( ) subject to m. e (2) L are 2L+1. = s SEG SEG Las Vegas 2012 Annual Meeting Page 1

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1 A new imultaneou ource eparation algorithm uing frequency-divere filtering Ying Ji*, Ed Kragh, and Phil Chritie, Schlumberger Cambridge Reearch Summary We decribe a new imultaneou ource eparation algorithm uing frequency-divere filtering. The method combine the array repone at different frequencie to uppre patial aliaing and convert the data eparation problem into a one-norm (l ) or zero-norm (l ) optimization problem. Synthetic and field data tet how that the algorithm work well with both patially aliaed and unaliaed data. The method require further work to reduce the computation cot. Introduction Seimic acquiition uing more than one ource, but firing the ource imultaneouly, i a technology of growing interet due to the reulting improvement in acquiition efficiency (Womack et al., 99; Bagaini, 2). Source eparation algorithm play an important role in imultaneou ource acquiition. Bealey et al. (998) decribe one method in which the ource are patially eparated. Alternatively, (i) by making the delay time between interfering ource incoherent, (ii) by collecting trace into a domain that include many excitation of each ource, and (iii) by aligning the trace uch that time zero correpond to the firing time for a pecific ource, then the ignal aociated with each ource can be eparated. Thi i baed on coherency, uing a random noie attenuation algorithm (Stefani et al., 27). Moore (28) reported that a modeling and inverion method generally give improved eparation reult. However, the modeling and inverion baed method perform le well with aliaed data. In thi paper, we decribe a new eparation algorithm baed on frequency-divere filtering technique for eparating imultaneouly acquired ource uing dithered time. The algorithm combine the array repone at different frequencie to uppre patial aliaing and convert the data eparation problem into a one-norm (l ) or zero-norm (l ) optimization problem. Method The eparation method can be applied to any number of ource but, to implify the decription, uppoe that we have two imultaneou ource, denoted S and S 2. Aume that S i fired regularly and S 2 i fired with a random timing delay with repect to the S firing time, and that the zero time of the trace correpond to the firing time for S (S -time). The timing delay for S 2, t, are known for each trace. The imultaneou ource eparation problem can be formulated a m min m2 ubject to m m2 ( A DA ) d ε 2 2, () where m, =, 2, i modeled data for each ource, A, =, 2 i an operator matrix contructed by uing multifrequency array teering vector at different lownee for each ource, Di an operator that hift the trace from S 2 time to S time, and ε i the noie level of the recorded data, d. The multifrequency array teering vector ued to contruct the matrix A, =, 2 in equation mut be capable of modeling the data. Aume that we ue 2M+ channel of data and 2+ number of frequencie, the element of the frequency-divere array teering vector at lowne p can be written a, ( ) ( ) ( ) 2 π,, x x, τ τ p m = j f l u b p e (2) where x m, m= M,,, M are poition vector of 2M+ channel of data, f l, l=, are 2+ frequencie, τ i the intercept time that repreent the arrival time at channel x of an event whoe phae function, u(p, x m x,τ ) can be linear or hyperbolic. The operator matrix A, =, 2 in equation i contructed by uing the multifrequency array teering vector defined a equation 2 with np lownee and nτ intercept time, written a, A, np nτ, ( b( p, τ ) b( p, τ ), =, 2 = Thee operator A, =, 2 only require that the data from each ource be coherent and fall within the defined lowne range and intercept time range. The operator D i a diagonal matrix contructed a D D= M O M. (4) D. (3) SEG a Vega 22 Annual Meeting Page

2 Simultaneou ource eparation algorithm Figure : Synthetic tet. The hot interval i 75 m and the bandwidth i 3-8 Hz. The f-k pectrum of the input data (fk-pc) how that the two linear event are aliaed. The etimate (pe and p2e) are very well eparated. The reidual of eparation (ubtracting pe and p2e from pc) i le than -4dB(%), indicating that nearly all the energy in the input data appear in either pe or p2e. The ource etimation error, pr = p-pe and p2r=p2-p2e, are alo le than -4dB(%), indicating that the two ource are nearly perfectly eparated. where j2πf lt M e D l = M O M, l=,, (5) j 2πf lt M e and t m, m= M,, M are the time delay for S 2. Once the model i known by olving equation, the eparated data are readily contructed uing the forward modeling. Although it doe not require the ource ignature to be known, the decribed method may improve the eparation becaue including each ource ignature in the multifrequency array teering vector improve the olution of equation. Data example Figure how a ynthetic common-offet gather ued to tet the eparation algorithm. The combined input data in S -time (pc) for the eparation algorithm are created by adding two ynthetic aliaing linear event correponding to each of the two ource (p and p2). The hot interval i 75 m and the bandwidth i 3-8 Hz. The f-k pectrum of the input data (fk-pc) how that the two linear event are aliaed. The eparation algorithm trie to recover p and p2 from pc. The eparated etimate (pe and p2e) appear nearly perfect. We found that the reidual of eparation (ubtracting pe and p2e from pc) i mall (le than -4 db or %), indicating that nearly all the energy in the input data appear in either pe or p2e. The ource etimation error, pr = p- pe and p2r = p2-p2e, are alo mall (le than -4 db), indicating that the two ource are nearly perfectly eparated. Figure 2 demontrate the performance of the method on a field data et. Figure 2a how a raw common-offet gather. The hot interval i 75 m. The amplitude vary greatly over hort ditance becaue of the complex geological tructure. Figure 2b how the eparated ource at S -time. We note that the teeply dipping event are well eparated and the amplitude variation i well preerved. Figure 2c how the SEG a Vega 22 Annual Meeting Page 2

3 Simultaneou ource eparation algorithm eparated ource 2 at S -time. We note that the coherent ignal i weak, indicating that the leakage from ource i weak. Figure 2d how the eparated ource 2 at S 2 -time. Again, we oberve that the teeply dipping event are well eparated and the amplitude variation i well preerved Figure 3 how the comparion with the reult obtained by uing Spare Contrained Time-Domain Radon Tranform (SCTDRT). Any coherent ignal appearing in (c) and (d) are leakage from ource. The weaker the coherent ignal in (c) and (d), the better are the eparation reult. The reult clearly how that the frequency-divere method outperform the pare contrained time-domain Radon tranform. Concluion A new imultaneou ource eparation wa decribed. The Figure 2: Field data et. (a) Raw common-offet gather. (b) Separated ource at S -time. The teeply dipping event are well eparated. (c) Separated ource 2 at S -time. The coherent ignal i weak, indicating that the leakage from ource i weak. (d) Separated ource 2 at S 2-time. The teeply dipping event are well eparated and the amplitude variation i well preerved. SEG a Vega 22 Annual Meeting Page 3

4 Simultaneou ource eparation algorithm Figure 3: Comparion. (a) Separated ource at S -time by pare contrained time-domain Radon tranform (SCTDRT). (b) Separated ource at S -time by frequency-divere filtering. (c) Separated ource 2 at S -time by SCTDRT. (d) Separated ource 2 at S -time by the new method. Any coherent ignal appearing in (c) and (d) are leakage from ource. The weaker the coherent ignal appear in (c) and (d), the better are the eparation reult. It i clear that the coherent ignal in (d) i ignificantly weaker than that in (c), particularly the teeply dipping event, indicating that the frequency-divere method out-perform the pare contrained time-domain Radon tranform. method doe not require the ource wavelet to be known. Synthetic and field data example demontrate that the new method can handle patially aliaed data. The comparion how that the frequency-divere method out-perform the parely contrained time-domain Radon tranform method. Acknowledgement We thank our colleague in the Geophyic Department of Schlumberger Cambridge Reearch and WeternGeco, epecially Craig J. Bealey, Ian Moore, and Nick Moldoveanu, for contructive and ueful dicuion and collaboration. SEG a Vega 22 Annual Meeting Page 4

5 EDITED REFERENCES Note: Thi reference lit i a copy-edited verion of the reference lit ubmitted by the author. Reference lit for the 22 SEG Technical Program Expanded Abtract have been copy edited o t hat reference provided with the online metadata for each paper will achieve a high degree of linking to cited ource that appear on the Web. REFERENCES Bagaini, C., 2, Acquiition and proceing of imultaneou Vibroei data: Geophyical Propecting, 58, 8. Bealey, C. J., R. E. Chamber, and Z. Jiang, 998, A new look at imultaneou ource: 68th Annual International Meeting, SEG, Expanded Abtract, Moore, I., B. Dragoet, T. Ommunden, D. Wilon, C. Ward, and D. Eke, 28, Simultane ou ource eparation uing dithered ource: 78th Annual International Meeting, SEG, Expanded Abtract, Stefani, J., G. Hampon, and E. F. Herkenhoff, 27, Acquiition uing imultaneou ource: 69th Conference and Exhibition, EAGE, Extended Abtract, B6. Womack, J. E., J. R. Cruz, H. K. Rigdon, and G. M. Hoover, 99, Encoding technique for multiple ource point eimic data acquiition: Geophyic, 55, SEG a Vega 22 Annual Meeting Page 5