3D predictive deconvolution for wide-azimuth gathers P.Hugonnet(1), J.L.Boelle(2), P.Herrmann(1), F.Prat(1), S.Navion(1) (1) CGGVeritas, (2) Total E&P Summary Usual pre-stack predictive deconvolution solutions for multiple attenuation are either monochannel or designed for 2D gathers. These existing 1D /2D solutions are suboptimal for today s modern acquisition geometries, which allow the construction of 3D, wide azimuth pre-stack collections. We therefore present a 3D pre-stack predictive deconvolution algorithm suited to today s WAZ HD HR gathers. It targets the attenuation of the multiples (either surface or internal) in horizontally layered media, and is applied on densely sampled, wide-azimuth gathers, from either marine, land, or OBC surveys (cross-spread gathers, receiver or shot gathers, mega bin, macro bin,...). As usual with the predictive deconvolution algorithms, it is the most useful for short to medium period multiples.
Introduction Predictive deconvolution is one of the most ancient tools used to attenuate the multiples and is still extensively used today. It is based on the periodicity property of the multiples, as opposed to the assumed random nature of the reflectivity series. Its most simple monochannel form is valid only for 1D (horizontally layered) media, and for zero offset traces or for plane-waves (hence the tau-p deconvolution). In practice however, it is well known that this 1D(operator)/1D(medium) deconvolution is pretty robust with respect to departures from these 1D assumptions. In 1995, Taner et al. [5] described a 2D/1D predictive deconvolution using 2D (t,x) operators, applied to 2D pre-stack shot gathers, thus correctly (and simultaneously) taking into account all the offsets. The 1D medium assumption was still made, though, making this algorithm theoretically equivalent to a tau-p deconvolution, but without the need of tau-p transforms and with more compact operators. In 2001, Bierstaker [2] extended it to the 2D/2D case : the principle remains the same, but 2D integral operators are applied to the whole pre-stack volume (time,cmp,offset) of a 2D line, and take into account the lateral variations of any 2D medium. This approach is surface-consistent. In 2005, Hugonnet and Tichatschke [3] extended the 2D/2D algorithm to the OBC case, and also showed that the internal multiples could also be attenuated up to a certain point. The 2D approaches are robust with respect to the departures from the 1D or 2D medium assumptions, but they are not adapted for the deconvolution of the wide-azimuth gathers we have to deal with today with modern acquisition geometries. That s why we have developed the 3D/1D version: 3D deconvolution operators for 1D media. 3D/1D predictive deconvolution The general formulation of the predictive deconvolution can be written, independently of the number of dimensions, as [3]: 2 min P D F D D F F D F (E1) F D are the data, F is the prediction operator, denotes the application of the operator to the data. F D models the source-side peglegs, D F models the receiver-side peglegs, and F D F is a second order term [1] (which can be neglected if the generators are not too strong). The result of the minimisation yields the deconvolved (primary) data P. Note that F is indeed the primary impulse response of the subsurface (i.e. the Green s function without a free surface), for a dipole source. In 1D/1D, D and F are single traces, and denotes the 1D convolution along the time axis. In 3D/1D, D and F are 3D (t,x,y) volumes, and denotes the 3D convolution (*) along time and space. D(t,x,y) is a pre-stack wide-azimuth gather, x and y being the inline and crossline offsets. Depending on the acquisition geometry, the gather can be a shot gather, a receiver gather, or a crossspread gather. Usually the most regularly sampled and least aliased collections are chosen. CMP gathers usually do not meet these requirements, but in the case of moderately structured geology, macro CMP gathers can be used. F(t,x,y) has exactly the same geometry as D(t,x,y). However, like any predictive deconvolution operator, it has to be lagged in time and to have limited lengths in time and space to make the whole process stable. The 3D convolution is commutative, so the first two terms in (E1) are equal. This results from the 1D assumption, which makes the source-side and receiver-side peglegs fully identical in any considered gather. The equation (E1) is hence turned into: 2 2 (E2) min P D 2 F * D F * D It can be solved by a conjugate gradient algorithm. F
For surface multiples attenuation, F(t,x,y) represents the impulse response of the generators. It has been shown [3] [6] that (E1) was also an exact formulation for the internal multiple attenuation under the 1D assumption : F(t,x,y) represents then virtual generators. While theoretically equivalent to a tau-px-py deconvolution, the 3D/1D deconvolution has the advantage of avoiding the forward and reverse tau-px-py transforms (which can be painful). Moreover, the by-design compact nature of the 3D deconvolution operator results in a more stable process. Synthetic data example We consider a synthetic (ray traced) 3D shot gather made of 161x161 traces. Both the Inline and Xline offsets ranges are [-2000;+2000]m, with 25m intervals. 3 Xline sections are displayed (fig. 1a) at y=0,+500,+1000m. There are 2 primaries at t=200ms and t=720ms, and all the surface multiples generated by the first primary. The 3D autocorrelation function of the gather is a also a (t,x,y) volume, and 3 Xline sections are displayed (fig.2a) at y=0,+250,+500m. It is dominated by an event at 200ms, which is the period of the multiples. The kinematics of this event is exactly the same than the generator s one. A deconvolution operator F(t,x,y) is designed and estimated around the 200ms period (fig.2b). The active part is [150;250]ms at zero offset. Again, the event present in the operator has the same kinematics as the generator. The operator is applied to the data (equ. (E2)) to get the multiples (fig.1b) and the primaries (fig.1c). The 3D autocorrelation function computed on the estimated primaries (fig.2c) has been blanked, except the central peak and an event at 520ms (which corresponds to the intercorrelation between the two primaries). Real data example The data in the next example are part of a wide-azimuth OBC survey. The nominal shot grid is 50x50m, and has been interpolated to 25x25m. The maximum Inline offset is 6000m, and the maximum crossline offset is 3750m. We use the up-going wavefield after PZ summation (deghosting) [4]. Fig.3a shows 3 shot lines extracted from one receiver gather (crossline offsets y=0,+1000,+2000m). A stack section (2D stack along an Inline above the displayed receiver) is displayed on fig.6a. The (t,x,y) deconvolution is applied in the receiver gather domain. The autocorrelation function (fig.3b, displayed for crossline offsets y=0,+500,+1000m) is dominated by an event at 170ms, which corresponds to the TWT-time of the water bottom. The deconvolution operator (fig.5) is designed and estimated around this period (active part [120;220]ms at zero-offset). It has the kinematics of the water-bottom, as if it was recorded by a surface acquisition. The stack of the operators along the line nicely reproduce the water bottom shape, as previously picked (fig. 4; note that no other significant generators are present in the operator stack apart from the water bottom itself). The gather and stack after deconvolution are displayed on fig.3c and 6b. The 3D autocorrelation function of the deconvolved gather has been cleaned up, as expected (fig.3d). Conclusions We have presented a 3D predictive deconvolution algorithm, which can be applied to any kind of dense and wide-azimuth gathers, and which is effective for the attenuation of the multiples (surface or internal) on all the offsets in horizontally layered media. Compared to the more conventional deconvolution in the tau-px-py domain we recommend the use of the more robust 3D pre-stack (t,x,y) deconvolution.
Acknowledgements The authors thank Total E&P and its partners on PL 040/043 for the permission to show the real data. References [1] M. M. Backus, 1959, Water Reverberations their Nature and Elimination : Geophysics, 24, no. 2, 233-261 [2] J. Biersteker, 2001, MAGIC: Shell's surface multiple attenuation technique : SEG, Expanded Abstracts, 20, no. 1, 1301-1304 [3] P. Hugonnet and C. Tichatschke, 2005, 2D Deconvolution for OBC Data and for Internal Multiple Attenuation Part 1 Theory : EAGE, Extended Abstracts, A026 [4] R. Soubaras, 1996, Ocean bottom hydrophone and geophone processing : SEG, Expanded Abstracts, 15, no. 1, 24-27 [5] M. T. Taner, R. F. O'Doherty and F. Koehler, 1995, Long period multiple suppression by predictive deconvolution in the x-t domain : Geophysical Prospecting, 43, no. 4, 433-468, [6] C. Tichatschke and P. Hugonnet, 2005, 2D Deconvolution for OBC Data and for Internal Multiple Attenuation Part 2 Practical Aspects and Examples : EAGE, Extended Abstracts, E024 (a) (b) (c) Figure 1 : (a) 3 Inline sections (y=0,+1000,+2000m) of a synthetic shot gather; (b) estimated multiples; (c) estimated primaries (a) (b) (c) Figure 2 : (a) 3D autocorrelation function of the input gather fig.1a; (b) 3D deconvolution operator; (c) 3D autocorrelation function of the deconvolved gather fig.1c Autocorrelations and operator plotted at y=0,+500,+1000m
(a) (b) (c) (d) Figure 3 : (a) 3D receiver gather from a wide-azimuth OBC survey (3 shot lines y=0,+1000,+2000m) and (b) its 3D autocorrelation function ( y=0,+500,+1000m). (c) the primaries estimated from the gather fig.3a and (d) their 3D autocorrelation function Figure 4 : stack of the deconvolution operators (exaggerated vertical scale), compared (on the top) with a previous picking of the water bottom. Note the 90 phase rotation of the signal (which is predicted by the theory). (a) Figure 5 : the deconvolution operator (y=0,+500,+1000m) (b) Figure 6 : (a) stack section before deconvolution (b) after deconvolution