is less pronounced, this up/down separation does a reasonable job.

11 Barak 11 P/S separation (a) (b) (c) (d) Figure 3: Inverted OBS data. A. Vertical displacements. B. Horizontal displacements. C. Pressure wave. D. Shear wave. The downgoing P-waves have been incorrectly reconstructed as upgoing P-waves and upgoing S-waves. The upgoing S-wave itself has been reconstructed well. [ER]

12 Barak 12 P/S separation (a) (b) (c) (d) (e) (f) Figure 4: Top: Observed vertical and horizontal particle velocities. Center: Decomposed P-wave particle velocities generated by injecting the inverted virtual source models. Bottom: Decomposed S-wave particle velocities generated by injecting the inverted virtual source models. [ER]

13 Barak 13 P/S separation (a) (b) (c) (d) Figure 5: A. Synthetic hydrophone data. B. Inverted hydrophone data. C. Synthetic+inverted hydrophone, resulting in mostly upgoing P-wave data. D. Syntheticinverted hydrophone, resulting in mostly downgoing P-wave data. [ER]

14 Barak 14 P/S separation DISCUSSION AND CONCLUSION The inversion reliably recreates the observed geophone data at the receiver locations. The dispositions of the virtual sources in relation to the receivers is a major factor in determining which wavenumbers can be modeled by the inversion. If some wavenumbers which exist in the field data cannot be modeled by a particular virtual source array, then the inversion will strive to find another explanation for their existence. This explanation will utilize the null-space, and produce undesirable results. The example I show above indicates that for a simple virtual source arrangement that can produce only upgoing waves, all downgoing P-wave displacements are explained by a combination of upgoing P-wave displacements and upgoing S-wave displacements. This is related to the fact that though the particle motion at the receiver level is well-constrained by the objective function, the vertical derivative of particle motion is not. Therefore, additional constraints must be used in the objective function, which will decrease the null-space. My conclusion is that a correct combination of constraints and virtual source locations will enable the inversion to recreate all wavenumbers correctly, and thus all wavemodes. The P and S separation operator uses a second order in space finite-difference approximation to the first derivative. This low-order approximation can generate shear-wave artifacts on the separated P-wave data, and vice versa. Improving the approximation order may reduce the artifacts. However, since a higher order approximation invariably uses a longer stencil, the effect of the recreated wavefield at greater distances from the receivers will come into play. Since we can only assume that the wavefield has been reliably recreated near the receivers, a higher order approximation may not be desirable. REFERENCES Amundsen, L., 1993, Wavenumber-based filtering of marine point-source data: Geophysics, 58, Barr, F. J. and J. I. Sanders, 1989, Attenuation of water-column reverberations using pressure and velocity detectors in a water-bottom cable: SEG expanded abstracts, 8, Claerbout, J. F. and S. Fomel, eds., 2011, Image estimation by example: Geophysical soundings image construction. Dankbaar, J. W. M., 1985, Separation of p- and s-waves: Geophysical Prospecting, 33, Dash, R., G. Spence, R. Hyndman, S. Grion, Y. Wang, and S. Ronen, 2009, Wide-area imaging from obs multiples: Geophysics, 74, Q41 Q47. Dellinger, J. and J. Etgen, 1990, Wave-field separation in two-dimensional anisotropic media: Geophysics, 55,

15 Barak 15 P/S separation Paffenholz, J., P. Docherty, R. Shurtleff, and D. Hays, 2006, Shear wave noise on obs vz data - part ii: Elastic modeling of scatterers in the seabed: EAGE extended abstracts, 68th conference and exhibition, A072. Ronen, S., L. Comeaux, and X. Miao, 2005, Imaging downgoing waves from ocean bottom stations: SEG expanded abstracts, 24, Schalkwijk, K. M., C. P. A. Wapenaar, and D. J. Verschuur, 2003, Adaptive decomposition of multicomponent ocean-bottom seismic data into downgoing and upgoing p- and s-waves: Geophysics, 68, Wapenaar, C. P. A., P. Herrmann, D. J. Verschuur, and A. J. Berkhout, 1990, Decomposition of multicomponent seismic data into primary p- and s-wave responses: Geophysical Prospecting, 38, Wong, M., B. L. Biondi, and S. Ronen, 2009, Inversion of up and down going signal for ocean bottom data: SEP-Report, 138, Zhang, Q. and G. McMechan, 2010, 2d and 3d elastic wavefield vector decomposition in the wavenumber domain for vti media: Geophysics, 75, D13 D26. Zhou, Y., C. Kumar, and I. Ahmed, 2011, Ocean bottom seismic noise attenuation using local attribute matching filter: SEG expanded abstracts, 81st annual meeting,