On the computation of wavenumber integrals in phase-shift migration of common-offset sections

Transcription

1 Computation of offset-wavenumber integrals On te computation of wavenumber integrals in pase-sift migration of common-offset sections Xiniang Li and Gary F. Margrave ABSTRACT Te evaluation of wavenumber integrals often occurs in migration and modelling algoritms in eploration seismology. Te integrals over offset-wavenumbers in pase-sift migration of common-offset sections are te specific case analysed in tis paper. Te accurate algoritm is numerically simple and easy to implement, but it is computationally very time consuming. Two approimation metods to compute te offset-wavenumber integrals are discussed in tis paper. Te first metod is te stationary-pase metod, wic is very often used to evaluate te offset-wavenumber integrals. Te second metod is called te trapezoidal Filon metod, wic as been nown in syntetic seismogram teory for decades. Numerical results from tese two metods suggest tat te stationary-pase metod as ig accuracy wen te wavefield etrapolation dept step-size is not very small relative to te source-receiver offsets of te input common-offset sections. Te trapezoidal Filon metod, on te oter and, is more suitable in te evaluation of offset-wavenumber integrals wen te offsets are large relative to te dept step-size. INTRODUCTION Migration from separated offset sections as many advantages in seismic data analysis and interpretation over conventional prestac migration scemes. One of suc advantages is te relative insensitivity to migration velocity errors, wic as been used to perform migration velocity analysis and obtain staced image by simple moveout correction on te image gaters (Sattlegger et al (1980), Deregowsi (1990), Kim and Krebs (1993), and Ferber (199)). Anoter advantage of common-offsetsection migration is tat it provides reflectivity information wit preservation of canges in te offset direction, wic is desired for AVO or AVA analysis (Eren and Ursin, 1995, 1999). AVO or AVA analysis after migration is a topic drawing more and more attentions (Moser et al., 1996 and Tygel et al., 1999). Most of te earlier wor on migration from common-offset sections utilised diffraction-summation type migration operators because of teir independence of te acquisition geometry. However, because diffraction summation metods ave difficulties to preserve te amplitude (reflectivity) information (among oter reasons), Fourier domain metods are still attractive. A sceme of common-offset-section pase-sift migration, wic involves offset-wavenumber integrals, was discussed by Popovici (199) and ten etended by Alalifa (1997) to a time migration for anisotropic media. Te strategies of te evaluation of te offset-wavenumber integrals in Popovici (199) and Alalifa (1997) are te same in tat tey bot use te stationary-pase approimation metod. Te main difference between tese evaluation tecniques lies in te different metods employed in finding te

2 Li and Margrave stationary-pase point. Popovici (199) uses a numerical algoritm to find te root (zero value point) of te pase derivative function, and tis algoritm is still epensive. Alalifa (1997) uses an algoritm to find te maimum of te pase function. By is own assessment, Alalifa s algoritm taes about 10% more computation time to migrate a single offset section tan te time required by te conventional zero-offset pase-sift metod. In tis paper, detailed numerical eperiments wit te stationary-pase metod for te approimation of te offset-wavenumber integrals will be conducted. In cases wen offsets are large relative to te wavefield etrapolation step-size, te stationarypase approimation results in significant errors. Fortunately, in tese cases, te trapezoidal Filon metod becomes appropriate. Te Filon metod is analysed, and suggested wen ig accuracy is required for large offsets. THE INVOLVED OFFSET-WAVENUMBER INTEGRALS Following te procedure of pase-sift migration sceme from common-offset sections, as presented in Popovici (199) and Li and Margrave (1999), an integral over offset-wavenumber (denoted as ) needs to be evaluated for given values of: te wavefield etrapolation dept step-size (denoted as z), te temporal frequency (denoted as ω), te wavenumber value in te CDP direction (denoted as ), te alf source-receiver offset (denoted as 0 ) of te input section, and te relevant migration velocity (denoted as v). Te integral can be eplicitly written as ω + i ( iz i ) + 0 d e d v v ω φ ep, (1) = were i is te square root of 1. Te direct discritization of tis equation results in a summation metod of te evaluation of te integral as iφ( ) iφ ( ( j e d )) e. () Te integral limits in (1) and te summation limits in () are set to (teoretically) include all te relevant values. However, in practice, te limits are often referred to te -range in wic te two square roots in equation (1) are bot real, i.e., te - range is determined by te following two inequalities: wic is equivalent to one condition j v ω, + v ω, (3a)

3 Computation of offset-wavenumber integrals ω. (3b) v Tis limitation (3b) of is used trougout tis paper ecept oterwise epressed eplicitly. Te direct summation metod in equation () is considered accurate for te evaluation of te integral in equation (1) trougout tis paper. Te accuracies of te stationary-pase metod and te trapezoidal Filon metod are analysed by comparing tese two metods wit te direct summation metod. STATIONARY-PHASE METHOD Te integrals for wic te stationary-pase metod was originally developed ave te following general form (Murray, 198, and Popovici, 199): b iλ ( ) ( ) () t = g t e f λ dt, () a were a, b, g(t), (t), t and λ are all real. As λ tends to infinite, tis integral can be approimated by I SP = g ( t ) 0 π λ " ( t ) 0 i[ λ( t0 ) + sign( " ( t0 )) π ] e. (5) Tis epression assumes tat function as only one turning point (a relative maimum or minimum), at t=t 0, in te integration range (a, b), wic means tat '( t ) 0, ' ( t0 ) 0 0 = ' and a < t 0 < b. (6) Tis approimation will be referred as I SP in tis paper for simplicity. If tere are more tan one turning point in te interval (a, b), integral () can be simply evaluated by splitting te integral range into one-turning-point segments. One special case is tat one of te integral limits, a, for eample, may be te only turning point in te integral range. In tis case te integral () can ten be approimated by a alf of te value of I SP in (5) (Murray, 198), and it is called te alf I SP approimation. Te stationary-pase I SP approimation for te specific wavenumber integral in equation (1) can be written directly from (5) as I SP π i[ φ ( 0 ) + sign( φ" ( 0 )) π ] = e (7) φ" ( ) 0 wit λ=1, g=1, and =φ, and integral variable being instead of t. Te value 0 is te stationary point in te integral range. For offset-wavenumber integral in equation (1), tere can only be one turning point because te second order derivative of

4 Li and Margrave function φ( ) is non-zero in te wole -range were te integral is evaluated. Eplicitly, te second derivative can be epressed as ( ) ( ) 3 3 v z v z v z v z ω ω ω ω, (8) wic is a summation of four terms wit te same sign (see also Popovici, 199). From epression (7), te evaluation of te integral in equation (1) over a certain range of is reduced to a problem by using te values of φ and φ at only one - location, at wic te derivative φ becomes zero. Altoug φ can be eplicitly epressed as a function of, tere is no straigtforward algoritm to find its zero point (Popovici, 199). Anoter way to locate te stationary-pase point (zero point of φ ) is to find were te function φ yields its maimum value (Alalifa, 1997). Figure 1. (a) sows several curves of φ as functions of wit different z value and all oter related quantities are fied. Only te dar line s turning point is indicated. (b) sows te real part (te dar curve) and te imaginary part (ligt grey curve) of ep(iφ) (corresponding to te dar line in (a)). Te stationary-pase point in (b) is indicated wit a vertical dased line.

5 Computation of offset-wavenumber integrals Figure 1 (a) sows some eamples of te pase function φ for different dept steps and offsets. Te turning points are indicated wit little vertical lines. Figure 1 (b) sows te real part (dar curve) and te imaginary part (grey line) of one of te integrands, ep(iφ), and te stationary-pase point is also indicated wit a dased vertical line. (Te turning point of φ and te stationary-pase point of ep(iφ) always represent te same ting in tis paper.) Figure 1 (b) demonstrates wy te integral of te eponential function ep(iφ) over te range is mainly determined by te beaviour of te function φ (and its derivatives) at te turning point. Te integrand, ep(iφ), oscillates rapidly between 1.0 and 1.0 at all locations ecept te ones close to te stationary-pase point. Te oscillatory values cancel eac oter out during te integration, and te main contribution to te integral comes from te points close to te stationary-pase point. SHORTCOMINGS OF STATIONARY-PHASE METHOD TO EVALUATE OFFSET-WAVENUMBER INTEGRALS In general, te stationary-pase metod as plausible accuracy for te approimation of te offset-wavenumber integral epressed in equation (1). However, attention sould be paid in cases wen te approimation accuracy may not be satisfactory. Te stationary-pase point tends to one of te integral-limits determined by (3b) wen z is relatively small respect to 0. Te asymptotic stationary-pase point is te value were one of te square roots in (1) equals zero. Tis can be seen by eamining te first order derivative of φ, wic is + φ' ( ) = z + 0. (9) ω + ω v v Wen te ratio 0 /z tends to be large, te value of te summation inside te bracet as to be large at te turning point to ensure φ to be zero at te turning point. Terefore, te stationary-pase point, i.e., te turning point, tends to be te value were one or bot of te square-root terms in te bracet yield very small values. In practice, wen te velocity vertical variation is significant (even witout considering lateral velocity variation), wavefield etrapolation wit appropriate accommodation of te velocity variation usually limits te dept step to a muc smaller size relative to te offset range. In suc cases, te stationary-pase point in te non-evanescent region of is very close to te critical point. In addition, te offset-wavenumber is always sampled wit a certain increment, and very often te stationary-pase point cannot be very well separated from te critical point. Figure sows some curves of φ for different step size z s, in a alf-offset range of 0 to 000 m. Te stationary-pase points of all te curves are indicated by "+" symbols.

6 Li and Margrave Figure. Curves of φ for different values of 0,wicissampledfrom0to000mwit100m increment, were (a) sows curves wit z = 50 m and (b) te curves wit z =00m.Allte oter quantities are fied at te same values for all te curves. In Figure, many of te stationary-pase points are practically (numerically) at te integral limit point. Tis results in integration values tat are closer to te alf I SP approimation instead of te full I SP approimation. Some eperiments wit I SP and alf I SP are sown Figure 3, were te approimation errors of te I SP and te alf I SP are plotted. Te alf I SP is a better approimation wen te stationary-pase points tend to one of te integral limits.

7 Computation of offset-wavenumber integrals Figure 3. Approimations of stationary-pase metod using full I SP and alf I SP approimations. (a) sows tree curves corresponding to real parts of accurate values (grey line), I SP values (dased dar line) and alf I SP values (dar solid line) for different 0 values (from 0 to 000 m) and fied z = 50 m; (b) is te approimation errors (real part only) of I SP (dased line) and alf I SP (solid line); and (c) is te absolute error of I SP (dased line) and alf I SP (solid line) relative to te absolute accurate values. In Figure 3(a), it seems tat bot te I SP and alf I SP approimations are quite good because no evident visual deviations can be recognised from te tree curves. In fact, even te approimation errors, as sown in Figure 3(b), are all small numbers, te approimations are unacceptable because te relative errors, as sown in Figure 3(c), are wildly large. Wit relative errors greater tan 100%, some of te I SP and alf I SP approimations involve more errors tan simply taing zero as te integral values. A conclusion can be drawn from te results in Figure 3 tat wen dept-step z is small relative to te alf-offset value 0, te stationary-pase metod sould not be used to evaluate te offset-wavenumber integrals epressed in (1). For eample, wen z = 50 m and 0 = 00 m, te stationary-pase point is very close to te integral limit (Figure (a)), and te integrand is basically oscillatory trougout te wole integral range. Te oscillatory curve sown in Figure (b) is te real part of te integrand function ep(iφ). Practically speaing, te pase is not very stationary at te stationary-pase point in tis eample, and te contribution from te local values of te integrand around te stationary-pase point does not dominate as muc as it does for an integrand function sown in Figure 1(b).

8 Li and Margrave Figure. Function φ (a) and ep(iφ) (real part only) (b) for z=50 m and 0 =00 m. Oter quantities are te same as in Figure 3. In tis case, te stationary-pase point is close to te integral limit. Figure 5: Migration dip-limits limit te offset-wavenumber integral to a smaller range and often mae te stationary-pase point lying outside te integral range. Te solid-line curves are drawn for 90-degree dip limit, te two vertical dased-lines indicate te integral range determined by a 60-degree dip-limit. Many of te stationary-pase points (plotted wit marers + ) lie outside te narrower integral range. In tis figure, z is fied at 500 m, and 0 ranges from 0 to 000 m. It can be seen tat all te approimations for 90-degree dip-limit cases are very accurate, wile te approimations for 60-degree cases become unacceptable wen 0 is greater tan 500 m. Furter more, wen 0 is greater tan 900 m, simply setting te integral value to be zero gives better approimations.

9 Computation of offset-wavenumber integrals For many practical reasons, migration algoritms often involve dip limits. As mentioned above, te widest range of te -integral (1) is determined by te values were te wave-components related to tese -values (and given and ω) are becoming evanescent. A migration dip-limitation of less tan 90 degrees limits to even narrower ranges. In suc cases, te stationary-pase points may often reside outside te integral range even for not very large 0 /z ratio. An eample wit 60- degree dip-limit is sown in Figure 5 were z = 500 m and 0 ranges from 0 to 000 m. Tis involves more difficulties for applying stationary-pase approimation to te offset-wavenumber integrals, or more strictly, te stationary-pase approimation is not valid any more. Te approimation results are sown in Figure 6. Figure 6: Stationary-pase approimations for 90-degree (a) and 60-degree (b) dip limits. Fortunately, in cases were 0 /z is relatively large, φ as a function of is (at least piecewisely) close to a straigt line, i.e., φ~- 0, wic can be seen grapically from te curves in Figure. Tis piecewise linear approimation of te function φ satisfies te conditions of te trapezoidal Filon metod for te evaluation of wavenumber integrals. TRAPEZOIDAL FILON METHOD Te trapezoidal Filon metod (Frazer, 1988) is an approimation algoritm to evaluate an integral wit a general form of Γ f sg ( ) ( p p e ) dp. (10) If te two functions f and g can be approimated piecewisely as linear function, ten trapezoidal Filon metod provides an approimation of te integral (10) as

10 Li and Margrave p f p 1 ( ) ( ) sg p δ ( p) p e dp = sδ sg sg δ ( ) ( f ) δ ( e ) δ fe ( g) sδ ( g), (11) were δ ( p) = p p1, and δ () u = u( p ) u( p 1 ) wit u representing any function of p (f, e sg and g). Te summation is over all te pieces (segments of p) on wic bot f and g can be approimated by straigt lines. Equation (11) is called te trapezoidal Filon approimation. For te integral epressed in equation (1), te variable p becomes, te function f taes te constant value 1, and g is equal to φ. Te constant s is simply te square root of 1. Te trapezoidal Filon approimation (11) becomes 1 iφ e ( ) δ ( ) d = iδ iφ δ ( e ) () φ, (1) and te only condition for te approimation is tat function φ can be approimated by linear function witin eac and every -segment ( 1, ). If possible, different sizes of interval ( 1, ) witin te integral range sould be more efficient. However, determining te lengt of eac ( 1, ) raises anoter problem and may also require more computation time. For simplicity, te following analysis of te Trapezoidal Filon metod is based on equal-size intervals trougout te wole integral range. Figure 7. Trapezoidal Filon and stationary-pase approimations wit dept step z = 50 m and te alf-offset 0 ranging from 0 to 000 meters. (a) sows te integral values evaluated by te accurate metod, two Filon approimations and te stationary-pase approimation. (b) sows te approimation errors of tese metods and (c) sows te relative errors in percentage of te accurate values.

11 Computation of offset-wavenumber integrals For dept step z = 50 m, wic was used for te results sown in Figure 3 and Figure, two different interval size Filon approimation results are sown in Figure 7 wit comparisons wit te results from accurate metod and te stationary-pase metod. In Figure 7 (a), te integral values from different metods are sown togeter, and no significant difference can be seen. In Figure 7 (b), on te oter and, te approimation errors relative to te accurate values sow tat te two Filon approimations are better tan te stationary-pase approimations. Te symbol Filon (3) indicates te results computed using te Filon metod wit linear-function approimation for every 3 consecutive samples, wile Filon (1) stands for te use of te Filon metod wit linear-function approimation for every 1 consecutive -samples. Figure 7 (c) sows te relative errors of te tree approimations. Note tat te Filon approimations ave muc better stability at large offsets. Figure 8. Te relative errors represented as percentages of te absolute values of te approimation errors (wic are comple values) over te absolute values of te accurate results (comple too). Figure 8 represents te relative errors of te Filon approimations wit four different interval sizes, wic are denoted as Filon (3), Filon (1), Filon (30), and Filon (60). Te results in tis figure are obtained wit z = 0 m, wic is more liely te size practically used for wavefield etrapolation wit strong vertical variation. For comparison, te stationary-pase approimations are sown as te dotted lines. Note tat te stationary-pase approimations are not acceptable in tis case, especially for large offsets. EFFICIENCY COMMENTS Altoug te main purpose in eperimenting wit different approimation algoritms is to find a metod wit less computation cost, te accuracy sould always be a concern. Te above analysis concentrates on te ability of te stationary-pase and te trapezoidal Filon metods to provide acceptable approimations of te offsetwavenumber integrals epressed in equation (1). Te eperiment codes were all

12 Li and Margrave implemented in Matlab, and tey may not be optimised. However, te following table of floating-point operations counted by Matlab function flops for different metods sould still provide an intuitive indication of ow fast or slow an algoritm migt be. Tese operations do not include te evaluation of φ as a function of, wic taes time especially for te direct summation metod. Table 1: Number of floating point operations for different algoritms Metods Number of operations Accurate summation 9559 Stationary-pase 671 Filon (3) 1959 Filon (1) 759 Filon (30) 1059 Filon (60) 559 Filon (100) 359 Anoter efficiency aspect of tese metods is te number of -samples were te function φ and/or its derivatives need to be evaluated. Basically, te stationary-pase metod needs te values of φ and its second derivative at only one sample (te stationary-pase point). Te computation cost of stationary-pase metod is te time used to find te stationary-pase point, wic can be epensive. For te trapezoidal Filon metod, te cost is directly related to te interval size were te linear function is used to approimate te function φ. Longer interval results in a faster algoritm, but provides less accuracy. Note tat in Figure 8, Filon (60) provides better approimation, ecept at 0 = 0, 100 and 00 m, and uses less operations tan te stationary-pase metod as sown in Table 1. MULTI-STEP WAVEFIELD EXTRAPOLATION In practical implementation of pase-sift migration algoritm, te wavefield at any dept may be directly obtained by a multi-step etrapolator, suc as te nonstationary migration filters in te v(z) f- algoritm (Margrave, 1998, Li and Margrave, 1998). In tese cases, te offset-wavenumber integral can be written as

13 Computation of offset-wavenumber integrals z ep i dz' i 0 d = v( z' ) v( z' ) + 0 ω ω + e iφ ( ) d. (13) Here te function φ and its derivatives may not ave analytical epressions, suc as equations (8) and (9). Terefore, te application of te stationary-pase approimations may require numerical evaluation of te function φ and its derivatives. Tis may involve some more computation cost tan te single-step wavefield etrapolator may. However, te step-size (still denoted as z) in equation (13) practically is muc larger tan te single dept step in equation (1), and te z-to 0 ratio is no longer a problem ecept for te image of very sallow layers. As sown in Figure 6(a), were z = 500 m and 0 ranges from 0 to 000 m, te stationary-pase approimation as very ig accuracy wen migration dip limit is not an issue. CONCLUSIONS Tis paper presents some numerical results of te evaluation of te offsetwavenumber integrals involved in pase-sift migration of common-offset sections. Te stationary-pase metod may not be appropriate wen wavefield etrapolation dept step is small relative to te offset values, altoug it as been used by many autors to approimate tese integrals. It also sows tat even wen step size is not small, migration dip limit may also introduce difficulties for applying stationarypase metod. Te eperiments on te trapezoidal Filon metod sow tat in te cases were te stationary-pase metod failed to provide accurate approimations, te Filon metod may be a better coice. Te cost of trapezoidal Filon metod can be comparable to stationary-pase metod, and migt be even faster. Te stationary-pase approimation of an integral of some wavenumber function depends on te values of te function at locations close to te stationary pase point. Tis implies tat te values of te function at locations far away from te stationarypase point are not taen into account at all. Our eperiments and te results of oter autors (e.g., Popovici, 199) sow tat stationary-pase approimation metod as very ig accuracy ecept for very small dept step. Multi-step etrapolation sceme, equation (13), can practically overcome te small dept-step etrapolation problem ecept te very sallow part of te imaging results. FUTURE WORK Many autors found tat te practical computation time may most be spent on searcing te stationary-pase point. We propose anoter way to use stationary-pase metod. Tat is, instead of locating te stationary-pase point itself, we could find a muc smaller integral range wic includes te stationary-pase point, and ten calculate te integral only over tis smaller range accurately. Tere sould be faster metods to find suc smaller ranges rater tan to locate te stationary-pase point. REFERENCES Alalifa, Tariq, 1997, Prestac time migration for anisotropic media: 67t Annual Internat. Mtg., Soc. Epl. Geopys., Epanded Abstracts,

14 Li and Margrave Deregowsi, S. M., 1990, Common-offset migrations and velocity analysis: First Brea, 8, no. 6, - 3. Eren, Bjorn O. and Ursin, Bjorn, 1995, Frequency-wavenumber constant-offset migration and AVO analysis: 65t Annual Internat. Mtg., Soc. Epl. Geopys., Epanded Abstracts, Eren, Bjorn O. and Ursin, Bjorn, 1999, True-amplitude frequency-wavenumber constant-offset migration: Geopysics, 6, Ferber, R.-G., 199, Migration to multiple offset and velocity analysis: Geopys. Prosp.,, no., Frazer, L. Neil, 1988, Quadrature of wavenumber integrals, III.3 in "Seismological Algoritms, Computational Metods and Computer Programs", Doornbos, D. J. (Ed.), Academic Press. Frazer, L. Neil and Gettrust, Josep F., 198, On a generalization of Filon's metod and te computation of te oscillatory integrals of seismology, Geopys. J.R. Astr. Soc., vol. 76, Kim, Young C. and Krebs, Jerry R., 1993, Pitfalls in velocity analysis using common-offset time migration: 63rd Annual Internat. Mtg., Soc. Epl. Geopys., Epanded Abstracts, Li, X, and Margrave, G. F., 1998, Prestac v(z) f- migration for PP and PS data, CREWES Researc Report, Vol. 10. Li, X, and Margrave, G. F., 1999, Prestac v(z) f- migration from common-offset sections, CREWES Researc Report, tis volume. Margrave, Gary F., 1998, Direct Fourier migration for vertical velocity variations: 68t Annual Internat. Mtg., Soc. Epl. Geopys., Epanded Abstracts,, Moser, C. C., Keo, T. H., Weglein, A. B. and Foster, D. J., 1996, Te impact of migration on AVO: Geopysics, 61, no. 06, Murray J.D., 198, Metod of stationary-pase, Capter of Asymptotic Analysis, Applied Matematical Sciences Vol. 8, Springer-Verlag New Yor Inc. Popovici, Aleander M., 199, Reducing artifacts in prestac pase-sift migration of common-offset gaters: 6t Annual Internat. Mtg., Soc. Epl. Geopys, Epanded Abstracts, Sattlegger, J. W., Stiller, P. K., Ecteroff, J. A. and Hentsce, M. K., 1980, Common-offset plane migration (COPMIG): Geopys. Prosp., 8, no. 6, Tygel, M., Santos, L. T., Scleicer, J. and Hubral, P. 1999, Kircoff imaging as a tool for AVO/AVA analysis, Te Leading Edge, August 1999, vol. 18, No. 8,