NON-LINEAR INVERSION OF SEISMIC DATA FOR THE DETERMINATION OF REFLECTOR GEOMETRY AND VELOCITY STRUCTURE

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1 JOURNAL OF BALKAN GEOPHYSICAL SOCIETY, Vol.8, No 3, Augut 005, p NON-LINEAR INVERSION OF SEISMIC DATA FOR THE DETERMINATION OF REFLECTOR GEOMETRY AND VELOCITY STRUCTURE Panteli M. Soupio (), Contantino B. Papaacho (), Gregory N. Toka (), Antoni Vafidi (3) () Laboratory of Geophyic & Seimology, Department of Natural Reource and Environment, Technological Educational Intitute of Crete, Chania, Greece () Geophyical Laboratory, Aritotle Univerity of Thealoniki, P.O. Box , Macedonia, Greece (3) Applied Geophyic Laboratory, Department of Mineral Reource Engineering, Technical, Univerity of Crete, Chania, Greece Abact: A nonlinear method i ued to compute firt eimic arrival and lection avel time for velocity model with complex lector geomey. The method ue a combination of raction and lection avel-time for imultaneou determination of velocity and interface depth of the model. The elatic wave aumed to be anmitted or lected at interface at which the raypath atify Snell law. The avel time of critically racted wave and derivative maix are computed by applying a revied ray bending method, upplemented by an approximate computation of the firt Frenel one at each point of the ray. Travel-time and raypath of lected wave are computed by applying a fat finite-difference cheme baed on a olution to the eikonal equation and following the avel-time gradient backward from the receiver to the ource, repectively. A damped leat quare inverion cheme i ued to reconuct the velocity model above the lector and the geomey of the interface by minimiing the difference between oberved and calculated (direct, racted and lected) avel-time. In order to reduce inverion artifact both damping and moothing regulariation factor are applied. The applicability of the propoed method i teted uing ynthetic data. Thi imultaneou inverion cheme i appropriate for tatic eimic correction, a well a determination of lateral velocity variation and lector geomey. INTRODUCTION Travel-time tomography i primarily ued by eimologit to tudy the 3-D velocity variation in the upper crut (e.g Papaacho and Nolet, 997). Thi method wa firt inoduced by Aki, Chritofferon and Huebye (977) who divided a horiontally layered media into a Carteian grid of cell with value of lowne aigned at each cell and calculated the lowne diibution by inverting the oberved avel time. The next tep wa the application of the ame method in problem with two type of unknown parameter, uch a, the earthquake parameter and the velocity ucture (e.g Aki and Lee, 976; Thurber, 983). The tomography method of Aki et al. (977) wa adapted to lection data by Bihop et al. (985) and Farra and Madariaga (988). Reflection tomography ha been ued for eimic data analyi primarily by peoleum companie, proceing a large number of recording for different ource - receiver location to etimate the uburface velocity ucture and lector location. Until very recently, application of avel-time inverion method to thee two different type of data (firt break and lection aveltime) had been carried out 74

2 Panteli M. Soupio et al independently. Recently, raction and lection data have been proceed imultaneouly in order to obtain a better knowledge of the tudy area (Zhang et. al., 998; Zelt et. al., 999) by applying imultaneou eimic raction and lection tomography for determination of hallow and/or deep velocity and interface ucture. In lection tomography, we can parameterie the lector and velocity field in ome uitable fahion and imultaneouly invert them. Thi approach i very atactive ince it eat both inverion parameter on the ame bai (Bihop et al., 985; Farra and Madariaga, 988; Hobro et al., 003; Trink et al., 005). However, becaue of the ade-off between media velocity and lector depth (the velocity-depth ambiguity e.g Stork and Clayton, 986; Bickel, 990; Line, 993; Ro, 994; Tieman, 994) the lector interface and velocity cannot be uniquely determined without a priori information on lector location. It i well known that raction tomography provide better longwavelength velocity information, with no or very little conol on the configuration of lecting interface. Theore, if both raction and lection data (wide-angle, normalincidence) are available along a profile, imultaneou inverion of both dataet can provide better conol on both velocity and lector geomey. In the preent tudy we examine poible improvement of the tandard imultaneou tomography uing ray theory. For the avel time of the critically racted wave we follow the approach developed in a previou work (Soupio et al., 00). Specifically, initial ray are calculated uing a fat, approximate ray acing method, known a the ray initialier (Thurber and Ellworth, 980), which i equivalent to olving a one-dimenional approximation to the actual threedimenional problem. Thi method provide an initial etimate of the minimum-time ray path in heterogeneou media for the final ray acing with a revied ray bending technique (Moer et al., 99). Moreover, knowing, that the propertie of the ray are influenced not only by the ucture along the ray, but alo by the ucture in ome "vicinity" of the ray, we incorporate phyical ray by etimating the approximate width of firt Frenel volume which i frequency dependent (Cerveny and Soare, 99, Woodward, 99). The avel-time of the lected wave are computed by a finitedifference algorithm which incorporate propagation of wave in complex velocity model (Hole and Zelt 995). Snell law for lection i ued in the vicinity of the lecting interface. The lector model i allowed to vary moothly with depth, which mean that the depth to the interface d i,k could take any real value and it i not conained to lie at grid node. The advantage of thi method i that avel-time are computed imultaneouly for all receiver. The raypath and the depth derivative are tored in order to conuct the derivative maix. A continuou lector i alo ued and repreented by a erie of upport point whoe depth varie a function of horiontal offet. Ray are aced backward in time from receiver location down to lecting interface and then back up to ource location to determine the expected avel-time (forward modeling). The reult i a ytem of linear equation coniting of two type of parameter, correponding to lowne and lector depth. It i generally aumed (Anderon et. al., 984) that the inverion problem i overdetermined and hence, can be olved in a leatquare ene by minimiing the data mifit. Since there i frequently a 75

3 NON-LINEAR INVERSION OF SEISMIC DATA certain degree of non-uniquene or ill conditioning in the problem, the tandard approach i to conuct a "leat-ucture" olution (Franklin, 970; Tarantola and Nerceian, 984; Contable et al., 987) by conidering additional conaint and uually minimiing the model and/or one of the model derivative norm. The invere problem i olved by either the Levenberg-Marquardt (L-M) method (alo known a the damped leat quare method-levenberg, 944; Marquardt, 963), or by uing both a damping factor and a roughne term, in the form of e.g. a econd derivative moothing filter to the lowne model (Contable et al., 987). To how the efficiency of our algorithm, we conider three problem: firt, we invert avel-time to olve only for the velocity ucture, then we invert firt arrival and lection aveltime to determine only the geomey of the optimum lector and finally we ued the avel-time to imultaneouly olve for depth and velocity ucture. We apply the method uing 3D ynthetic data. The propoed algorithm i ueful for hallow eimic urvey (Soupio, 00) a well a, in the determination of deeper ucture depending on the cale of the problem. FORWARD MODELING Model parameteriation In local inverion tudie, a flat layered Earth model i commonly ued (Aki and Lee, 976). A imilar aumption wa alo adopted here becaue of the mall cale of our tudy area and the convenience of the carteian coordinate ytem, which ignificantly peed up all computation, epecially thoe concerning the 3-D ray acing. Our model i compoed of a threedimenional (3-D) grid of block of ame ie, with value of lowne aigned at the grid point. The lowne at a point which i located between node to 8 i calculated by ilinear interpolation, a given by the following equation, lowne=wf *(wfy *(wfx *n+wfx * n)+wfy *(wfx *n3+wfx *n4))+ wf*(wfy*(wfx*n5+wfx*n6)+wfy *(wfx*n7+wfx*n8)) () where n,n,,n8 denote the lowne value at the node of the grid block and wf(x,y,), are the weight. Either mooth velocity model or model with velocity dicontinuitie can be eaily ued (Hole and Zelt, 995). We alo incorporated a velocity dicontinuity interface, which i defined by a et of point connected by ilinear interpolation. The main advantage of the method i that the lector can be located anywhere in the model (Hole and Zelt, 995) and not necearily lie on the lowne grid node. Ray Tracing The ucceful olution of the forward modelling involve the accurate calculation of the avel-time for the two type of wave (racted and lected) and the etimation of the partial derivative of the avel-time with repect to both velocity and depth at the node (baed on the tarting velocity and lector model). Ray for the firt arrival are calculated uing a revied bending algorithm (Moer et al., 99), which ha uperior convergence propertie. In an effort to alleviate problem uch a convergence and execution peed aociated with the bending method, we incorporated the approximate ray acing cheme of Thurber and Ellworth (980). In thi approach, an equivalent one-dimenional (-D) ucture i calculated from the 3-D model between each ource-receiver pair. The velocitie at the grid point 76

4 Panteli M. Soupio et al that fall within the model are harmonically averaged, which i equivalent to arithmetically averaging the lowne. Thee average value are ued a layer velocitie for the onedimenional model, where the direct and all poible racted wave are conidered. The direct ray i determined uing a combination of the fale poition method and the ecant method (Pre et al., 988), while racted ray are handled uing time term (Officer, 958). The minimum avel-time path from thi ray acing routine i ued a the initial ray path gue for our bending ray acing routine (Moer et al., 99). In thi technique the ray i repreented by a erie of upport point which decribe the ray path with the ue of Beta-pline (Pereyra, 99). The avel time along the Beta-pline curve can be calculated with the apeoidal rule. Finally, we defined the width of the approximate Frenel volume for each point along the raypath from the ource to the receiver, a uggeted by Soupio et al. (00). The firt Frenel i incorporated by applying a imple algebraic manipulation on the element of the raypath maix, M. Thi operation diibute the value of the maix element which correpond to ray point to it Frenel volume. Τhe implementation of the propoed equation (Soupio et al., 00), i much more imple for the calculation of the Frenel volume, ince we only need the raypath connecting the given ource and receiver and the dominant frequency of our ignal for the wavelength etimation. The ray acing for the lected wave i performed uing a fat finite difference cheme (Hole and Zelt, 995). After pecifying the velocity model, the avel time of the incident wave to the lecting interface i firt calculated. Since wave that turn at or below the lector may arrive earlier but urely do not repreent the lected arrival, velocitie below the interface are et to be lower than velocitie above the lector (Podvin and Lecomte, 99). Uing Snell law, lected avel-time are aigned at the node immediately above the lector. The final lected avel-time i calculated by adding the avel-time from the lection point to the receiver, to the one previouly calculated. Raypath are determined and partial derivative of avel-time (maix M) are calculated from the pre-calculated lection aveltime a well a the velocity field and the lector geomey (Williamon, 990). Thi algorithm ha been modified to etimate the derivative moving backward in time. INVERSION PROCEDURE For a two-parameter model including lowne and lector depth, the avel-time deviation δt i given a follow, M 0 δ δt = or, M M δ δt M x = δt (3) where M and M are maice (coniting of lowne partial derivative) for racted and lected wave repectively, and M i a maix of depth partial derivative for lected wave, x i the correction vector which contain information about lowne and lector depth, δ and δ are the velocity and lector correction vector and δt i a vector of the difference between calculated and oberved aveltime. The maix M of equation (3) conit of the partial derivative with repect to both parameter (velocitylector) that have already been calculated. An important iue i the relative weight of the lected and 77

5 NON-LINEAR INVERSION OF SEISMIC DATA racted arrival, a well a the adeoff between the lowne, δ, and lector depth, δ, correction. In the preent work we follow the Bayeian approach (e.g. Tarantola, 987), where equation (3) i written a: lector depth,, and the lowne the lection, σ, avel-time error. Cd M Cm Cm x = Cd δt Cd M CThee m x = error Cd δtconol the relative (4) where C d and C m are the data and model a priori covariance maice and variance, ( = v /v ) where v i the velocity variance. Moreover, the data error can be decribed by the raction σ, and weighting of the two data type. Hence, in order to conol the weight of the two data type (raction-lection), we ue an additional relative-weight factor, x = Cx x are the normalied model r w, which i calculated a: perturbation. The objective function that we are effectively minimiing i raction _ weight σ σ rw = = = given below, lection _ weight σ T T S = ( δt Mx) Cd ( δt Mx) + γ ( x Cm x) σ (7) (5) If we replace C where, γ i a Lagrangian multiplier d and C m in equation (4) by the correponding variance we which conol the ade-off between the derive the following equation: normalied model-norm (meaure of the olution x length) and the normalied fit S δ M + 0 = δt between the oberved and the calculated σ σ data. S δ δ We conider the implet and M + M = δt σ σ σ uually adopted cae, where velocity and lector node have independent (8) and contant a priori error, v and, where δt contain the avel-time repectively, equation (5) can be reidual, M, M and M are anformed to: lowne and lector derivative T T T S = ( δt Mx) Cd ( δt Mx) + γ ( V xv xv + x maice. x ) If we multiply equation (8) (6) with σ and ue equation (7), we rewrite where the perturbation vector x ha equation (8) in maix notation a been plit into two vector, x v and x, follow, each containing the velocity and M x = δt lector depth perturbation to be (9) determined. Thu, the model variance i where, decribed only by the variance of the t t t L 0 0 L 0 n M M O M M M O M t K t K t K L 0 0 L 0 n t t t = M rw rw L r t w t t m rw rw L r w n M M O M t L tl tl M M O M rw rw L r t w L tl tl m rw rw L r w n 78

6 Panteli M. Soupio et al (0) ˆ δt ˆ δt M M ˆ n = xˆ, δtk = δt ˆ δt. rw ˆ δt. rw M M ˆ m δtl. rw () where m i the number of lector node, n i the number of lowne node, K i the number of raction avel-time, L i the number of lection avel-time and ŝ and ẑ denote the normalied (with repect to tandard error) perturbation of lowne and lector depth, repectively. In the current work we preent reult uing the LSQR algorithm where originally propoed by Paige and Saunder (98). The algorithm baed on a bidiagonaliation of the ytem uing Lanco method (Lanco, 950), followed by a QR decompoition to find the olution. In order to peed up convergence and tabilie the inverion everal approache can be adopted, a previouly decribed, uch a damping or moothing. In the preent work we mooth the lowne perturbation uing an appropriate Laplacian operator (Lee and Croon, 989) and the lector geomey by a econd-order difference operator (Ammon and Vidale, 990). The Laplacian operator add the following ytem of equation to the avel time inverion problem, ( 6 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) i, j, k i, j, k i+, j, k i, j, k ˆ = () ˆ i the normalied lowne i, j, k perturbation at the (i,j,k) node and d i the grid pacing. Similarly the econd-difference operator add the ytem of equation, = 4 ˆ ˆ ˆ ˆ ˆ = ˆ i, j i, j i+, j i, j i, j+ where the (3) ˆ indicate the lector i, j depth perturbation at the (i,j) horiontal poition. In order to ue the above equation in -D problem, we imply ormulate them a, ˆ ˆ ˆ ˆ ˆ 0 and 4 i, k i, k i+, k i, k i, k+ = ˆi ˆ i ˆ i+ = 0, repectively. Moreover, in mot cae the derivative maix ha mall ingular value and i typically highly illconditioned. One approach in dealing with thi i to ue damping factor and augment the ytem of equation by a 0 d i, j+, k i, j, k i, j, k+ = 0 et of additional conaint. Since, we are uing normalied variable, we can imply write thee conain a ε I n m where ε i the damping value, I i the unit maix and the indice or er to lowne or lector depth. Typically, ε hould be et to (Franklin, 970) [or σ in our cae ince the ytem (8) wa multiplied by σ to define ytem (9)], if a priori data and model error are correct. Uually ε i et to value higher than to account for outlier, not expected by the Gauian diibution. The final augmented ytem of equation can be written a, M δtˆ ε I n 0 0 ˆ 0 ε = I m 0 ˆ λs λ D 0 (4) 79

7 NON-LINEAR INVERSION OF SEISMIC DATA where ε and ε are the damping factor, λ and λ are the moothing factor (concerning lowne and lector repectively), S i the Laplacian operator (equation ) and D i the econd difference operator (equation 3). NUMERICAL EXAMPLE In order to demonate the effectivene of the propoed technique we have teted our algorithm uing four model. In thee tet we elected a pecific velocity model for the tudy area and olved the forward problem. Then, the ynthetic racted and lected avel-time are inverted back to a velocity ucture and a pecific lector. Since our ynthetic data are error-free we expect that the input (ynthetic) model hould be recovered. Hence, comparion of the original velocity and what our algorithm recover permit u to analye the efficiency of the inverion cheme. Trying to chooe the proper regulariation parameter we ued the Picard condition and the L-curve criterion a preented by Soupio et al. (00). Thu, for all the tet decribed in thi ection, we conclude that a et of value of λ =5.0 and λ =3.0 (λ and λ are decribed above) hould be elected for the final olution ince our ynthetic data were error free. Alo, a imilar value ε=5.0 wa ued for the damping factor. In the preent work we have teted our method for the following cae: o Auming that lector i completely defined by a priori information, thu we invert only for the velocity field, o Auming that velocity field i completely defined by a priori information (velocity analyi etc.), thu we invert only for lector geomey and, o Simultaneou determination of the velocity model including the lector geomey. o A ynthetic tet with 3-D ource-receiver geomey for a complex geological ucture (anticline). Reconuction of velocity model The tarting model conit of a four-layer background velocity model where the velocity increae linearly with depth according to the following gradient and table (): 0.075Km / Km (5) and a flat lector at a depth of 70 Km (figure ). Model A ha a high velocity block (+0%) at depth from 0 to 40 Km. For the next model (B) we ued a low velocity anomaly in the ame region and with the ame amplitude (- 0%). In the lat model (C), two velocity anomalie (poitive and negative) were ued. A grid with 4 node in the x direction, in the y direction and in the direction (depth), reulted in a total of 86 velocity node (figure ). For the tet we ued 9 ource with 0 Km hot pacing and 0 geophone with geophone pacing 0 Km, reulting in *380 ray (direct, racted and lected wave). For thee ray, ynthetic avel time were calculated. In the inverion we et the r w parameter to a minimum (near ero) value in order to calculate the velocity field mainly from the racted wave. Figure () how the raypath coverage of the racted and lected wave for the tudy area. The ace of the raypath of racted wave are hown by the dotted line avelling through the model. Reflected raypath are preented by the dot on the grid line a the interection of the raypath with the grid. 80

8 Panteli M. Soupio et al Figure. Source-Receiver geomey and the velocity diibution ued for the ynthetic tet. Ray coverage correponding to three (A, B & C) different velocity model. The rectangular block define the region of velocity anomaly, which erred to the background velocity that increae with depth. The racted raypath avelling through the model from and to the urface are alo defined. The black ace on the grid line repreent the interection of the raypath of lected wave with the model grid. It i obviou that the raypath recognie the exitence of the velocity anomalie. Figure () how the final model from the optimiation-invere proce. The contour repreent the percentage (%) of the velocity anomaly compared to the background velocity model. The high velocity rectangular one i reconucted quite well. 8

9 NON-LINEAR INVERSION OF SEISMIC DATA Figure. Cro-ection of the final inverted velocity model a calculated from inverion of firt arrival and lection aveltime. The contour repreent the percent (%) of reconucted velocity model. The conained lector i preented by a olid line at 70 Km. A ditortion of the hape of the anomalie i found and it amplitude ha been underetimated. A) Poitive velocity anomaly i reconucted quite well (>50%), B) The negative velocity anomaly i poorly identified (<50%) ince the ray coverage in the area wa poor and C) The amplitude of the high velocity anomaly i better reconucted than the low velocity anomaly (Wielandt, 987). For model A, the hape of the high velocity anomaly contour i influenced by the ource - receiver geomey. Within the rectangular block, deviation of 0-4% are oberved in the reconucted model compared to the input model. Higher deviation are oberved for model B (-8%) and C (6-0%), due to poor coverage in thoe region (figure ). The high velocity anomaly i better reconucted becaue the ray method work well for poitive anomalie but i clearly inufficient in the preence of negative anomalie (Wielandt, 987). Reconuction of lection interface Uing the ame ource-receiver geomey but a narrower grid ( node in X, node in Y and node in Z) we have inverted the avel-time for 8

10 Panteli M. Soupio et al interface geomey auming that velocity field i a priori known. We ued three ideal lector (figure 3) with different geomey to check the algorithm efficiency. In all cae, a flat lector wa conidered at the depth of 70 Km for the tarting model. Thu, we had to invert for unknown lector node parameter (figure 3). In thi tet, we maximie the effect of lection derivative and minimie the raction, maximiing the r w parameter value a erred in the previou tet. Thu, only the lection avel-time calculated for each ideal lector were the input in the inverion proce. Figure (3) preent the ray coverage from the lected wave a etimated olving the forward problem with finite difference. Low raypath coverage at the edge of the model (lector) i oberved due to the ource/receiver and lector geomey. The ame problem appeared in cae B and C in the area where the lector exhibit, a yncline and a vertical fault, repectively. Thu, low quality of the reconucted image i expected in thoe place. Figure 3. Ray coverage from lection wave a calculated from the forward modelling. The raypath of lected wave are preented by the dot on to the grid line. Low raypath coverage at the edge and in the area of lector anomaly i oberved due to ource-receiver geomey and lector anomaly. 83

11 NON-LINEAR INVERSION OF SEISMIC DATA Figure (4A) how the tomographic reult for the dipping lector. At the end of third iteration, lector i reconucted by 95%. Figure 4. Reconuction of lector, by inverting lection avel time. The ideal lector i decribed by dahed line and the continuou line denote the tarting lector geomey. Solid bullet preent the final lector. A) Dipping layer. The ue model i reconucted by 95%, B) Syncline model, which i finally reconucted by more than 87% and C) Reflector with of a fault-like model, which i poorly reconucted due to low ray coverage cloe to the fault region. 84

12 Panteli M. Soupio et al Figure (4B) preent the tomographic image in cro ection for the yncline model which i ueful in the reconuction of tectonic ucture. The reult how that lector geomey i well reconucted. The reconucted model how dicrepancie from the actual one at low raypath coverage region uch a the edge and the middle of the model. For the lector with fault, ranging between the depth of 60 km and 80 km, the algorithm recover it depth quite well (figure 4C). However, ome region (edge of lector) were reconucted poorly, due to low ray coverage (Figure 3C). The r.m. error wa reduced from 8.96 (initial model coniting of a flat lector model) to.3. Reconuction of the velocity model and the lector geomey The input model i 00 Km long and 00 Km deep, with a background velocity model which conit of four layer, whoe velocity increae linearly with depth (equation 5 and table ) and a flat lector at a depth of 70 Km. The ame velocity anomaly a in figure (A) with a lower amplitude (0%) and the three initial lector of figure (3) are alo ued for the calculation of critically racted and lected wave. We have ued a narrow a well a a dene grid with /4 node in the x direction, in the y direction and / in the direction (depth), reulting in a total of 3/86 velocity node and /4 lector node. 9 ource with 0 geophone along the top of the model were ued, thu reulting in 760 (*380) avel time obervation. Figure (5) preent the raypath coverage from the imultaneou olution of the forward problem uing firt arrival and lection aveltime. Poor ray coverage i alo oberved in region near the lector. Figure (6A), (6B) and (6C) how the inverion reult. We hould note that contour repreent the percentage (%) of the reconucted velocity model. Thu, the high velocity region i reconucted quite well, ince the ray ample better the fater region. Regarding the lector, the inverion recovered it geomey quite well except for the fault model due to low ray coverage. It i important to notice that the main velocity model characteritic, uch a the amplitude and hape of the anomalie were better determined than in the previou tet. 3D ynthetic tet for the modelling in complex geological area The 3D model wa defined a 6 Km and 00 Km deep uing the ame background velocity model a preented in table (). Table. Starting background velocity model Layer Average Velocity (Km/) Depth We ued almot the ame model a in figure () with 3D ource and receiver geomey. A high velocity anomaly (0%) lying on the top of the flat initial lector and an ideal lector like a anticline geological ucture are ued for the calculation of the ynthetic avel time. The initial lector model wa a flat dicontinuity at a depth of 60 Km. We have ued a narrow 3D grid with 6 node in the x direction, 6 85

13 NON-LINEAR INVERSION OF SEISMIC DATA node in the y direction and in the direction (depth), reulting in a total of 5376 velocity node and 56 lector node. A total number of hot point (iangle) diibuted on the top of the model were recorded by 0 urface geophone (quare) a i hown in figure (7). The final data et conited of 840 firt break of P wave and lection aveltime from the initial lector. Figure (7) preent in 3D the tomographic image from the imultaneou inverion of firt arrival and lection aveltime for the lat iteration. The reult how that the initial model i well reconucted. Applying appropriate regulariation parameter ha deflected ome artefact. It wa intereting that in the area of interet both the amplitude and the hape of anomaly (velocity/lector) are ignificantly reconucted for more than 50%. CONCLUSION We preent a tomographic method for joint etimation of the velocity field and lector poition uing firt arrival and lection aveltime. We have alo hown that our algorithm i capable of producing reliable tomographic image uing ynthetic data. Thi method may be alo helpful in conucting complex geological model with local bodie, yncline, anticline and fault. The propoed method i alo able to perform an accurate determination of the hallow velocity model and correct the large time hift (tatic) in lection, reulting in better lector continuity. Moreover, we can ue tomographically determined velocitie to image eimic data applying depth migration, avoiding tacking and normal moveout (pretack migration). The propoed method incorporate the ue of an efficient ray acing technique and a finite difference code for olving the forward problem in conjuction with a combination of inverion tabiliation approache. In pite of the amount of data, the propoed method and algorithm are quite efficient, ince they require relative mall computation time. REFERENCES Aki, K., and W. H. K. Lee, 976, Determination of threedimenional velocity anomalie under a eimic array uing firt p arrival time from local earthquake:, A homogeneou initial model: J. Geophy. Re., 8, Aki, K., Chritofferon, A., and Huebye, E. S., 977, Determination of the three dimenional eimic ucture of the lithophere: J. Geophy. Re., 8, Anderon, D. L., and Diewonki, A. M., 984, Seimic Tomography: Scientific American 5, No 0, Ammon, C. J., and Vidale, J. E., 990, Seimic avel-time tomography uing combinatorial optimiation technique: Sei. Re. Lett., 6, 39. Bickel S. H., 990, Velocity-Depth ambiguity of lection aveltime: Geophyic, 55, Bihop, T. N., Bube, K. P., Cutler, R. T., Langan, R. T., Love, P. L., Renick, J. R., Shuey, J. T., Spindler, D. A. and Wyld, H. W., 985, Tomographic determination 86

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