Automatic selection of reference velocities for recursive depth migration


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1 Automatc selecton of mgraton veloctes Automatc selecton of reference veloctes for recursve depth mgraton Hugh D. Geger and Gary F. Margrave ABSTRACT Wave equaton depth mgraton methods such as phaseshft plus nterpolaton, extended spltstep Fourer, or Fourer fnte dfference plus nterpolaton requre a lmted set of reference veloctes for effcent wavefeld extrapolaton through laterally nhomogeneous velocty models. In basc mplementatons, reference veloctes are selected as ether a lnear or a geometrc progresson spannng the range of model veloctes. However, t s unlkely that the model veloctes are dstrbuted lnearly or geometrcally. If the reference veloctes can be dstrbuted statstcally to more closely approxmate the actual dstrbuton, the accuracy of the extrapolaton step can be mproved. In ths paper, we present a modfcaton to a prevously publshed algorthm for statstcal selecton of reference veloctes (Bagan et al. 1995). Key features of our automatc reference velocty selecton algorthm are 1) dvson of the velocty dstrbuton nto clusters 2) entropy based statstcal control to determne the mnmal number of reference veloctes requred wthn a cluster, 3) a novel greedy search that selects reference veloctes wthn each cluster at or near peaks n the probablty dstrbuton, and 4) calculaton of a sngle reference velocty f the velocty range wthn each cluster s less than a mnmum threshold. We show that our method s superor to Bagan et al. s method, whch ncludes only step (2) above, and  n place of step (3)  selecton of reference veloctes by pecewse constant nterpolaton of the probablty dstrbuton of the underlyng veloctes. Our automatc velocty selecton algorthm can be run usng a sngle parameter the approxmate desred velocty step expressed as a percentage although the maxmum step and mnmum threshold can be specfed f the defaults are not sutable. The automatc velocty selecton algorthm s mplemented as part of a prestack PSPI depth mgraton of the 2D Marmous model. The resultng mages are clearly superor to mages created usng a lnear or geometrcal dstrbuton of a smlar number of reference veloctes. The algorthm s sutable for 3D data, where the tradeoff between accuracy and effcency s more pronounced. INTRODUCTION Wave equaton depth mgraton methods such as phaseshft plus nterpolaton (Gazdag and Sguazzero, 1984), extended spltstep Fourer (Kessnger, 1992), or Fourer fnte dfference plus nterpolaton (Bond, 2002) requre a lmted set of reference veloctes for effcent wavefeld extrapolaton through laterally nhomogeneous velocty models. All of these authors suggests that a geometrc dstrbuton can produce a good set of reference veloctes. Kessnger (1992) mentons that, by emprcal testng, a percentage velocty ncrement of 15% provdes a good compromse between accuracy and effceny. Another standard method calculates a lnear dstrbuton of veloctes (e.g. Ferguson and Margraves, 2002), whch may have an advantage of ncreasng the accuracy of hgherangle wavefeld propagaton wth depth. CREWES Research Report Volume 16 (2004) 1
2 Geger and Margrave example velocty profles FIG. 1. The Marmous velocty model. Red lnes mark velocty profles used n ths study. Bagan et al. (1995) propose a statstcal method for optmal selecton of reference veloctes. The optmal choce of reference veloctes wll be a balance between accuracy and effcency. In ths study, we do not determne a quanttatve method for evaluatng optmalty. Instead, we compare the selected reference veloctes aganst the probablty dstrbuton of the veloctes over the range for the profle. Fgure 1 s a colour contour of the Marmous velocty model. The red lnes ndcate profles used n ths study. too few reference veloctes? ~3000 ms 1 (~230%) ~40 ms 1 (~ 2%) too many reference veloctes? ~320 ms 1 (~ 17%) ~70 ms 1 (~ 4%) ~70 ms 1 (~ 4%) ~80 ms 1 (~ 5%) FIG. 2. How should reference veloctes be chosen for wavenumberspace doman extrapolators such as PSPI? Emprcal testng (Kessgner, 1992) suggests that an ncrement of approxmately 15% provdes a reasonable compromse between accuracy and effcency. In selecton of the boundng veloctes creates too large an ncrement (230%), whereas n a careful selecton of veloctes that closely matches steps n the velocty profle may produce too many veloctes (steps of 2%, 4%, 4%, and 17%). 2 CREWES Research Report Volume 16 (2004)
3 Automatc selecton of mgraton veloctes Keepng Kessnger s emprcal percentage ncrement of 15% n mnd, we can easly determne when we mght have too few veloctes (Fgure 2. However, a good ft to a velocty profle, as shown n Fgure 2b, mght not satsfy Kessnger s emprcal ncrement. How then should we select veloctes to create a best ft to the underlyng probablty dsrbuton? In ths paper, we revew the method of Bagan, show that ther reference veloctes are not necessarly near the peaks n the probablty dstrbuton of the veloctes over a layer, and propose a new statstcal method that produces a more optmal dstrbuton of reference veloctes. We test our method aganst Bagan s method, and aganst the lnear and geometrc methods, usng velocty profles from the Marmous acoustc model (Versteed and Grau, 1991). LINEAR AND GEOMETRIC METHODS The lnear method s descrbed as follows: 1) choose an approxmate velocty spacng dv, 2) Calculate nv=round((v max v mn )/dv) and 3) determne references veloctes as equal ncrements of v step =(v max v mn )/nv. However, what s a good choce for dv? One opton s a value close to Kessnger s emprcal percentage. A queston remans, however: Wll the selected ncrement be optmal reasonable for both low and hgh veloctes? Fgure 4 llustrates the geometrc method appled to a Marmous velocty profle. too few reference veloctes? ~3000 ms 1 (~230%) ~40 ms 1 (~ 2%) too many reference veloctes? ~320 ms 1 (~ 17%) ~70 ms 1 (~ 4%) ~70 ms 1 (~ 4%) ~80 ms 1 (~ 5%) FIG. 2. How should reference veloctes be chosen for wavenumberspace doman extrapolators such as PSPI? Emprcal testng (Kessgner, 1992) suggests that an ncrement of approxmately 15% provdes a reasonable compromse between accuracy and effcency. In selecton of the boundng veloctes creates too large an ncrement (230%), whereas n a careful selecton of veloctes that closely matches steps n the velocty profle may produce too many veloctes (steps of 2%, 4%, 4%, and 17%). CREWES Research Report Volume 16 (2004) 3
4 Geger and Margrave The geometrc method s descrbed as follows: 1) choose an approprate percentage step v prcnt, 2) calculate v() = (1+v prcnt )*v(1). 3) The percentage can be adjusted by solvng a smple logarthmc equaton f reference veloctes are desred at both the mnmum and maxmum veloctes. As mentoned prevously, Kessgner (1992) recommends v prcnt =0.15. Fgure 4 llustrates the geometrc method appled to the deeper Marmous velocty profle. ~500 ms 1 (~10%) ~500 ms 1 (~20%) ~750 ms 1 (~15%) ~370 ms 1 (~15%) (detal for subsequent fgures) FIG. 3. Standard methods for selectng reference veloctes nclude a lnear nterpolaton between the mnmum and maxmum veloctes wthn a layer or cluster, and a geometrc nterpolaton based on a fxed percentage ncrement from the mnmum velocty. Emprcal testng (Kessgner, 1992) suggests that an ncrement of approxmately 15% provdes a reasonable compromse between accuracy and effcency. The green box n fgure shows the velocty profle used n Fgure 4. THE METHOD OF BAGAINI ET AL. (1995) The method of Bagan et al. (1995) can be descrbed as follows: 1) choose a prelmnary dv (geometrc?); 2) determne equally spaced bns over v mn :v max (e.g. Fgure 4a, where nb temp = 6)bn the veloctes to gve probablty densty P, where ΣP = 1, 4) determne theoptmal number of bns nb opt by statstcal entropy S=ΣP logp, where nb opt = round(exp(s)+0.5) (e.g. nb opt = 5 n Fgure 4); 5) calculate the cumulatve probablty dstrbuton Y, where Y = S P  each optmal bn wll hold 1/nB opt of the cumulatve probablty dstrbuton (e.g. 0.2 n Fgure 4; 6) start at v mn and lnearly nterpolate the poston of the reference veloctes from the temporary bn boundares (e.g. at 0.2, 0.4, 0.6, 0.8 n Fgure 4. 4 CREWES Research Report Volume 16 (2004)
5 Automatc selecton of mgraton veloctes FIG. 4. Bagan et al s (1995) method for selectng reference veloctes. the veloctes are bnned to create a dscrete probablty dstrbuton. An optmal number of reference veloctes nb opt s calculated usng a measure of statstcal entropy. the dscrete cumulatve probablty dstrbuton s dvded nto nb opt equal ncrements by lnearly nterpolatng from the bn boundares. Is ths method optmal? A lnear ncrement of the cumulatve probablty dstrbuton does not necessarly produce reference veloctes on or near the peaks n the probablty dstrbuton. In the worst case scenaro, none of the reference veloctes wll fall on peaks and the extrapolaton through that layer wll requre large nterpolatons. OUR NEW GREEDY PEAK SEARCH METHOD We propose a new method that specfcally searches for peaks n the probablty dstrbuton. We use Bagan s measure of statstcal entropy to determne an optmal number of reference veloctes. In all lkelhood, however, the number of peaks n the probablty dstrbuton wll not match the statstcally optmal number of number of reference veloctes. Our method uses a varety of technques to poston the reference veloctes closest to the peaks. In short, we collapse the closest peaks, startng wth the peaks of lowest probablty densty. A new reference velocty s determned as a weghted dstance between the two orgnal peaks, except at the ends where the values of the mnmum and maxmum veloctes are preserved as the boundng reference veloctes. CREWES Research Report Volume 16 (2004) 5
6 Geger and Margrave We also use a clusterng algorthm to dvde the velocty profle up nto clusters that mght be separated by a much larger percentage dfference than acceptable. Wthn each cluster, we can apply any of the above specfed methods. The general clusterng method s descrbed as follows: 1) sort the veloctes n the profle 2) select clusters where the percentage ncrement n sorted veloctes exceeds some maxmum value 3) each th cluster s now defned by a mnmum velocty. Startng wth the fst cluster (lowest velocty), determne M v a. If b. If M m % thresh m v and maxmum velocty v v 1+ v, then v m v M and ths cluster can safely be v avg m M v v extrapolated at the sngle velocty ( ) 2 = +. Set all veloctes wthn these bounds to the average velocty v n m, M avg xy v v = v, add ths use avg velocty to the velocty vector v = v, and ncrement k = k+ 1. M v M m c c 1+ c % max k, then the boundng veloctes for the cluster v m and are a vald step wthout any ntermedate veloctes M m % c. If c c > 1+ cmax there wll be at least three veloctes n the cluster (ncludng the boundng veloctes). Use the lnear, the geometrc, or the Bagan et al. (1995) measure of statstcal entropy to determne an optmal number of bns n the cluster. 4) Dependng on the method chosen n 3), calculate the reference veloctes wthn the cluster and cycle through all clusters determned n 2) In Fgure 5, we compare the selecton of reference veloctes wthn a cluster. The greedy peak search method shown n Fgures 5c and 5f does a better job of locatng the reference veloctes near the peaks n the probablty dstrbuton. MARMOUSI EXAMPLE We appled the varous technques to the select reference veloctes for PSPI mgraton of the Marmous data set. The results of the PSPI mgraton usng the clusterng algorthm wth the Bagan method are shown n Fgure 6. Fgure 6a s the bandlmted reflectvty calculated as usng gradents of the velocty and densty model. The modfed Bagan method obvously produces an excellent mage of Marmous (Fgure 6. Even the elusve target zone at depth (Fgure 6d) compares well wth the reflectvty (Fgure 6d). Ths research s ongong. More results wll be presented at the meetng 6 CREWES Research Report Volume 16 (2004)
7 Automatc selecton of mgraton veloctes FIG. 5. ac) Probablty dstrbuton of veloctes wthn a cluster from a selected layer of the Marmous (blue bars) and selected reference veloctes usng varous methods (red lnes). df) the velocty profle wth dstance (blue dotted lne), selected reference veloctes (red dotted lnes) and boundng veloctes (red lnes). Note that n and d) the reference veloctes by the statstcal entropy method of Bagan et al (1995) are not necessarly at the peaks n the dstrbuton. In and e), large gaps are lnearly nterpolated. In c) and f) the new greedy peak search method selects reference veloctes at the peaks n the probablty dstrbuton. The new method should produce more accurate wavefeld extrapolaton for the same computatonal cost, and hence better mages. CREWES Research Report Volume 16 (2004) 7
8 Geger and Margrave c) d) FIG. 6. Bandlmted reflectvty for Marmous acoustc model. Depth mage by PSPI mgraton wth crosscorrelaton magng condton usng modfed method of Bagan et al (1995), where reference veloctes are selected usng clusterng and statstcal entropy wthn clusters. c) and d) Zoom of full bandwdth reflectvty and PSPI mage at the target zone. The target s accurately maged, although the lmted bandwdth of the data results n a rngy appearance. 8 CREWES Research Report Volume 16 (2004)
9 CONCLUSIONS Automatc selecton of mgraton veloctes An optmal selecton of reference veloctes wthn a depth layer can maxmze the accuracy and effcency of wavenumber/spatal doman wavefeld extrapolators such as PSPI. A lnear or geometrc dstrbuton does not take nto account the underlyng dstrbuton of the veloctes. Hence, large nterpolatons between reference wavefelds may be requred, whch can result n reduced accuracy and a poor mage. The statstcal method of Bagan et al. (1995) dstrbutes the reference veloctes based on equal ntervals of the cumulatve probablty dstrbuton of the veloctes wthn a layer. On average, ths wll reduce the sze of the nterpolatons, but may stll produce a reference velocty dstrbuton that requres some large nterpolatons. We propose a modfcaton to the method of Bagan et al., whereby the veloctes wthn a layer are frst separated nto clusters dvded by large gaps n the dstrbuton, an optmal number of veloctes wthn a cluster s chosen based on Bagan et al. s method of statstcal entropy, a greedy search algorthm s used to select reference veloctes wthn each cluster as close as possble to the peaks n the probablty dstrbuton, and as weghted averages between smaller pars of peaks f there are more peaks than the optmal number of veloctes. A sngle reference velocty s selected for clusters or layers where the spread of veloctes s less than a specfed threshold. The new greedy peak search method was tested on the Marmous acoustc data set usng a PSPI waveequaton prestack shotrecord mgraton algorthm. The depth mages from the new peak search method are clearly more accurate than those produced by a geometrc dstrbuton. Surprsngly, a lnear dstrbuton s shown to produce reasonable mages n the shallow secton that are smlar to the new algorthm and much better than thosee produced usng a geometrc dstrbuton. At present, we do not have a good explanaton for ths result. In the shallow secton, the peak search method and the Bagan et al. method yeld smlar mages. The benefts of our new method are most pronounced n the deeper secton, where focusng and postonng are better than any of the methods used for comparson. REFERENCES Bagan, C., Bonom, E., Peron, E., 1995, Data parallel mplementaton of 3D PSPI: 65 nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Bond, B., 2002, Stable wdeangle Fourer fntedfference downward extrapolaton of 3D wavefelds: Geophyscs, 67, Ferguson R. and Margrave, G., 2002, Prestack depth mgraton by symmetrc nonstatonary phase shft:geophyscs, 67, Gazdag, J. and Sguazzero, P., 1984, Mgraton of sesmc data by phase shft plus nterpolaton: Geophyscs, 49, Kessnger, W., 1992, Extended spltstep Fourer mgraton: 62 nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Stoffa, P., Fokkema, J., de Luna Frere, R., and Kessnger, W., 1990, Spltstep Fourer mgraton: Geophyscs, 55, Versteeg, R., and Grau, G., Eds., 1991, The Marmous experence: Proc Eur. Assn. Expl. Geophys. Workshop. CREWES Research Report Volume 16 (2004) 9
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