Asymptotic Analysis of the Paradox in Log-Stretch Dip Moveout
aa r X i v : . [ m a t h . A P ] M a r Asymptotic Analysis of the Paradoxin Log-Stretch Dip Moveout
Xin-She Yang
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK.
Binzhong Zhou
CSIRO Division of Exploration and Mining, P.O.Box 883, Kenmore, QLD 4069, AUSTRALIA.
Abstract
There exists a paradox in dip moveout (DMO) inseismic data processing. The paradox is why Notforsand Godfrey’s approximate time log-stretched DMOcan produce better impulse responses than thefull log DMO, and why Hale’s f-k DMO is correctalthough it was based on two inaccurate assumptionsfor the midpoint repositioning and the DMO timerelationship? Based on the asymptotic analysisof the DMO algorithms, we find that any form ofcorrectly formulated DMO must handle both spaceand time coordinates properly in order to deal withall dips accurately. The surprising improvementof Notfors and Godfrey’s log DMO on Bale andJakubowicz’s full log DMO was due to the equivalentmidpoint repositioning by transforming the time-related phase shift to the space-related phase shift.The explanation of why Hale’s f-k DMO is correctalthough it was based on two inaccurate assumptionsis that the two approximations exactly cancel eachother in the f-k domain to give the correct final result.
Citation detail:
X. S. Yang and B. Zhou, Asymp-totic analysis of the paradox in log-stretch dip move-out,
Geophys. Res. Lett. , (3), 441-444 (2000). INTRODUCTION
Dip Moveout (DMO) technique has been widely usedin seismic data processing over the past decade andmany different algorithms have been developed [
Hale,1984; Bale and Jakubowicz, 1987; Notfors and God-frey, 1987; Liner, 1990; Gardner, 1991; Black etal., 1993; Zhou et al., 1996 ]. Hale’s DMO is ac-curate for all reflector dips and has become an in-dustrial standard, but it is computational expensiveand temporally non-stationary. In all the methods,a logarithmic time-stretching technique [
Bolondi et al., 1982 ] is widely used, probably due to its com-putational efficiency and easy implementation. Theimpulse response produced by Bale and Jakubow-icz’s full log-stretch operator (hereafter referred to asBale’s full log DMO) in the frequency-wavenumber (f-k) domain is inaccurate although no approximationswere made in the mathematical formulations. Theimpulse response is surprisingly improved by Notforsand Godfrey’s approximate log-stretch scheme (here-after referred to as Notfors’ log DMO). Bale’s inaccu-rate DMO impulse responses reveal that Hale’s DMOderivation had some inappropriate assumptions. Toimprove the DMO impulse responses, an exact log-stretch DMO was derived by transforming Hale’stime log-stretch impulse response into the Fourier do-main [
Liner, 1990 ] .
Black et al. [1993] providedanother derivation of the more reliable f-k DMO bytreating the reflection-point smear from dipping re-flectors correctly. Based on Black’s DMO relation-ship,
Zhou et al. [1995, 1996] presented a log-stretchf-k DMO which is similar to Gardner’s DMO [
Gard-ner, 1991 ] but from a derivation with more directphysical insight into the DMO process. All theseDMO schemes were developed to try to generate bet-ter impulse responses and to improve the computa-tional speed, but have different corrections of Hale’ssubtle flaw and improving Bale’s inaccurate impulseresponses.Despite extensive studies and the routine use of theDMO algorithms, there still exists a paradox whichhas not yet been explained satisfactorily. The para-dox is why Notfors’ approximate log DMO can pro-duce better impulse responses than Bale’s full logDMO, and why Hale’s f-k DMO is correct althoughit was based on two inaccurate assumptions. Linerpointed out [
Liner, 1990 ] that Bale and Jakubow-icz’s DMO derivation implicitly assumes that theFourier transform frequency in the log-stretch domainis time-independent. Black et al. attributed Hale’ssubtle flaw to the lack of midpoint change and the1onsequently improper treatment of the reflection-point smear from dipping reflectors. Zhou et al.suggested that the approximations from the non-repositioned midpoint and from the incorrect timerelationship before and after DMO may counteracteach other in the f-k domain to make the final resultcorrect, but they may not cancel each other in the log-stretch f-k domain. In this paper, we analyse theseviewpoints and compare DMO algorithms to find outwhat is really responsible for the inaccurate impulseresponses of Bale’s full log DMO scheme. For con-venience in the following discussion, only common-offset DMO forms will be analysed although simi-lar methodology can be also applied to common-shotDMO analysis.
ASYMPOTIC ANALYSIS OFPHASE SHIFT FUNCTIONS
We assume a constant-velocity medium and followthe notations used earlier [
Hale, 1984; Notfors andGodfrey, 1987; Black et al., 1993; Zhou et al., 1996 ].The heuristic DMO mapping can be expressed by P ( t , x , h ) = P n ( t n , x n , h ) , (1)where P n ( t n , x n , h ) is a normal-moveout (NMO) cor-rected input section and P ( t , x , h ) is a zero-offsetDMO output section. h is the half offset, t n and x n are time and space in the NMO-corrected domain re-spectively, and t and x are the time and space inDMO-corrected domain, respectively. The time log-stretch transform pair [ Bolondi et al., 1982; Bale &Jakubowicz 1987 ] τ = ln( t/t c ) and t = t c e τ , (2)transforms the time coordinates from ( t , t n ) to thelog-stretch variables ( τ , τ n ). t c is the minimum cut-off time introduced to avoid the singularity of thelogarithm at zero.The DMO processing algorithm can be divided intothe following five steps [ Liner, 1990; Gardner, 1991;Hale, 1991 ]: 1) transform NMO corrected input data P n ( t n , x n , h ) to P n ( τ n , x n , h ) by the time log stretchrelation (2), 2) 2-D FFT of P n ( τ n , x n , h ) over τ n and x n to P n (Ω , k, h ), 3) use the phase factor for filteringin the (Ω , k ) domain to get P (Ω , k, h ), 4) inverse 2-DFFT of P (Ω , k, h ) over Ω and k to P ( τ , x , h ), and5) inverse log stretch of P ( τ , x , h ) by the inverserelation of equation (2) to get the output P ( t , x , h ).Of all the available log stretch DMO algorithms, thesame transform relation (2) is used in steps 1) and 5),and the two-dimensional FFT transform pair is also used in steps 2) and 4). The main difference betweenalgorithms is thus the multiplying phase factor in step3). The following discussion will therefore focus onthe analysis of the geometry-determining phase factorin different log-stretch DMO algorithms. As we willsee, all the phase shift functions depend mainly onthe variable ξ = hk Ω . (3)Mathematically speaking, ξ can vary from 0 (zero dipat the centre of the DMO ellipse) to ∞ (very shallow,steeply dipping). Bale’s Full Log DMO
From Hale’s DMO formulation,
Bale and Jakubowicz [1987] derived the full log stretch DMO P (Ω , k, h ) = e − i Ωln(1 − ξ ) P n (Ω , k, h ) , (4)However, this algorithm does not produce good im-pulse responses. Liner [1990] attributed this featureto the fact that Bale and Jakubowicz’s derivation im-plicitly assumes that Ω is not time-dependent. Infact, Bale’s phase shift functionΦ F = −
12 Ωln(1 − ξ ) , (5)is only physically meaningful when ξ < ξ ≥ inverted Gaussian shape. Notfors’ Log DMO
An approximation to (4) was derived by
Notfors andGodfrey [1987] P (Ω , k, h ) = e i Ω( √ ξ − P n (Ω , k, h ) , (6)which is based on two approximations: (1) the inde-pendence of Ω and τ n , and (2) ( hk/ Ω) <<
1. Withthese approximations, Bale’s phase shift function be-comes Notfors’ phase shift functionΦ N = Ω( p ξ − , (7)which is meaningful for any value of ξ (all reflectordips). This derivation implicitly extends the defini-tion range of ξ although the approximations are basedon ξ <<
1. It is this second approximation thatmakes Notfors’ log DMO produce better impulse re-sponses than Bale’s full log DMO (4).2 iner’s Log DMO
The exact log stretch DMO derived by
Liner [1990]is P (Ω , k, h ) = e i [Ω∆ s − ky s ] (1 + β s ) / P n (Ω , k, h ) , (8)where β s = y s /h, (9)∆ s = 12 ln(1 − β s ) , (10)and the stationary point is y s = h ξ (1 − p ξ ) . (11)Liner’s derivation introduces log stretch variablesinto Hale’s ( t, x ) elliptical response rather than intothe NMO equation, as was done earlier by Baleand Jakubowicz [1987]. The formula (8) does yieldthe correct impulse response geometry [
Liner, 1990;Zhou et al., 1996 ]. Zhou’s Log DMO
Zhou et al. [1996] presented an accurate log stretchDMO P (Ω , k, h ) = e i Ω[ √ ξ − − ln[ ( √ ξ +1)] P n (Ω , k, h ) , (12)which is equivalent to Gardner’s results except for asign difference on the phase term due to a differentdefinition of the 2-D Fourier transform in Gardner’sformulation. Zhou et al. [1996] derived their resultbased on the DMO relationships given by
Black etal. [1993]. It is easy to check that the phase shiftin equation (12) is identical to that in equation (8).Therefore, Zhou’s log DMO and Liner’s log DMO willproduce the same impulse response geometry. Theonly difference is that Zhou’s DMO will yield slightlylarger amplitudes than Liner’s DMO because (8) hasan amplitude factor 1 / (1 + β s ) / ≤
1. For con-venience in the following analysis, we will refer toLiner’s log DMO and Zhou’s log DMO as the exactlog DMO, and will use the simpler phase shift expres-sionΦ E = 12 Ω[ p ξ − − ln[ 12 ( p ξ + 1)] , (13)which is accurate for all reflector dips.All the phase shift functions (5), (7) and (13) aremonotonically increasing, therefore, we discuss onlytwo asymptotic cases. Case 1: ξ << For ξ →
0, equations (5), (7), (13) all becomeΦ
F,N,E ≈
12 Ω ξ , (14)which means that all these log-stretch DMO algo-rithms are asymptotically equivalent near zero dips( ξ →
0) and will yield virtually the same impulse re-sponse geometry near the center of the DMO ellipse.
Case 2: ξ >> As the reflector dips become shallow and steep (for ξ >> N ≈ Ω ξ, (15)and Φ E ≈ Ω ξ (1 − ln ξ ξ ) , (16)respectively. Using ln ξξ → ξ → ∞ , we can write(15) and (16) as a single asymptotic formΦ N,E ∼ Ω ξ, (17)which means that Notfors’ log-stretch DMO and theexact log-stretch are asymptotically equivalent butthey are not approximately equal due to the term O ( ln ξξ ) in (16). This implies that the exact logDMO is able to deal with all reflector dips correctly,but Notfors’ log DMO mishandles the shallow, steepdips although it is an improvement on Bale’s DMOscheme. THE EXPLANATION OF THEPARADOX
Based on the above analysis, we can explain the para-dox of why Notfors approximate log DMO improvedBale’s full log DMO, and find out the real flaw inHale’s formulation. The f-k integral for common-offset in
Hale’s
DMO [1984] is P ( ω, k, h ) = Z Z P ( t , x , h ) e i ( ωt − kx ) dx n dt n = Z Z A e i ( ωt n A − kx n ) P n ( t n , x n , h ) dx n dt n , (18)where A = r ht n dt dx ) = r hkt n ω ) . (19)3nd the DMO mapping (1) and the following relationsof time and space coordinate are used t = At n and x = x n , (20)which are not strictly correct and make the DMO op-erator amplitude-unpreserved. The similar time andspace relationships were used by Bale and Jakubow-icz [1987] in their full log DMO derivation.
Black etal. [1993] derived a new amplitude preserving DMOintegral P ( ω, k, h ) = Z Z P ( t , x , h ) e i ( ωt − kx ) dx n dt n = Z Z A − A e i ( ωt n A − kx n ) P n ( t n , x n , h ) dx n dt n . (21)by substituting (1) and basing on Black’s correct re-lationship of time and space coordinates before andafter DMO processing t = t n A and x = x n − A h t n kω , (22)where the definition of A is the same as in (19).From equation (22), we can easily derive the DMOimpulse ellipse (Zhou et al., 1995) while the rela-tionships in equation (20)give an impulse point. As A ≥
1, Black’s correct relationship (22) implies theinequality t ≤ t n . (23)It is clearly seen that phase shifts in Black’s DMO(21) and in Hale’s DMO (18) are identical [ Blacket al., 1993 ] due to the identical phase factorexp[ i ( ωt n A − kx n )]. This implies that the two approx-imations resulting from the inaccurate relationships(20) exactly cancel each other in the f-k domain, andthus make the final result correct. However, the coun-teraction of the two approximations does not give ex-act cancellation in Bale’s log-stretch DMO algorithmbecause only the time coordinate is stretched and the space coordinate remains unchanged. Therefore, weexpect that the phase shift in Bale’s log DMO is es-sentially due to the log-stretch traveltime relationship τ − τ n = −
12 ln(1 − ξ ) . (24)This implies the incorrect inequalities τ ≥ τ n and t ≥ t n , (25)which contradict the correct relation (23). Notforsand Godfrey [1987] used the approximate version of(24) τ − τ n = p ξ − , (26) which yields the same incorrect inequality as in (25).From the general forms of log-stretch DMO algo-rithms, we know that the phase shift functionΦ = k ( x n − x ) − Ω( τ n − τ ) , (27)consists of two parts: a space-related phase shift k ( x n − x ) and a time-related phase shift Ω( τ n − τ ).The asymptotic analysis in the previous section re-veals that the space-related phase shift is alwaysgreater than the time-related phase shift. Compar-ing the phase shift functions (7) and (13), we see thatNotfors approximate log DMO virtually transformedthe time-related phase shift into its space-relatedcounterpart, and thus equivalently repositioned themidpoint x = x n + hξ (1 − p ξ ) , (28)where the second term on the right hand side is simi-lar to the expression (11) of the stationary point y s inLiner’s exact log DMO. It is this equivalent midpointrepositioning that makes Notfors’ approximate logDMO overcome Hale’s subtle flaw and subsequentlyproduces a better match to the geometry of the im-pulse response DMO ellipse, but the Notfors’ widergeometry of the DMO ellipse shows that the shallowsteep dips are still improperly treated. CONCLUSIONS
The analysis of different log-stretch DMO algorithmshas shown that any form of correctly formulatedDMO must handle both space and time coordinatesproperly in order to deal with all dips accurately.The log-stretch DMO algorithms presented by
Liner [1990] and
Zhou et al. [1996] are able to treat allreflector dips correctly. Bale and Jakubowicz’s fulllog DMO is only approximately correct for the nearzero dips, and it becomes singular for shallow, steepdips. Notfors and Godfrey’s approximate log DMOis a great improvement in handling all dips, but itsimpulse response still departs from the exact DMO el-lipse, especially for shallow steep dips. The surprisingimprovement of Notfors and Godfrey’s log DMO onBale and Jakubowicz’s full log DMO was mainly dueto the equivalent midpoint repositioning by equiv-alently transforming the time-related phase shift tothe space-related phase shift. The explanation of whyHale’s f-k DMO is correct although it was based ontwo inaccurate assumptions is that the two approxi-mations exactly cancel each other in the f-k domainto give the correct final result.4 cknowledgements.
The authors wish to thankthe anonymous referees for their very helpful com-ments. This research was supported by CSIRO Ex-ploration and Mining in Australia and the Cooper-ative Research Centre for Mining Technology andEquipment.
REFERENCES
Bale, R., and Jakubowicz, H., Post-stack prestackmigration, 57th Ann. Mtg.,
Soc. Exp. Geophys. ,Expanded Abstract, 714-717, 1987.Black, J. L., Schleicher, K. L., and Zhang, L.,True-amplitude imaging and dip moveout:
Geo-physics , , 47-66, 1993.Bolondi, G., Loinger, E., and Rocca, F., Offset con-tinuation of seismic sections, Geophys. Prosp. , , 813-828, 1982.Gardner, G. H. F., Interpolation, crossline migra-tion and inline depth migration of 3-D marinesurveys, The SAL Annual Progress Review , ,57-69, 1991.Hale, D., Dip moveout by Fourier transform, Geo-physics , , 741-757, 1984.Liner, C. L., General theory and comparativeanatomy of dip moveout, Geophysics , , 595-607, 1990.Notfors, C. D., and Godfrey, R, J., Dip moveout inthe frequency-wavenumber domain, Geophysics , , 1718-1721, 1987.Zhou, B., Mason, I. M., and Greenhalgh, S. A.,An accurate formulation of log-stretch dip move-out in the frequency-wavenumber domain, Geo-physics , , 815-821, 1996.Zhou, B., Mason, I. M., and Greenhalgh, S. A., Ac-curate and efficient shot-gather dip moveout pro-cessing in the log-stretch domain, GeophysicalProspecting ,43