Multiple attenuation with a modified parabolic Radon transform B. Ursin*, B. Abbad, NTNU, Trondheim, Norway, M. J. Porsani, UFBA, Salvador, Brazil.

Transcription

1 Multiple attenuation with a modiied parabolic Radon transorm B. Ursin* B. Abbad TU Trondheim orway M. J. Porsani UFBA Salvador Brazil. Copyright 009 SBG - Sociedade Brasileira de Geoísica This paper was prepared or presentation during the 11 th International Congress o the Brazilian Geophysical Society held in Salvador Brazil August Contents o this paper were reviewed by the Technical Committee o the 11 th International Congress o the Brazilian Geophysical Society and do not necessarily represent any position o the SBG its oicers or members. Electronic reproduction or storage o any part o this paper or commercial purposes without the written consent o the Brazilian Geophysical Society is prohibited. Abstract We present a ast and eicient reuency-domain implementation o a modiied parabolic Radon transorm (modiied PRT) based on a singular value decomposition with applications to multiple removal. With a change o variable the problem is transormed into a comple linear system involving a single operator ater merging the curvature-reuency parameters into a new variable. A comple singular value decomposition (SVD) can be applied to this operator and the orward modiied parabolic Radon transorm is computed or all reuency components within the signal bandwidth by means o comple back-substitution only. The standard PRT can be obtained by interpolation in the modiied transorm domain. The method is also able to resolve the multiple energy rom the primaries when they interere in a small time interval and resists well to AVO eects in the data. The proposed modiied PRT was successully applied to a deep-water seismic line at the Gul o Meico to attenuate water-bottom multiples and subseuent peg-legs originating rom multiple paths in the water column. The modiied parabolic Radon transorm The inverse parabolic Radon transorm is deined as a decomposition o seismic data in the curvature-intercept time domain using a set o curvature values k 1. For discrete seismic data D n t recorded at oset n and time t this is deined by a sum along parabolic paths in the domain (ampson 1986) Dn tmj tjn n1 (1) j1 where M represents the parabolic Radon transorm and intercept time and being the at curvature number o Radon curvatures while k denotes the number o seismic traces. Using the linearity property o the Fourier transorm euation (1) can be epressed in the reuency domain as n j i jn d m e. () j1 Euation () is a comple linear system or a given reuency. In matri notation this can be written as d L m (3) d and The elements o the matri L have the orm where m are reuency-dependent vectors. j k i kj 1 ; 1 L e k j. (4) The above ormulation o the PRT reuires solving the inverse problem in (3) or each reuency component in the signal bandwidth to compute the transorm because the comple operator L is reuency-dependent leading to a dierent matri or each spectral component. To remove the reuency-dependence in the transorm L we introduce a new variable with operator m unit. Writing () in terms o the variable leads to the ollowing system o comple euations n j j1 i jn d m e. (5) This can be written in a more compact orm as d L m (6) where the comple matri jk L deined as i kj 1 ; 1 L e k j (7) Euations (5) or (6) correspond to a modiied PRT which can be computed much aster than the standard PRT because the inverse o the matri L has to be computed only once given a data acuisition geometry n1 and a chosen discretization n j j 1. The standard PRT in the reuency domain m can be obtained by interpolation in the modiied PRT domain. The minimum-norm solution In a high-resolution modiied PRT we choose the number o -values. Then the problem is underdetermined and we choose the minimum-norm solution which can be written as 1 m M L L L d. (8) We use the SVD decomposition or the operator L

2 URSI ABBAD AD PORSAI 0 L U V (9) where diag 1 with 1 0 to obtain a stable estimate mm V U d 0 (10) where is the pseudo-inverse o with components 1 k i k k k 1. (11) 0 i k and The transormations m V m give d U d m M k M M kd k k 1. 0 k 1 (1) We see that the last components o are zero corresponding to the minimum-norm properties o the solution. m M m M Multiple attenuation We may attenuate multiple relections in the domain by partly correcting the seismic data or normal moveout eects such that the primary relections appear with less moveout than multiple relections. The main assumption on seismic data input to the transorm is that residual moveouts o both primary and multiple relections have parabolic behavior as a unction o oset (ampson 1986). The data are transormed to the domain using an inverse operator mˆ L d (13) deined in euation (10). An estimate o the primary relections are obtained by iltering the data mˆ F m ˆ where the ilter is P F 1 or 0 0 or 0 (14) where 0 is a curvature parameter that separates primaries and multiples. The primary relections are then obtained by transorming back to the domain d L F L d G d. (15) P The matri G L F L has a dimension. It behaves as a ilter applied to seismic data in the reuency-oset domain to remove multiples directly rom the data. The multiples can be also estimated using the ollowing relationship dm I G d. (16) where I is the identity matri. For a given reuency the operator G depends only on the data acuisition parameters our choice o parameters in the -domain and the choice o parameters or the pseudo-inverse. Figure 1. The modiied PRT or the gather with AVO eects included along the events. Eamples A comple synthetic eample that consists o a large number o events (0 parabolas) interering in a narrow time window is illustrated in Figure 1a. This model contains 1 primaries with negative curvatures (Figure 1b) and 8 multiples with positive curvatures (Figure 1c). We included AVO eects along the parabolic moveouts according to the unction Aya0 a1yay ; y (17) ma where y is the normalized oset a0 a1 and a are real coeicients. The modiied PRT is implemented using a ine sampling leading to an under-determined problem to obtain a dense Radon space (50- traces or 100 seismic traces) which makes demultiple more eicient. The curvature was scanned or in the range 0.30s 0.30s which gives a parameter varying in

3 MULTIPLE ATTEUATIO BY A MODIFIED PRT - - the interval [-4.5 km 4.5 km ]. We used symmetric moveout range or scan to allow the mapping o all the events in the gather. 3 Figure 3. Average absolute error on the AVO inverted curves in the eample o Figure 1 over the 1 primary events using both the modiied PRT and the IRLS methods. Figure. Theoretical and inverted AVO curves in the eample o Figure 1 or the 1 primary events using both the modiied PRT and the IRLS methods. We used the minimum-norm solution which converts the problem into a hermitian inverse problem o dimension as provided by euation (8). The eigenvalue decomposition o the minimum-norm operator gives a 4 condition number eual to or 50. To account or this the inverse operator was stabilized using 3 a truncation limit at 10 which reduces the numerical rank o L by ignoring small singular values. The mapping o the whole gather in Figures 1a in the domain is shown in Figures 1d. Primary and multiple energy are mapped in distinct areas o the domain. Each event with parabolic moveout in the time-oset domain is transormed along a radial line in the domain passing through the origin and characterized by the respective event curvature as the slope value with a reuency etent shaping the event wavelet bandwidth (spectrum). According to this events with positive curvatures 0 are mapped into the positive space while negative space contains events with 0. Figures 1e and 1 illustrate how the events in the gathers o Figures 1b and 1c are transormed in the modiied PRT domain. Since primaries have negative curvatures in the eample a mute zone was established at 0 to separate primaries rom multiples. The reconstructed gathers ater inverse transorm are shown in Figures 1g 1h and 1i they show the events in the original gather with their true curvature and amplitude characteristics. The reconstruction errors or the three gathers are depicted in Figures 1j 1k and 1l. This error is very small or the whole gather and is mainly due to the use o truncation in estimating the pseudo-inverse o L. All PRT methods suer such residual errors ater reconstruction due to the ill-conditioning o the inverse problem. The residual errors rom separating primaries and multiples show inverse signs on both gathers occurring mainly along the ais 0 thus generating horizontal low-amplitude artiacts in the iltered gathers. The amplitude eects on the 1 primary events (in Figure 1b) are plotted in Figure (curve in black). These show signiicant amplitude changes with normalized oset and several zero-crossings can be observed. The coeicients ( a0 a1 a) are given on each plot or the respective event. The event intererence between primaries and multiples and between primaries alone leads to amplitude distortion on individual events in the gather. Thus the amplitudes o the events being interered add to each other within the crossing area hence altering the AVO curve and making the amplitude inversion inaccurate. The grey curves in Figure show the practical amplitude unctions or each event o the primaries that accounts or the intererence with the other primaries. Outside the intererence zone the theoretical and practical amplitude curves are eual. The recovered amplitude unctions o the primaries ater demultiple are also drawn in Figure (curves in blue). Schonewille and Zwartjes (001) proposed a sparsenessconstrained reuency-domain algorithm resisting well to AVO eects where the weighting matri is obtained by averaging the model within the signal bandwidth at the previous iteration to constrain the solution at the current iteration. Their method can be implemented using the iteratively reweighted least-suares (IRLS) algorithm combined with a comple version o the LSQR solver. Both algorithms give comparable results but the modiied PRT algorithm has the advantage to be etremely ast.

4 URSI ABBAD AD PORSAI Figure 4. The modiied PRT or a real data gather rom the Gul o Meico. The amplitude curves recovered rom applying the IRLS method are shown by the red curve in Figure or the 1 events ater a single iteration. From the curves we observe that both methods provide a good reconstruction o the practical amplitude unctions along the primaries at most osets despite the strong intererence in the treated eample. The recovered amplitudes are very accurate at most osets. At medium osets the suggested approach allows or more accurate conservation o the AVO curve than does the IRLS method. At near osets the IRLS method has generally smaller amplitude errors but the error is larger than the modiied PRT or some primaries (events 6 and 7 or instance). The same observation can be made at very ar osets where the IRLS method perorms slightly better. The proposed method proves higher accuracy in amplitude recovery than the IRLS approach at most osets. This observation is urther conirmed on Figure 3 which depicts the average absolute amplitude error or the 1 events in Figure 1b. The superiority in perormance o our approach (blue curve) is again demonstrated ecept at the etreme osets in the data. The grey and black curves in Figure show the absolute errors o the AVO unction using and 5 iterations in the IRLS approach respectively. With urther iterations in the IRLS method increased accuracy is achieved at most osets. owever a loss o accuracy can also be observed at intermediate oset ranges when comparing results ater and 5 iterations. There is a signiicant reduction o the errors associated with most osets than the irst iteration (red curve) but these errors still larger than the modiied PRT errors at most osets which has the added advantage to be very ast. The modiied PRT method was also tested on a marine seismic line rom the Gol o Meico characterized by a deep sea-bottom with important amplitude variations with oset and water-bottom multiples. Figure 4c shows a gather rom the survey and illustrates the strong intererence in the area and the large number o events recorded. onhyperbolic velocity analysis was perormed on the data to generate post-mo data to be input or the modiied PRT algorithm. A velocity spectrum computed using the semblance coeicient is given in Figure 4d. The decrease o stacking velocities ater 4s is an obvious indicator to the presence o water-bottom multiple relections and peg-legs originating rom the thin layers just below the sea-bed. Figure 4b shows the gather ater applying nonhyperbolic MO corrections with parameters derived rom velocity analysis. The primary energy corrected with the eact moveout parameters (MO velocity and eective anellipticity) is well lattened in the ull oset range whereas multiple energy has positive residual moveouts that can be well approimated using parabolic curves as observed in the lower part o the gather ater 4s. To accelerate computations the modiied PRT was applied in the time range 1.5s 6.0s thus concerning only the area below sea-bottom where events are recorded. The moveouts were scanned or in the time range 0.05s 0.70s at the maimum oset in the gather. To implement the transorm the maimum signal reuency was set to 60z which gives a range in - - the interval -0.17km 1.79km. To conserve the whole amplitude inormation in the primaries we used a ilter cut-o at a positive curvature value s/ km thus corresponding to a radial mute zone in the domain deined by the euation -3 = The respective domains or the dierent wave modes are given in Figures 4e and 4i or the primary and the multiple modes respectively where the separation between the trends is obvious. The data gathers corresponding to these domains are computed through euation (6) and are illustrated in Figures 4 and 4j. Inverse nonhyperbolic moveout corrections help in reconstructing the original gathers iltered into primary (Figure 4g) and multiple (Figure 4k) modes. These gathers show that multiples are concentrated in the lower part o the gather due to multiple paths in the watercolumn and peg-legs in subseuent thin layers below the sea-bed. The velocity spectra or the iltered primaries and multiples are respectively depicted in Figures4h and 4l. Ater demultiple signiicant primary energy is recovered ater 4 s and can be clearly recognized in the primary gather (Figure 4g) and also in the velocity spectrum (Figure 4h). 4

5 MULTIPLE ATTEUATIO BY A MODIFIED PRT encountered in deep marine eplorations or in 3D acuisitions. The modiied PRT method satisies eactly the sampling conditions and oers a wider range at all reuencies than does the standard PRT. 5 Acknowledgements This project received inancial support rom the orwegian University o Science and Technology (TU) PetroMa the Federal University o Bahia (UFBA) CP Statoilydro ASA through the VISTA project and the orwegian Research Council through the ROSE project. BA wishes to thank Eric Duveneck or early support on this project and Martin Landro or ruitul discussion. Reerences ampson D Inverse velocity stacking or multiple elimination Journal o the Canadian Society o Eploration Geophysicists ugonnet P. and G. Canadas 1995 Aliasing in the parabolic Radon transorm: The 65th Annual International Meeting SEG Epanded Abstracts Schonewille M.A. and A. J. W. Duijndam 001 Parabolic Radon transorm sampling and eiciency: Geophysics Schonewille M.A. and P. Zwartjes 001 igh-resolution transorms and amplitude preservation: The 7th Annual International Meeting Society o Eploration Geophysicists Epanded Abstracts Figure 5. Processing results o the Gul o Meico data set. a) Stacked section without demultiple. b) Stack obtained with multiple removal using the modiied PRT. Figure 5a shows the stack obtained rom nonhyperbolic velocity analysis without any demultiple. The stack illustrates the main eatures o the area mainly a salt body. The lower part o the stack is dominated by multiple relections ater 3.5s. The modiied PRT is applied to this seismic line in the aim to attenuate water-bottom multiples and subseuent peg-leg relections. The obtained stack is depicted in Figure 5b and demonstrates how the algorithm allowed or eicient demultiple. The multiple energy in the section o Figure 5a was mostly removed particularly the relected waves with multiple paths in the water layer located in time between 3.5 s and 4.0 s. The multiple energy located in later times is due to the peg-leg multiples and was successully attenuated. Conclusions A ast implementation o a modiied parabolic Radon transorm is proposed in this paper. The transorm is linked to the standard PRT by means o interpolation. The new method shows high perormance in conserving accurate AVO eects ater the demultiple process which has several practical beneits. The method allows also or time savings especially in large data sets such those