( x) SUMMARY (2) α z. (3) Now approximating z derivative by forward difference

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1 Depth etrapolatio of seismic wavefields usig cubic splie approimatio Suhas Phadke* ad Dheera Bhardwa Cetre for Developmet of Advaced Computig, Pue Uiversity Campus, GaeshKhid, Pue 4007, Idia SUMMARY Depth etrapolatio equatio used for seismic migratio is ofte solved by fiite-differece techique. The most commoly used migratio method is based upo the Crak- Nicolso implicit scheme. Some modificatios to this scheme are i practice which improve the impulse respose of the migratio operator. I this paper we propose a differet approach where the wavefield is approimated by cubic splie fuctio. The approimatio to secod derivative resultig from this approach is similar to that of the /6th trick of Claerbout. There is aother splie parameter θ which cotrols the ature of impulse respose. The scheme is ucoditioally stable for 05. θ 0.. As the value of θ icreases the dispersed evaescet eergy gets more ad more atteuated. The method is demostrated by calculatig the impulse respose ad by applyig it to a sythetic data set. INTRODUCTION Seismic migratio is a key step i imagig of the udergroud geological structures. For laterally varyig velocity structures, the methods based upo the parabolic approimatio (Claerbout 985) of the wave equatio are most commo i use. Several migratio methods have bee proposed which use fiite-differece approimatio to the oe way wave equatio for dowward etrapolatio of the wavefield. Both implicit (Claerbout, 985) ad eplicit (Hale, 99) methods are i practice. Eplicit methods eed special care i their implemetatio because of the stability criteria, whereas implicit methods are ucoditioally stable. I this paper we have developed a implicit scheme for the depth etrapolatio of seismic wavefield usig cubic splie approimatio. The first part of the paper gives a mathematical descriptio of the method. Net we demostrate the usefuless of this method by showig the impulse resposes. The method is fially tested by applyig it to a sythetic data set, followed by discussio ad coclusios. MATHEMATICAL FORMULATION Wave etrapolatio equatio, which is accurate for migratig dips upto 45 degree is derived from the dispersio relatio kv α kv z ω ω kv β () ω where ω is the agular frequecy, v is the wave velocity ad k ad k z are horizotal ad vertical waveumbers respectively. For 45 degree accuracy α ad β are ad respectively. Accuracy for larger dips ca be obtaied by choosig α ad β i some optimal way (Yilmaz, 987). I terms of the retarded wavefield (Claerbout 985) the differetial equatio resultig from the diffractio term is give by im Q Q + Q 0 m z () α z α where m (ω/v) ad i is the square root of egative oe. The other term called thi les term is solved aalytically. Equatio () is usually solved usig some fiite differece approimatio. Here we preset a differet approach for solvig equatio (). Rewrite equatio () as β z i Q αm im Q + α Q 0 (3) Now approimatig z derivative by forward differece approimatio ad ( Q ) by secod order derivatives ( M ) of cubic splie fuctio S ( ) (where is mesh iterval) iterpolatig Q ( 0,,,... J), we get αm + + ( M M ) im ( Q Q ) z α z + + θm + ( θ) M 0 (4) where, θ [0,] is a cubic splie parameter ad z is icremet i z-directio. Now the cotiuity coditio for secod order derivative of cubic splie fuctio S ( ) (Ahlberg et. al., 967) gives 6 ( ) ( Q Q Q ) + + M + 4M + M + (5) or ( ) 6 ( ) ( Q Q Q ) + + M M + M + 6M + δ M 6 + 6M δ Q ( ) δ M Q δ + 6 where δ Q Q Q + Q (7) +. (6)

2 Depth Etrapolatio usig Cubic Splies Approimatio of the secod derivative by the epressio give i (6) is the same as that of the /6 trick of Claerbout (985). He calls it a less obvious epressio that offers more accuracy at less cost, without givig a eplaatio for the derivatio of the epressio used. Here we have derived it by approimatig the wavefield by cubic splie fuctio approimatio. This approimatio gives better results because cubic splies provide better approimatio to a give fuctio (Ahlberg et. al., 967). Now substitutig M from equatio (6) i equatio (4), ad multiplyig by z + δ 6, we obtai ( )( ) θ( z) δ + im δ + + δ Q + Q + Q α 6 im δ θ( z) δ Q + Q δ Q α 6 (8) 0. Distace (km) 0.. θ 0. Usig (7) i (8) we obtai after some algebraic maipulatios Distace (km) AQ+ + BQ + AQ AQ + BQ + AQ (9) θ Figure : Impulse respose of migratio algorithm for differet values of θ. N as the value of θ icreases. Steep dips are also atteuated with the icrease i where 06 im θ( z) A + 6α im θ ( z) B 3 α im ( θ)( z) A 6α im ( θ)( z) B + 3 α

3 Depth Etrapolatio usig Cubic Splies i Figure for differet values of θ. For θ oe ca observe a large amout of dispersed evaescet eergy. Also at steep dips there is dispersio of low ad high frequecies. For θ 055. the dispersed evaescet eergy has bee atteuated to a great etet. Also all the dips are correctly positioed. For θ 06. ad θ 07. the dispersed evaescet eergy is further reduced. Steep dips are also atteuated, but the remaiig dips are correctly positioed alog cocetric semicircles. Oe has to decide o a optimal choice for θ, which reduces the udesired evaescet eergy as well as correctly positios all the required dips. θ 06. seems to be the best choice from the practical poit of view. SYNTHETIC DATA EXAMPLE Net we applied the migratio method based upo the cubic splie approimatio to a sythetic data set. The velocity model used for geeratig the sythetic data is show i Figure. The model comprises of a weathered layer o top of a dippig layer ad a graba like structure. Sythetic seismograms were calculated for this model for 46 source positios with a source iterval of 50m ad a receiver iterval of 5m. A higher order fiite differece modellig algorithm based upo acoustic wave equatio was used with a grid spacig of 5m. A Ricker wavelet with a domiat frequecy of 30Hz was used as the source fuctio. Distace (km) m/s 500 m/s 850 m/s 3400 m/s Equatio (9) is a tridiagoal system of equatio. The solutio of (9) gives the wavefield at depth (+) i terms of the wavefield at depth (). This implicit scheme is ucoditioally stable for 05. θ 0.. Absorbig boudary coditios are used o the sides of the model. IMPULSE RESPONSE I order to test the etrapolatio method described above, a 45 degree migratio program was developed. The iput to the migratio program was a sectio cotaiig three bad limited Ricker wavelets i the cetre of the sectio. The domiat frequecy of the wavelets was 30 Hz. Spatial samplig of 8m ad a samplig rate of ms was used. The migratio of this data set yields the impulse resposes show Figure : Velocity model used for the geeratio of sythetic data set. The acoustic wave modellig code was implemeted i a distributed computig eviromet usig PVM (Parallel Virtual Machie) message passig calls (Geist et. al. 994). For both, modellig ad migratio eamples preseted i this paper we have used 8 UltraSparc Workstatios etworked usig a fast etheret switch. This is a part of the facility called PARAM OpeFrame. Each UltraSparc Workstatio has a 00 MHz CPU with 8 MB of RAM ad Solaris operatig system.

4 Depth Etrapolatio usig Cubic Splies Distace (km) Distace (km).5.0 Two-way time (sec) Figure 3: A CDP stacked sectio of the sythetic data set. A CDP stacked sectio of the sythetic data set is show i Figure 3. Net this stacked data set was migrated usig the above algorithm. The migrated output sectios for two differet values of θ are show i Figure 4. O both the migrated sectios all the evets are properly imaged. Oe ca otice that for θ 05. there is more evaescet eergy o the migrated sectio as compared to the migrated sectio with θ 06.. A user ca decide o the appropriate value for θ. DISCUSSION AND CONCLUSIONS I this paper we have developed ad demostrated a implicit scheme for the depth etrapolatio of the seismic wavefield usig cubic splie approimatio. The approimatio to the secod derivative is derived from the cotiuity coditio for the secod order derivative of cubic splie fuctio. The differece approimatio to the oe-way wave equatio resultig from this approach has aother splie parameter θ, which cotrols the dispersed evaescet eergy. As the value of θ icreases more ad more evaescet eergy gets atteuated. Steep dips also get atteuated with icrease i the value of θ. A proper choice of this parameter helps i reducig the udesired oise o the migrated sectios. This approach ca be easily icorporated i D prestack migratio algorithm as well as i 3D prestack ad poststack migratio algorithms. ACKNOWLEDGEMENTS Authors wish to epress their gratitude to the Departmet of Sciece ad Techology (DST), Govermet of Idia, for fudig the seismic data processig proect. Authors also wish to thak the Cetre for Developmet of Advaced Computig (C-DAC), Pue for providig the computatioal facilities o PARAM OpeFrame ad permissio to publish this work..5 θ Distace (km).5.0 θ Figure 4: The migrated seismic sectios of the stacked data show i Figure 3 for θ 05. ad θ 06.. Notice the reductio i evaescet eergy for θ 06.. REFERENCES Ahlberg, J. H., Nilso, E. N., ad Walsh, J. L., 967, The theory of splies ad their applicatios, Academic Press. Claerbout, J. F., 985, Imagig the Earth s iterior, Blackwell Scietific Publicatios. Geist, A., Begueli, A., Dogarra, J., Jiag, W., Machek, R., ad Suderam, V., 994, PVM: Parallel Virtual Machie, A users guide ad tutorial for etworked parallel computig, MIT Press, Cambridge, Massachusetts. Hale, D., 99, Stable eplicit depth etrapolatio of seismic wavefields, Geophysics, 56, p

5 Yilmez, O., 987, Seismic Data Processig, Society of Eploratio Geophysicists. Depth Etrapolatio usig Cubic Splies