G048 Resolution Analysis of Seismic Imaging SUMMARY

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1 G048 Reolution nalyi of Seimic Imain R.S. Wu (Univerity of California), X.B. Xie (Univerity of California, Santa Cruz), M. Fehler (Lo lamo National Laboratory) & L.J. Huan (Lo lamo National Laboratory) SUMMRY We define and formulate the reolution of an imain ytem baed on the invere theory and local anle domain decompoition of Green function. The reolution defined in thi way include both the effect of acquiition ytem and imain (miration) proce. The theory and method are baed on wave theory and no aymptotic approximation i made in the calculation. It repreent and quantifie the actual reolution we oberved in the mirated imae. The reolution takin into account only the influence of the acquiition ytem (frequency band and patial aperture) and nelectin the factor uch a the error in backpropaation (miration) can be conidered a the reolution limit of the ytem, the bet reolution an acquiition ytem can et. Theoretical analyi and numerical example are iven to how the importance of propaator accuracy in the evaluation of reolution in complex media. EGE 68 th Conference & Exhibition Vienna, utria, June 2006

2 Introduction Spatial reolution ha been tudied by Beylkin et al. (1986) baed on eneralized Radon tranform and the mappin of domain interation (frequency band and patial aperture) into a pectral coverae in patial frequency domain (wavenumber domain). The mappin i done by imple ray-tracin which doe not take into account of the frequency dependence and other wave phenomena. Since then, the topic ha been invetiated by many author from different point of view (Chen and Schuter, 1999, 2001; Geliu et al., 2002; Gibon and Tzimea, 2002). While thee analye are ueful tool for the reolution problem, mot analye are formulated to calculate the influence of acquiition ytem to reolution, and did not look at other factor in the imain/miration proce, uch a the accuracy of the propaator, error in velocity model, etc. Some author called uch a reolution a the reolution limit, the bet reolution of an acquiition ytem can et. In thi reearch, we formulate the reolution of imain ytem, which include the acquiition ytem and imain (miration) proce. The analyi i baed on the invere theory and the local anle domain decompoition of Green function. Thi reolution of imain ytem repreent and quantifie the actual reolution we oberved in the mirated imae and provide a theoretical bai for the etimation of different factor influencin the reolution and imae quality in complex media. The other important feature of our formulation i that the reolution problem i treated totally baed on wave theory and no hih-frequency aymptotic approximation i made. Formulation of patial reolution baed on invere theory Spatial reolution can be tudied under the eneral frame of inverion theory. Reolution operator or the dicrete form, reolution matrix ha been defined to quantify the reolution of parameter inverion by a particular inverion cheme (ki and Richard, 1980; Tarantola, 1987). The reolution matrix ha dependence on both the acquiition ytem and the inverion cheme. ume the acquiition proce can be modeled by Fm = d (1) where F i the forward modelin operator and d i the data. If we adopt a pecific inverion operator B to apply to the data we can et a et of model parameter m I, which i different from the real m, mi = Bd (2) Subtitutin (2) into (1) we et the relation between the inverted model and the real model mi = BFm, (3) and the reolution matrix (or operator) i defined a R = BF (4) For an exact inverion of a well-poed problem, we hould have R = I (5) where I i the identity matrix. For mot the cae, the reolution matrix i not an identity matrix and the preadin of the matrix element alon the diaonal ive ome quantitative meaure of the parameter reolution of the inverion. For the ake of implicity, here we will ue the noiele formulation. For the tochatic approach (Tarantola, 1987) a imilar derivation can be obtained. The imain problem can be formulated a a pecific invere problem. We aume the media can be decompoed into a mooth variation of velocity and a harp jump of impedance (dicontinuitie). The velocity ditribution of the backround media can be derived with different approache and i aumed known in the imain problem. The unknown in the imain problem are the parameter trenth and their location (ditribution). For the problem of patial reolution, we aume the catterin coefficient everywhere are unity and therefore are known. Then the reolution matrix i totally defined by the patial reolution. In the followin, we ue the miration operator (imply backpropaation interal) a the inverion operator (here the imain operator) to how the effect of different factor to the final reolution. We can write the pace-domain formulation for modelin (acquiition proce) and imain (miration) a: EGE 68 th Conference & Exhibition Vienna, utria, June 2006

3 2 x u( ω, x; x) = k dx' GM( ω, x'; x) ( x) GM( ω, x; x') (6) x ( x) V G (, ; ) I ω x x I( x) = dω dxw ( ω, x) GI ( ω, x; x)2 dxw( ω, x) u( ω, x, x ) (7) where u i the cattered wave field oberved on the urface at x excited by a ource on the urface at x. G M i the Green function of the modelin (acquiition) proce, which may include all the factor (eometric preadin, intrinic and catterin attenuation, boundary catterin, etc.) for the real heteroeneou media; G I i the Green function of inverion proce, which could be quite different from G (ee Fiure 1). The interation on the M receiver aperture and ource aperture are for the pretack miration proce. W are the weihtin function for interation. The interation on the receiver aperture i the Rayleih interal which imulate the backpropaation proce. The interation on frequency i from the imain condition which tate that the downward extrapolated ource field and the cattered field will meet at zero time at the catterin point. In (7) we ued the crocorrelation imain condition. Other imain condition for correctin the imain amplitude can be alo applied, but the eneral concluion of reolution analyi will not be influenced. Fiure 1. Schematic diaram howin the modelin (data acquiition) and imain (inverion) procee. Write modelin and imain proce (6) and (7) into operator form, reultin in, = ( ω, x ( x0 U( ω, x x ) F, x x S ) (8) I( x) = B( x ω, x, x ) U ( ω, x, x ) (9) where F i the acquiition (modelin) operator and B i the imain operator which invert the data U into the imae I. Therefore the reolution operator i obtained a R( x, x0 ) = B( x ω, x, x) F ( ω, x, x x (1 The kernel for the reolution operator can be obtained a 2 = dωk dxw ( ω, xx, ) GI ( ω, xx ; ) GM( ω, x0; x) (11) ( ω, x; x) 2 dxw( ω, x, x) GM( ω, x; x For zero-offet (zero receiver aperture) acquiition, or monotatic meaurement in radar terminoloy, the above defined reolution matrix i deenerated to 2 = 2 dωk dxw ( ω, xx, ) GI ( ω, xx ; ) (12) ( ω, x; x) GM( ω, x0; x) GM( ω, x; x Note that the amplitude function in (12) i different from the explodin-reflector modelin. In the latter cae, (12) i further implified to

4 G (, ; ) 2 I ω x x = 2 dωk dxw ( ω, xx, ) GM( ω, x; x The above derived reolution matrix (operator) i in fact the patial reolution matrix (reolution operator) for the whole acquiition and miration proce (the imain ytem). The matrix element R( xx, i called the reolvin kernel of the reolution operator, which i the point preadin function (PSF)of the imain ytem. In thi way the point preadin function (impule repone) i defined under the uidance of eneral inverion theory. If we et the catterer ditribution ( x') = δ ( x' x0 ) in equation (6) and ubtitute it into equation (7), we ee the equivalence of the imain proce to the calculation of reolution matrix. Thi ive u the numerical procedure of calculatin the reolution matrix or PSF for any imain ytem. If we aume that an exact Green function G i ued a the invere propaator M G, I then the effect of imperfect propaator can be eliminated, and the reolution i totally determined by the data aperture. The reolution derived thi way i a theoretical limit of the acquiition ytem imilar to the reolution tudied by previou invetiation (Beylkin et al., 1986; Geliu et al., 2002; Gibon and Tzimea, 2002). (13) nular-pectral repreentation of point preadin function (PSF) nular-pectral repreentation of reolution of PSF i more intuitive. We can ee directly the information coverae in the local anle domain. We perform local 3D Fourier tranform on R( xx, with repect to x with coordinate center at x 0 : iki ( x x0 RKx (, ) = dxe W( xx, v (14) where K i the 3D wavenumber vector and W( x, x i a 3D window function to localize. For implicity, we et the weihtin function W and W in miration a unity. Subtitutin (11) into (14) we obtain 2 RKx (, = dωk dk dxg I ( ω, x0, K; x) GM( ω, x0; x) f (15) ( ω, x0, K K; x) 2 dx GM( ω, x; x where G( ω, x0, K; x') i the beamlet decompoition (or local plane-wave decompoition) of the Green function (Wu and Chen, 2002) around x, and x i the ource or receiver poition. 0 The influence of propaator accuracy to the reolution To how the difference between the real reolution for an imain ytem and the theoretical reolution limit, we ive an example of imain uin explodin reflector data enerated by a full-wave finite difference alorithm, in a random medium (Fiure 2). We ue both the ray-kirchhoff miration method and the dual-domain one-way wave propaator method a the miration operator. Since the medium ha random heteroeneitie with cale comparable to the wavelenth of the central frequency, the ray approximation of the Green function can produce lare error for lon rane propaation. Thi error ha evere conequence on the reolution matrix of the imain ytem. hown in Fiure 2, both the multi-arrival and firt-arrival ray-kirchhoff miration ive ditorted point preadin function, and the horizontal reolution are alo deenerated (bottom panel of Fiure 2). Concluion The reolution of an imain ytem defined in thi paper baed on the invere theory include both the effect of acquiition ytem and miration proce. It repreent and quantifie the actual reolution we oberved in the mirated imae. If we conider only the influence of the acquiition ytem, the reolution obtained i the reolution limit of the EGE 68 th Conference & Exhibition Vienna, utria, June 2006

ytem. Theoretical analyi and numerical example have hown the importance of propaator accuracy to the evaluation of reolution in complex media.

The reearch i upported by the DOE/BES Project jointly conducted at the Univerity of California, Santa Cruz and Lo lamo National Laboratory and the

and Miller, D., 1986, Spatial reolution of miration alorithm, coutic imain, v. 14, 155-168. Chen, J. and Schuter, G.

, 2002, nalyi of the reolution function in eimic pretack depth imain, Geohpy. Propect., 50, 505-515. Gibon, R.L. Jr. and Tzimea, C.

, 1987, Invere Problem Theory, Method for data fittin and model parameter etimation, Elevier, New York. Wu, R.S. and L.

5 ytem. Theoretical analyi and numerical example have hown the importance of propaator accuracy to the evaluation of reolution in complex media. cknowledement. The reearch i upported by the DOE/BES Project jointly conducted at the Univerity of California, Santa Cruz and Lo lamo National Laboratory and the WTOPI Reearch Conortium at UCSC. Reference ki, K. and Richard, P., 1980, Quantitative Seimoloy, W.H. Freeman and Company. Beylkin, G., Otitalio, M. and Miller, D., 1986, Spatial reolution of miration alorithm, coutic imain, v. 14, Chen, J. and Schuter, G., 1999, Reolution limit of mirated imae, Geophyic 64, Geliu, L.-J., Lecomte, I. nd Tabti, H., 2002, nalyi of the reolution function in eimic pretack depth imain, Geohpy. Propect., 50, Gibon, R.L. Jr. and Tzimea, C., 2002, Quantitative meaure of imae reolution for eimic urvey dein, Geophyic, 67, Tarantola,., 1987, Invere Problem Theory, Method for data fittin and model parameter etimation, Elevier, New York. Wu, R.S. and L. Chen, 2002, Mappin directional illumination and acquiition-aperture efficacy by beamlet propaator, 72nd nnual Internat. Mt., Soc. Expl. Geophy., Expanded btract, Fiure 2: Top: velocity model of point catterer in a random medium; Middle: point preadin function uin the multi-arrival and firt-arrival ray-kirchhoff miration operator; Bottom: horizontal reolution curve of the ame acquiition ytem but different miration operator

Removal of water layer multiples and peg-legs by wave-equation approach

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