Hgh resoluton 3D Tau-p transform by matchng pursut Wepng Cao* and Warren S. Ross, Shearwater GeoServces Summary The 3D Tau-p transform s of vtal sgnfcance for processng sesmc data acqured wth modern wde azmuth recordng systems. Ths paper presents a 3D hgh resoluton Tau-p transform based on the matchng pursut algorthm. 3D sesmc data are frst transformed to the frequency doman, and a sparse set of f-p x -p y coeffcents are teratvely nverted by matchng pursut. When the data are spatally alased, the nverson results are weghted by a total energy mask. The numercal examples demonstrate ths scheme acheves a hgh resoluton Tau-p transform that represents the nput data accurately and handles spatal alasng properly. Introducton The Tau-p transform, also called the lnear Radon transform, has been a very mportant tool n sesmc data processng. Its applcatons range from lnear nose removal to de-ghostng and data nterpolaton/regularzaton. The Tau-p transform s usually formulated as an nverse problem to solve for the Tau-p coeffcents that best represent the nput data under certan crtera, Lm d. (1) In the above equaton, m s the tau-p model and L s the transform operator. Unlke the Fourer transform, the bass for the Tau-p transform s not orthogonal. Therefore tradtonal L2 norm based nverson algorthms provde a low resoluton transformaton, especally when the data samplng and aperture are not deal. Varous authors developed 2D hgh resoluton Radon transform schemes (Thorson and Claerbout 1985; Sacch and Ulrych 1995; Cary, 1998; Trad et al., 2003, Wang et al., 2009) by mposng sparsty constrants n the tme or frequency doman, and the above ssue has been effectvely mtgated for 2D sesmc data. Though 2D Tau-p transform has been wdely used n sesmc processng, ts 2D plane wave assumpton cannot handle the 3D curvature varatons wth azmuth present n 3D sesmc data. Therefore the Tau-p transform needs to be extended to 3D to process 3D sesmc data, especally those data acqured wth wde azmuth recordng systems. Zhang and Lu (2013) developed an accelerated hgh resoluton 3D Tau-p transform wth an effcent spatal transform scheme followed by teratve thresholdng of the nverted Tau-p coeffcents. Wang and Nmsala (2014) proposed a fast progressve sparse Tau-p transform: a low-rank optmzaton s appled for nverson stablty and effcency, and alasng s handled by progressvely drvng the nverson for hgh frequences wth the low frequency results. Ths paper proposes a matchng pursut method for 3D Tau-p transform. The matchng pursut algorthm provdes an approxmate L0 norm soluton to nverse problems, and has been very successful wth FK doman sesmc nterpolaton/regularzaton (Xu et al., 2005;Ozbek et al., 2010; Nguyen and Wnnett, 2011; Hollander et al., 2012; Schonewlle et al., 2013). We choose a matchng pursut approach because of ts proven performance n the FK context, and because t has a straghtforward mplementaton. Also, to properly handle spatal alasng, the total energy of each p trace s computed and used as a mask for the matchng pursut soluton. Method The matchng pursut algorthm bulds up a sparse soluton to the nverse problem by teratvely pckng the strongest component n the model resdue. For the Tau-p transform mplementaton, matchng pursut s mplemented on frequency slces after the nput data are transformed nto the frequency doman. In the frequency doman, the terms n equaton 1 are formulated as: nput data d d f, x, y ), where ( ( x, y ) represent the spatal coordnates of trace number ; j j m m ( f, p x, p ) s the Tau-p coeffcent for the jth p y par. The transform operator L s expressed as L e, j j j 2 f ( p x x p y y ) The adjont transform s formulated as j j H f p x p y m ~ j j ( x y ( f, p, p ) L d e ) x y. (2) 2 d ( f, x, y ), (3) The algorthm s mplemented as follows: 1. Intalze the data resdual wth the nput data 2. Calculate the adjont model by applyng the adjont of operator L to the resdual 3. Select the strongest component n the adjont model and add t to the estmated soluton 4. Inverse transform the selected component nto the data (space) doman, and update the data resdual by subtractng ths transformed component 5. Go to step 2 f resdual level s not low enough. Otherwse ext teraton. To avod a large number of temporal Fourer transform operatons, the nput sesmc data are transformed nto the
Hgh resoluton 3D Tau-p transform frequency space doman, and matchng pursut teratons are mplemented for each frequency slce. In 3D sesmc acquston, the sesmc data are usually alased n the crosslne drecton, and spatal alasng brngs a serous challenge to the Tau-p transform. To overcome ths ssue, for each p x, p y trace we compute the total energy durng the matchng pursut teratons, creatng a mask over the entre p x, p y space. Then n later teratons, ths mask s used as a model-space weght for the nverson. Ths process s repeated perodcally throughout the nverson teratons untl a satsfactory resdual s acheved. Fgures 1-3 show a smple synthetc example to compare Tau-p transform results obtaned wth the matchng pursut algorthm and a Cauchy-lke sparse nverson scheme wth some smlartes to that of Sacch and Ulrych (1995). As llustrated n Fgure 2, the matchng pursut algorthm delvers a Tau-p transform wth hgher resoluton and more accurate ampltude. Also these two Tau-p panels are transformed back to sesmc traces to check the fdelty of the forward transform by checkng the round-trp errors. Fgure 3 shows the round-trp errors scaled by 10 dsplayed at the same range as Fgure 1. The worst error ampltude of the Cauchy-lke approach s over ten tmes that for the matchng pursut scheme. Ths test demonstrates that relatve to the Cauchy-lke approach shown here the matchng pursut algorthm provdes a Tau-p transform that has hgher resoluton and represents the data more accurately. Fgure 2: Tau-p panels for a) matchng pursut method and b) Cauchy-lke method Fgure 3: Tau-p transform round-trp error for a) match pursut method and b) Cauchy-lke method Fgure 1: synthetc data wth 4 lnear events Fgures 4-6 show a 3D synthetc example to llustrate the performance of ths ant-alasng scheme. The trace spacng n the nlne and crosslne drecton s 10 m, and the wavelet s a Rcker wavelet wth maxmum frequency 50 Hz. The dense crosslne samplng n Fgure 4 s decmated to 30 m to generate an alased nput for the Tau-p transform, and the Tau-p coeffcents are used to nterpolate the data back to the dense geometry. Fgure 5 shows that wthout the ant-alasng scheme, a sgnfcant amount of error s ntroduced at the nterpolated trace locatons. By contrast, the error at the nput trace locatons s very low, whch ndcates a good data match n Tau-p transform. When the ant-alasng step s appled, the nterpolaton error s greatly reduced, as Fgure 6 shows.
Hgh resoluton 3D Tau-p transform Feld data example Fgure 4: nlne and crosslne vews of the synthetc data pror to crosslne downsamplng from 10 m to 30 m. Fgure 5: nterpolaton error for Tau-p wthout ant-alasng We appled ths sparse Tau-p transform algorthm to a deep-water 3D marne shot gather to test ts performance on nterpolaton. Acquston conssted of ten cables wth a spacng of 150 m, and 12.5 m channel spacng along the cable. For the test, the data s frst decmated by a factor of two n the cable drecton to generate a sparse dataset. As the three-slce vew n Fgure 7 shows the dataset s badly sampled n both the nlne and crosslne drecton. The FK spectrum n Fgure 8 shows the data s clearly alased n the nlne drecton. The hgh resoluton 3D Tau-p transform s appled to the decmated data, and the resultant Tau-p coeffcents are used to nterpolate back to the orgnal dense samplng n the cable drecton, and compared wth the true dense data. Fgures 9 and 10 demonstrate the nterpolaton results for a near and a far cable. Note that although ths s a 3D transform, the nterpolaton was performed only n one (the nlne) drecton, not the very sparse crosslne drecton. We also test the nterpolaton performance for dfferent percentages of the Tau-p coeffcent threshold. As shown n Fgure 11 for the center part of the data (trace 25-80) the RMS nterpolaton error drops to 10-15% f we use 0.5% of the Tau-p coeffcents, and ncreasng the threshold percentage to 1% brngs lttle mprovement to the error level. There s no sgnfcant dfference for the near and far cables. The valleys of the "V" shapes are locatons where nput traces are avalable and we have data control to reach the best nterpolaton ft. The nterpolaton error on the near traces s at or below 10%. Ths s a 2 to 1 nterpolaton, and the crosslne drecton s very badly alased (wth 150 cable spacng). Overall we thnk the nterpolaton performance s reasonable consderng the nput data s serously alased n both drectons. Fgure 6: nterpolaton error for Tau-p wth ant-alasng Fgure 7: 3D sparse nput data for nterpolaton The average cable spacng s 150, and after decmaton the group nterval s 25 m.
Hgh resoluton 3D Tau-p transform Fgure 8: FK spectrum of the decmated nput n the nlne drecton Fgure 11: nterpolaton error vs traces for two dfferent threshold percentages (0.5% and 1%) of Tau-p coeffcents for nterpolaton Conclusons Fgure 9: Interpolaton for a far cable: true dense cable (left), nterpolated dense cable (center) and the nterpolaton error (rght). A hgh resoluton 3D Tau-p transform scheme s developed to process sparse 3D sesmc data. Wth the matchng pursut algorthm and an ant-alasng strategy based on total energy n the Tau-p plane, a hgh resoluton Tau-p representaton of the alased 3D data can be acheved. Numercal examples demonstrate mproved resoluton n the Tau-p transform and a more accurate data representaton. The applcaton of ths scheme to feld data nterpolaton/regularzaton shows promsng results. Acknowledgments The authors would lke to Sergo Gron for many helpful dscussons and Shearwater Geoservces for the permsson of publsh ths work. Fgure 10: Interpolaton for a near cable: true dense cable (left), nterpolated dense cable (center) and the nterpolaton error (rght).
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