Frkchet derivatives for curved interfaces in the ray approximation

Transcription

1 Geophysical Journal (1989) 97, Frkchet derivatives for curved interfaces in the ray approximation Robert L. Nowack and Jeff A. Lyslo Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN 47907, USA Accepted 1988 November 24. Received 1988 October 10; in original form 1988 May 12 SUMMARY The sensitivity of ray and beam theoretical seismograms to changes in velocity models and curved interfaces is discussed in this paper. Previous results from Nowack & Lutter (1988) give the derivatives of travel-time and ray amplitude with respect to changes of smoothly varying velocities. These derivatives are required to perform linearized maximum likelihood inversions for structure. In the ray approximation, smooth interfaces are incorporated by applying Snell s law locally, correcting wavefront curvature, and using local plane-wave reflection and transmission coefficients. The partial derivatives for travel-time are directly calculated along the original ray trajectory using Fermat s principle. For perturbations of the ray amplitude of a reflected/transmitted ray, the ray shift of the perturbed two-point ray trajectory must be accounted for. The approach followed here is to calculate the approximately perturbed two-point ray using perturbation theory without additional ray tracing. The perturbed ray amplitudes are then computed directly, including modified reflection/transmission coefficients and geometric spreading, along this approximate twopoint ray. Several numerical experiments are conducted which invert for velocity and interface shape using both travel-time and amplitude in order to test the derived partial derivative operators. Travel-time and amplitude inversion results are also contrasted with amplitude being less sensitive to larger scale features and more sensitive to heterogeneity curvature. Key words: FrCchet derivatives, ray theory, seismic inversion 1 INTRODUCTION In this paper, the sensitivity of ray theoretical seismograms to changes in smoothly splined velocity models with curved interfaces is investigated (Fig. 1). Previous results of Nowack & Lutter (1988) give the partial derivatives of travel time and geometric spreading amplitude for smoothly varying velocity with no interfaces. This was based on the perturbation theory results of Farra & Madariaga (1987). These partial derivatives are required in order to perform linearized inversion for structure. Curved interfaces can be included within the ray method by applying Snell s law locally, correcting for wavefront curvature and using local plane-wave reflection/transmission coefficients. The FrBchet derivatives investigated here give changes in the ray field due to changes in both velocity as well as interface shape. The partial derivatives for travel-time are directly calculated along the original ray trajectory using Fermat s principle. This is a major computational reduction and is used in most travel-time tomography algorithms. For perturbations of the total ray amplitude of a reflected/transmitted ray, the ray shift of the perturbed two-point ray trajectory must be accounted for. The approach followed here is the calculation of the approximately perturbed two-point ray using perturbation theory without additional ray tracing. The perturbed ray amplitudes are then calculated directly including modified reflection/transmission coefficients and geometric spreading along this approximate two-point ray. Alternative formulations of the inverse problem using reflected/transmitted waves have been considered by other investigators. Bishop et af. (1985) did a combined inversion for velocity and interface parameters using just travel-times and piecewise linear interfaces. In contrast, here sensitivity operators for inversion are formulated using both travel-time and amplitude for splined velocity nodes and smoothly splined interfaces. The travel-time partials described are currently being used to analyze direct and reflected data from the 1986 Ouachita PASSCAL experiment for both velocity and shape of interfaces (see Lutter, Nowack & Braile 1989; Lutter & Nowack 1989). For plane layered structures, MacDonald, Davis & Jackson (1987) inverted both travel-time and amplitude data. They found that near the critical distances, more complete expansions for spherical waves are required for finite frequencies. Our initial analysis here is to derive ray amplitude partial derivatives for velocity and interface shape. These partials can then be directly incorporated to construct more complete Gaussian beam amplitude partials in critical regions. In the following sections, the FrCchet derivatives for travel-time and ray amplitude are formulated. Several trial inversions are then performed in order to test these partials as well as to contrast travel-time and amplitude within linearized inversions. 491

2 498 R. L. Nowack and J. A. Lyslo Figure 1. Illustrative model with smoothly splined velocity heterogeneity and a curved interface. 2 TRAVEL-TIME SENSITIVITY OPERATORS In this section, the FrCchet derivatives of travel-time with respect to velocity and interface shape are formulated. The travel-time along a ray can be written I:, T = L(u(x), x, X] ds, (1) where L = u(~)(.i~.i~) ~ and u(x) is the slowness. The first variation of the travel-time can be written where 6u is the perturbation of the slowness. Using integration by parts this can be rewritten as Note that in the last term (iixi)l Z = 1. For a valid ray, according to Fermat s principle, the travel-time is stationary, giving 6T = 0. For fixed endpoints, the first term on the right hand side of equation (2) is zero. Assuming no slowness perturbations, this results in the ray equations or - d (+) =-. a3 axi du (4 Assuming a ray with a free endpoint, such that 6xi is along the wavefront, then once again, 6T = 0. The first term in equation (2) then gives dl - 6Xi = 0. ax;

3 Frichet derivatives 499 Therefore al dx - = u(x) 2 = p; ds ax; is perpendicular to the wavefront, where pi is the slowness vector tangent to the ray. For a ray with fixed endpoints, the perturbation of the travel-time from equation (2) for a variation in material slowness, u(x), is 6T = l 6u(x) ds, which is computed along the original unperturbed ray trajectory. This is an important simplification in the calculation of sensitivity of travel-time from changes in material slowness which is ultilized in most travel-time inversion algorithms. For a reflected/transmitted ray with fixed endpoints incident on a smooth boundary, Zo, then equation (2) can be written where s, refers to the incident side of the boundary and s,' refers to the reflected/transmitted side of the boundary. For a valid ray at a fixed boundary, with 6xi tangent to the boundary, then the travel-time is stationary giving 6T = 0. This results in Snell's law or continuity of locally tangential components of the slowness vector at the boundary. Assume now, for a reflected/transmitted ray that the boundary, Z,,, is perturbed to Z,,.First, a local coordinate system is constructed at the point of incidence, x,, with respect to the unperturbed boundary, Zo, with the normal, h, pointing into the medium which contains the incident ray. To first order, equation (3) can be written (3) can then be computed in a straightforward manner. The partial derivative of travel-time with respect to a vertical perturbation of a boundary node of an interface can then be written using equations (4) and (5) as For the special case of a piecewise linear boundary and an incident reflected ray, this reduces to the result given by Bishop et al. (1985, equation B-10). 3 AMPLITUDE SENSITIVITY OPERATORS In this section, the FrCchet derivatives of amplitude with respect to variations in material slowness and interface shape are formulated. In the ray approximation, the amplitude of a multiply reflected and transmitted ray can be written (see cerveng 1985a) 1 ~ ( 0 s = ) ~ ( ~ s ) c ( ~ s II ) [R(~;)G~(~~)IW? (7) i=n where C is the complete receiver matrix, R is the reflection/transmission matrix, Y is the source matrix and GL is the rotation matrix at each interface. A(0,) can be written 1 A(0s)= [v(os)p(0,) det Q(OS)l1'* where 0, is at the receiver and 0, is the ith reflection/transmission point. The unprimed values are on the incident ray side of the boundary and the primed values are on the reflected/transmitted side. Q(0,) is derived from where ph is the normal h component of the slowness vector and 6h is the normal component of the boundary perturbation at the point x,. 8 is the angle of incidence, 8' is the angle of reflection/transmission and 7 is for reflection or transmission. u(x,) is the slowness on the incident side of the boundary and u'(x,) is the slowness on the reflected/transmitted side. For example, for an unconverted reflected ray, then 6T = -2u(x,) cos 8 6h. This can be written in a global coordinate system with vertical z down by using dh -- dz(x,) - * cos /% is the local dip angle of the boundary from the horizontal at x,. The + sign is for an upgoing incident ray and the - sign for a downgoing incident ray. Finally, this can be related to the node points of an interpolated boundary, where in the following examples a splined boundary is assumed. The spline coefficients for a small perturbation of an interface node are first computed and stored. The values of where n(oi, Q,-J are made up of fundamental propagator solutions to the dynamic ray equations, with n(oi, 0,) = I. F(0,) are the transformation matrices at the interfaces (see cervenf 1985a). With appropriate initial conditions, det Q(0,) is the geometric spreading. The problem now is to find the partial derivatives, du(o,)/av,, for a perturbation of the amplitude with respect to a velocity or slowness node perturbation in one of the layers, and also to find the partial derivatives, du(o,)/az,, for a perturbation of the amplitude with respect to an interface node perturbation. The total derivative of ray amplitude with respect to a heterogeneity involves two parts, first the variation of the amplitude along the original ray plus the effect of ray shift due to the heterogeneity. Previous work by Farra & Madariaga (1987) gave results for perturbation in the geometric spreading part of the amplitude due to a variation in a smooth velocity node. This approach was used to approrimately compute ray theoretical seismograms for a perturbed velocity structure using only rays traced in an initial velocity structure. Nowack & Lutter (1988) used these results for several numerical inversions for velocity using both tfavel time and geometric spreading amplitudes.

4 500 R. L. Nowack and J. A. Lyslo First, the perturbation of the reference ray due to a material slowness perturbation is investigated. For a direct ray, the change in the transverse ray position, Sq,, and the transverse deviation of the slowness vector, Sq,, in ray centered coordinates can be written using the linearized dynamic ray equations as (see Farra & Madariaga 1987; Nowack & Lutter 1988) d - SX, = A SX, + SB, ds where 0 A= [,, - 2u-'u,uq "-'I 0 and SB = [auq- u-luq O SU I and uq = dulaq. The solution can be written in terms of the ray propagator of the unperturbed medium as SX,(s) = ln(s, s') SB(s') ds'. (9) Upon reflection from an unperturbed interface, SX,(s:) = -SX,(S,). An important aspect of the equations for SX,(s) is that the new approximate ray will not in general hit the receiver. However, the ray propagator can be used again to derive an approximate two-point ray SX - p P 6PZP. This involves readjusting the ray at the source so as to approximately hit the receiver. In 2-D, the new initial conditions are bp(so) = 0 (10) 6~2p(s0) = &r(sf)/n(l, 2 I sf, SO), where n(1, 2 1 s, so) is the (1,2) component of the n(sf, so) matrix, and N is the number of reflections along the ray. Using these initial conditions, then m+.(s) = (-l)"n(st so) ~Xzp(S0) + W ( S ) (11) where n is the number of reflections from so to s. This will only be approximate since the propagator used is from the unperturbed medium. For the case of a perturbation of interface shape, only one term is required and this is evaluated at the interface. For a shift of the interface in the direction of the local normal by 6h, at the point of incidence, s,, then for a reflected ray &,(s,) = +2 sin 8 Sh(s,) + O(Ss,), where h is positive in the direction of the medium containing the incident ray. For a change in slope of the interface at s, due to a perturbation of an interface node, then for a reflected ray 6p,(s,) = -2u(su) + o(6sq)9 where Sa is the rotation of the interface clockwise about the local coordinate system normal and tangent to the interface at s,. For a transmitted ray analogous formulas can be given. The above two formulas for Sq, and Sp, are only approximate since they do not include variations due to slowness heterogeneity directly at the boundary. For smoothly varying slowness at the boundary they should be accurate to first order; however, more complete formulas are required. In order to relate these changes to a vertical position at an interface node, ah(s,)/az(xnd,) and aa(s,)/az(x,,,,) are computed from a stored set of perturbed spline coefficients along each interface. Since a perturbation of an interface node will both shift and rotate the interface locally at the incident ray point, both Sq, and Sp, must be taken into account. The perturbed reference ray due to a perturbation of an interface node for reflection can then be written where N, is the reflection number at interface s, and O<i< N- Nu. For s <s,, the reference ray is not perturbed. Since this perturbed ray will not in general hit the receiver, it must be readjusted at the source in order to construct the approximate two-point ray. In 2-D, with initial conditions given by equation (lo), then the approximate two-point ray can be computed using equation (11). For the two-point ray, the entire ray trajectory has been changed by perturbing the shape of the interface. In the following, several examples will be shown illustrating several of the above concepts. In Fig. 2, the ray shift of a direct ray resulting from a velocity heterogeneity is shown. The model is from 0 to 30 km in depth and -1 to +1 km laterally. The initial unperturbed ray starts at 30 km and goes vertically up to the surface in a homogeneous 6kms-' medium. A smoothly splined velocity node centered at 15 km depth and -2 km in x is now increased by 2 per cent above the background velocity. The solid line in Fig. 2 marked the 'perturbed ray' is the exact ray in this new medium with the same initial takeoff angle as the unperturbed ray. The dashed line is the approximate perturbed ray computed using equation (9). Note that this ray no longer hits the receiver at (0,O). The solid line in Fig. 2, noted as the 'two-point' ray, is the exact two-point ray from source to receiver. The dot-dash line is the approximate two-point ray computed using equation (11). The discrepancy between the exact and approximate two-point ray results from using the initial unperturbed ray propagator, n(s, so), in equation (11). Figure 3 shows a similar calculation using a reflected ray with coincident source and receiver at (O,O), and a reflector at 30km. The vertical line is the original unperturbed downgoing and upgoing ray. The solid line, noted as the 'perturbed ray', is the exact ray in a perturbed medium with a smoothly splined velocity node centered at (-2,15) increased by 1 per cent above the background velocity of 6kms-'. The approximate perturbed node is within a line width of the exact perturbed ray for this case.

5 s h Y 5 n 3.3 Q 0 DIRECT RAY SHIFT DUE TO VELOCITY NODE VARIATION REFLECTED RAY SHIFT DUE TO VELOCITY NODE VARIATION 1 0 perturbed ray h E 24 v 5 n 0 Q F%gnre 2. Ray shift for a direct ray due to a smoothly splined velocity node perturbation at (-2.15) increased by 2 per cent above the background. The exact unperturbed ray is the vertical solid Line. The exact and approximate perturbed ray have the same initial takeoff angle as the unperturbed ray. The exact and approximate two-point ray have a modified initial takeoff angle in order to hit the receiver. -1. x(km) F w 3. Ray shift for a reflected ray from an interface at 30 km resulting from a smoothly splined velocity node perturbation at (-2,15) increased by 1 per cent above the background. The unperturbed, exact and approximate perturbed and exact and approximate two-point rays are the same as in Fig

6 REFLECTED RAY SHIFT DUE TO INTERFACE VARIATION I I 0' E Y 5 U I I- a W FI X(km) Flguc 4. Ray shift for a reflected ray from a 30km interface with a small perturbation, in tilt of the interface, The unperturbed, exact and approximate perturbed and exact and approximate two-point rays are the same as in Fig Figare 5. Ray diagram with source at (20,O) and 24 rays reflected from a 30 km Moho with a small 0.5 km corrugation. 70 DISTANCE IN KM 1 Lto

7 Frichet derivatives 503 The perturbed ray is bent away from the higher velocity node, and has a resultant endpoint ray shift, no longer hitting the receiver. Also, this ray is no longer incident vertically on the boundary resulting in changes to the reflection coefficient within the amplitude calculation. The solid line in Fig. 3 noted as the 'two-point' ray is calculated exactly, while the dot-dash line is the approximate two-point ray using equation (11). The two-point ray is now incident on the interface at near vertical incidence, which is more appropriate for the perturbed reflection coefficient. Figure 4 is similar to the previous figures only now with a variable lower reflecting interface. The unperturbed reflected ray for a coincident source and receiver and boundary at 30 km is shown by the solid vertical line. The solid line marked 'perturbed ray' is the exact ray for a slightly tilted lower interface. Within the line width is the approximate perturbed ray computed from equation (12). This ray is coincident with the downgoing unperturbed ray but diverges for the upgoing segment and no longer hits the receiver. The solid line noted as the 'two-point' ray is the exact ray for a tilted lower boundary. The entire ray trajectory has now been modified and the reflection point has moved updip. The dash-dot line is the approximate two-point ray computed from equation (11). Since the two-point ray is the only ray that gives the correct reflection/transmission coefficient changes, it is the appropriate ray for amplitude sensitivity calculations. The sensitivity of the geometric spreading part of the amplitude in equation (7) was investigated by Farra & Madariaga (1987) and Nowack & Lutter (1988). The geometric spreading, Q(s), can be computed in the perturbed medium using the linearized dynamic ray equations. The solution can be written in terms of the ray propagator in the perturbed medium, X(s) = n'(s, so)x(s,). This propagator can be approximated using a Born linearization and the initial medium propagator, n, as n'(s, so) = n(s, so) + n(s, s') GA(s')n(s', so) ds'. This approximate formulation can be used to compute the perturbation of the geometric spreading. This approach was followd by Farra & Madariaga (1987) for forward modeling and by Nowack & Lutter (1988) for inverse modeling. A ONE NODE INTERFACE VARIATION I I I I I I I I b) - I I I I I I I I Figure 6. Trial inversion for 8 interface nodes. True model has one interface node elevated by 0.1 km and is noted by solid line. The dashed line in (a) is the travel-time inverted result and the dashed line in (b) is the log-amplitude inverted result.

8 504 R. L. Nowackand J. A. Lyslo problem with this expression is that it gives the appropriate geometric spreading for the perturbed ray in Figs 2-4 but not the two-point ray. Just considering the geometric spreading part of the amplitude for a small perturbation, this is still a satisfactory approximation. However, for reflected/transmitted rays, the perturbed reflection/ transmission coefficients for the two-point ray must be used in the complete amplitude calculation or for the calculation of partial derivatives. As a result, the strategy followed in this paper is to first construct velocity and boundary perturbation templates which can be overlayed on different node points of the model. This is then used to compute perturbed two-point ray trajectories using the approximate formulations above without additional kinematic ray tracing. The linearized dynamic ray equations and reflection/transmission coefficients are then solved for this approximate two-point perturbed ray trajectory. Finally, alj(os)/3m is formed from the initial and perturbed amplitudes where m is either a velocity node or boundary node perturbation. This is currently implemented as a subroutine directly within SEIS83 (see cervenj & PSenEik 1984). Once a ray trajectory, travel time and amplitude have been computed, the subroutine PARTIAL is called and do-loops over velocity nodes and interface nodes are performed in order to obtain approximate two-point travel-time and amplitude partial derivative operators. These partial derivatives are then appropriate for input into linearized inversion programs using either travel-time or ray amplitude. 4 INVERSION FOR INTERFACE SHAPE AND VELOCITY In order to test the travel-time and amplitude partial derivatives described in the previous sections, several simple test inversions are performed. Rays reflected from a 30 km deep Moho are used. Only ray theoretical amplitudes will be considered here appropriate for high-frequency waves. At lower frequencies, a Gaussian beam formulation should be used which gives more complete results at critical distances and at caustics for finite frequencies (see Cerveny, Popov & PSenh k 1982; Nowack & Aki 1984; cervenj 1985a,b). Since a superposition of ray solutions is used in the Gaussian beam approach, the ray amplitude partials in the previous section can be directly incorporated into Gaussian beam amplitude partials. In Fig. 5, the source geometry from a single source gather is shown. 24 reflected rays are incident on a Moho with a TWO NODE INTERFACE VARIATION I I I

9 Frkchet derivatives 505 small 0.5 km bump. Even this small a heterogeneity on the interface can cause significant amplitude focusing. Inversion for interface shape is now attempted using a model consisting of the vertical position of eight splined interface nodes from 20 to 90 km with a horizontal spacing of 10 km. The synthetic data consist of 24 reflected travel times and amplitudes from a single shot gather with a source position at (20,O). The true model has a single interface node elevated by 0.1 km and is shown by a solid line in Fig. 6 (a and b). The initial model assumes a flat interface. The travel-time inverted result using the previously described partial derivatives is shown by the dashed line in Fig. 6(a). The inverted result using log-amplitude is shown in Fig. 6(b). For this example, the travel-time gives the correct result and the amplitude gives a slightly lower result by several percent. In Fig. 7, the true model consists of two of eight interface nodes elevated by 0.1 km and is shown by the solid line. The dashed line in Fig. 7(a) shows the travel-time inverted result. The log-amplitude inverted result is shown by the dashed line in Fig. 7(b). For this example, the log-amplitude gives a slightly better result than the travel-time. In Fig. 8, the true model consists of a row of seven and eight interface nodes from 20 to 80 km elevated by 0.1 km. This is the entire portion of the boundary which is illuminated by the given rays. The travel-time inversion result shown in Fig. 8(a) does reasonably well using only rays from this single shot gather. The travel-time solution is however about 10 per cent low for the interface nodes at 70 and 80 km. The log-amplitude inverted result is shown in Fig. 8(b). In case of a row perturbation of interface nodes, the amplitude result only partially reconstructs the anomaly. The ends of the row are partially reconstructed but not the middle flat portion of the anomaly. Even at interfaces amplitude is more sensitive to model curvature and not DC shift. This is analogous to inversion results for velocity nodes found by Nowack & Lutter (1988). Using more iterations of an amplitude inversion, more of the middle of the anomaly would possibly be brought out due to increased model curvature in this area. Finally, an example is given using the same ray geometry of 24 rays from one source gather reflected from a 30 krn Moho. In this case, focusing from velocity heterogeneities is inverted in order to test the velocity partial derivatives. In Fig. 9, ray focusing due to a single velocity heterogeneity at (50,25) is increased in velocity by 2 per cent above background. The velocity node spacing is 10 km laterally and 5 km in depth. Ray focusing can occur both on the downgoing and upgoing legs of the ray. Note that the entire two-point ray is modified by the presence of the heterogeneity. This changes reflection coefficients as well as receiver corrections. In the formulation of the previous ROW INTERFACE VARIATION I I I I I I I I

10 506 R. L. Nowack and J. A. Lyslo 0 ' E: Y z H I I- LL w r3 20 LtO Lto DISTANCE IN KM Figure 9. Ray diagram with source at (20,O) and 24 rays reflected from a 30 km Moho. A single smoothly splined velocity node at (SO, 25) is increased by 2 per cent resulting in ray focusing. section, this is taken in account along the approximate perturbed two-point ray trajectory. In Fig. 10, two of eight velocity nodes from 20 to 90 km at a depth of 25 km, just above the Moho, are increased 1 per cent above the background. Fig. lo(a) shows the travel-time inverted result using the 24 reflected rays and eight velocity node unknowns. The log-amplitude inverted result. is shown in Fig. 10(b). For this case, both the travel-time and amplitude successfully invert for the desired smoothly splined heterogeneities. However, for the case of a row of increased velocity nodes, the results are similar to the row of interface nodes in Fig. 8. The travel-time does a better job on a larger scale row perturbation. The amplitude appears to be more sensitive to the velocity curvature transverse to propagation and less sensitive to large-scale DC velocity changes. 5 CONCLUSION In this paper, the sensitivity of ray theoretical travel-times and amplitudes to changes in smoothly splined velocities and curved interfaces have been investigated. Partial derivatives for both travel-time and amplitude are formulated which can be directly used within linearized inversions for structure. Several trial inversions have been successfully performed in order to test the partial derivative operators. Also, inversions based on travel-time and amplitude are contrasted showing the amplitude to be less sensitive to larger scale features and more sensitive to medium curvature. Areas for further investigation include the study of resolution and tradeoff between the velocity and interface medium parameters using both travel-time and amplitude. The addition of amplitude may give better depth perception along rays than just travel-time alone. Next, the ray amplitude partials for finite frequencies will have to be generalized to Gaussian beam partials which are more complete in singular regions such as caustics and at critical distances. Analysing reflected data with significant caustic formation will also cause difficulties in picking travel-times and amplitudes. Bording et al. (1987) have partially avoided this problem by using travel-times to perform a tomographic step and then migration to reposition the reflector within the new velocity model. Finally, since caustics are easily generated in media with curved interfaces, further investigations are required to study the degree to which linearity of the partial derivatives hold in near critical regions. ACKNOWLEDGMENTS The authors thank the two reviewers for useful and constructive comments. This work was supported by NSF Grant No. EAR

11 Frkchet derivatives 507 a> Figure 10. Trial velocity inversion using the 24 reflected rays in Fig. 9. The true model now has two velocity nodes increased by 1 per cent. (a) The travel-time inverted result and (b) the log-amplitude inverted result. For this case, both results are very close to the true model.

12 508 R. L. NowackandJ. A. Lyslo 10. (Conrinued). b)

13 Frkchet derivatives 509 REFERENCES Bishop, T. N., Bube, K. P., Cutler, R. T., Langan, R. T., Love, P. L., Resnick, J. R., Shuey, R. T., Spindler, D. A. & Wyld, H. W., Tomographic determination of velocity and depth in laterally varying media, Geophysics, 50, Bording, R. P., Gersztenkorn, A,, Lines, L. R., Scales, J. A. & Treitel, S., Applications of seismic travel-time tomography, Geophys. 1. R. astr. SOC., 90, eervenf, V., 1985a. The application of ray tracing to the numerical modeling of seismic wavefields in complex structures, in Handbook of Geophysical Exploration, Section I; Seismic Exploration, eds Helbig, K. and Treitel, S., Volume 15A on Sebmic Shear Waves, pp , ed., Dohr, G., Geophysical Press, London. eervenf, V., 1985b. Gaussian beam synthetic seismograms, I. Geophys., 58, Cervenf, V., Popov, M. M. & PSenEik, I., Computation of wave fields in inhomogeneous media-gaussian beam approach, Geophys. 1. R. astr. Soc., 70, brvenf, V. & PSenEik, I., Documentation of Earthquake Algorithms. SEIS83-Numerical Modeling of Seismic Wave Fields in 2-D Laterally Varying Layered Structures by the Ray Method , ed. Engdahl, E. R., Report SE-35, Boulder, Colorado. Farra, V. & Madariaga, R., Seismic waveform modeling in heterogeneous media by ray perturbation theory, J. geophys. Res., 92, Lutter, W. J., Nowack, R. L. & Braile, L. W., Seismic imaging of upper crustal structure using travel times from the PASSCAL Ouachita experiment, J. geophys. Rex, in press. Lutter, W. J. & Nowack, R. L., Inversion for Crustal Structure from the PASSCAL Ouachita experiment, 1. geophys. Res., in press. Macdonald, C., Davis, P. M. & Jackson, D. D., Inversion of reflection travel times and amplitudes, Geophysics, 52, Nowack, R. L. & Aki, K., The 2-D Gaussian beam synthetic method: testing and application, J. geophys. Res. 89, Nowack, R. L. & Lutter, W. J., Linearized rays, amplitude and inversion, Pure appl. Geophys., 128,