Pre-stack deghosting for variable-depth streamer data. R. Soubaras* (CGGVeritas) Summary

Transcription

1 Pre-stack deghosting for variable-depth streamer data R. Soubaras* (CGGVeritas) Summary Variable-depth streamer acquisition is an acquisition technique aiming at achieving the best possible signal-to-noise ratio at low frequencies by towing the streamer very deeply, but by using a depth profile varying with offset in order not to limit the high frequency bandwidth. Previous papers have shown how a joint deconvolution allows the post-stack deghosting of such variable-depth streamer acquisitions, and the purpose of this paper is to show how this can be generalized to a multichannel joint deconvolution that allows pre-stack though post imaging deghosting of such acquisitions. After computing migrated gathers as well as mirror migrated gathers, a deghosted gather is computed through joint deconvolution by assuming a parametrical form in offset for the events. This method preserves the AVO behaviour. Synthetic and real data examples are shown.

2 Introduction For a conventional streamer acquisition, receiver deghosting is done at preprocessing stage by including the zero-offset receiver ghost into the far-field signature. z being the depth of the streamer and c the water velocity, the receiver ghost: G( f) = 1 e 2 jπ2 z f/c (1) This vertical propagation approach can be refined by taking into account the angle of propagation of the wavefield: G( f,k x,k y ) = 1 e 2 jπ2 z f 2 /c 2 k 2 x k2 y (2) However, while it is easy to take into account non vertical propagation in the inline direction x (parallel to the streamers), it is much more difficult to take into account the crossline direction y, due to the coarse y sampling that a multi-streamer acquisition performs. For a variable depth streamer acquisition (Soubaras 2010, Soubaras and Dowle 2010), the receiver deghosting in performed after the imaging stage, which allows a real 3D deghosting as well as an optimal signal-to-noise deghosted output. The deghosting is a dual input process, the two inputs being the migration and the mirror migration. Two cases can be considered: the post-stack deghosting, where the inputs are the stacked images, and the pre-stack case, where the inputs are the two common image gathers. Post-stack deghosting by joint deconvolution The normal migration stacks coherently the primary events, the ghosts events being imperfectly stacked in such a way that the migration has a residual ghost wavelet that is causal. The mirror migration stacks coherently the ghosts events with their polarity reversed, in such a way that the primary events are imperfectly stacked in such a way that the mirror migration has a residual ghost wavelet that is anticausal. The proposed deghosting method uses this "binocular vision" of two images of the same reflectivity with a different viewpoint to extract the true amplitude deghosted migration, that would have been obtained by a conventional migration if the water-surface was non-reflective. Considering d 1 (t) and d 2 (t) two given signals, find a signal r(t), a normalized minimum phase operator of given length g min (t) (normalized meaning g min (0) = 1), and a maximum phase normalized operator of given length such as: d 1 (t) = g min (t) r(t) (3) d 2 (t) = g max (t) r(t) Although the joint deconvolution problem as stated by equation (3) looks like the conventional deconvolution model, it has totally different mathematical properties. It is a well-posed problem, which means it has a unique solution, even when the minimum phase and maximum phase properties are marginally respected (meaning the operators have perfect spectral notches) (Soubaras, 2010). A mathematical proof can be sketched as following: suppose we have two solutions (g min,g max,r) and (G min,g max,r) of problem (3), we therefore have: G min (t) R(t) = g min (t) r(t) G max (t) R(t) = g max (t) r(t) (4) therefore: and: R(t) r 1 (t) = G 1 min (t) g min(t) R(t) r 1 (t) = G 1 max(t) g max (t) G 1 min (t) g min(t) = G 1 max(t) g max (t) (6) The l.h.s. of equation (6) is causal because a minimum phase signal is causal and has a causal inverse. The r.h.s is anticausal because a maximum phase signal is anticausal and has an anticausal inverse. The (5)

3 only way a signal can be both causal and anticausal is by having a non-zero value only at lag 0, therefore being proportional to a Dirac function λ δ(t). Because of the normalizations, λ must be 1. Therefore: G min (t) = g min (t) G max (t) = g max (t) R(t) = r(t) (7) which proves that the joint deconvolution has an unique solution. The joint deconvolution is used to deghost post-stack by using, in overlapping space-time windows, the migration image as d 1 (t) and the mirror migration image as d 2 (t), the deghosted migration being recovered as r(t). Pre-stack deghosting by multichannel joint deconvolution The joint deconvolution allows to deghost after stack a variable depth streamer acquisition. The deghosting can also be performed before stack, on common image gathers (that is before stack but after imaging), by using a multichannel joint deconvolution model. The multichannel joint deconvolution considers the data d 1 (t,h), which is the pre-stack gather after normal imaging and d 2 (t,h) the pre-stack gather after mirror imaging. h is the offset dimension or any other dimension of a pre-stack gather such as angle. Noting the reflectivity gather by r(t,h) we can write the following multichannel joint deconvolution model, by introducing normalized minimum and maximum phase operators g min (t,h) and g max (t,h) d 1 (t,h) = g min (t,h) r(t,h) d 2 (t,h) = g max (t,h) r(t,h) (8) Written as this, the multichannel deconvolution problem is already well-posed, as it consists of N h separate joint deconvolutions. However, solving the problem this way results in amplification of the noise due to each ghosts notch. In order to take advantage of the notch diversity, we must couple these otherwise independant problems in h. This can be done by imposing to the reflectivity gather r(t,h) a parametrical variation in h: r(t,h) = p i=0 a i (t)t i (h) (9) where T i (h) is typically a set of orthogonal polynomials. Inverting for a i (t), g min (t,h) and g max (t,h) gives a reflectivity model r(t,h) that can be considered as the deghosted gather. It is however better to use r(t,h), g min (t,h) and g max (t,h) to produce a ghost model and a mirror ghost model: G 1 (t,h) = g min (t,h) r(t,h) r(t,h) G 2 (t,h) = g max (t,h) r(t,h) r(t,h) (10) which can be subtracted to the gather d 1 (t,h) and mirror gather d 2 (t,h). The two deghosted images can be averaged, combining the pre-stack image produced by the direct arrival and the pre-stack image produced by the arrival which is reflected by the water-surface. Synthetic data example Figure 1-a and 1-b shows a synthetic gather and mirror gather. These have been computed by modeling variable depth streamer shot records using two events at 3.5 and 3.6 seconds in a given velocity model and performing the migration and mirror migration with a non-exact velocity model. Random noise has also been added. Because the reflectivity has two events, it cannot be considered as white. On the gather, the two overcorrected black events corresponding to the direct arrivals can be seen, as well as the 2 white events corresponding to the ghosts. On the mirror gather, we can check that the two black events have exactly

4 the same moveout curve as the two black events on the normal gather, the perturbing white events being precursors. This is due to the fact that on the mirror gather the black events are the ghosts and the white ones direct arrivals. The important fact to notice is that, when the velocity is wrong, the residual moveout of the primary arrival on the normal gather is exactly the one of the ghost arrival on the mirror gather. Figure 1-c and 1-d show the ghost model and mirror ghost model, and Figure 1-e the deghosted output. Figure 1-f is the reference gather that has been computed by modeling the data with no reflecting watersurface. No random noise has been added to this reference gather. The very good match between the deghosted gather 1-e and the reference gather 1-f can be checked. The red curve on Figure 1-e is the RMO curve of the event at 3.5 seconds, along which we can plot in Figure 1-g the AVO behaviour of both the deghosted (in black) and reference gather (in red). The deghosted gather reproduces very well the AVO response of the reference gather. Real data example A variable-depth data acquisition was performed in the Gulf of Mexico. The bandwidth achieved on this dataset was six octaves [2.5 Hz Hz]. Figure 2 shows the migration and Figure 3 the mirror migration. Pre-stack deghosting was done on this dataset and Figure 4 shows the stack of the prestack deghosted gathers. The effect of the pre-stack deghosting can be seen on the interleaved gathers. Amplitude variations and residual moveout have been preserved on the deghosted gathers. Conclusion We have described how the joint deconvolution can be used to perform the pre-stack receiver deghosting of variable-depth streamer data. The method uses the common image gathers after migration and mirror migration. This deghosting method is true amplitude, preserves AVO behaviour and residual moveout curves. Tests on synthetic data has shown that it can recover the AVO curve of the reference unghosted data. References Soubaras, R. [2010] Deghosting by joint deconvolution of a migration and a mirror migration. 80th SEG Annual Meeting, Expanded Abstracts 29, Soubaras, R. and Dowle, R. [2010] Variable-depth streamer - a broadband marine solution. First Break, 28(12), s a) b) c) d) e) f) g) 4.0s km Fig. 1: Synthetic data: a) gather, b) mirror gather, c) ghost model, d) mirror ghost model, e) deghosted gather, f) reference gather, g) AVO for the RMO curve of f) (black=deghosted, red=reference)

5 Fig. 2: Migration and gathers. Fig. 3: Mirror migration and gathers. Fig. 4: Deghosted migration and gathers.