We P9 16 Eigenray Tracing in 3D Heterogeneous Media

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1 We P9 Eigenray Traing in 3D Heterogeneous Media Z. Koren* (Emerson), I. Ravve (Emerson) Summary Conventional two-point ray traing in a general 3D heterogeneous medium is normally performed by a shooting method. The loation and the slowness omponents are speified at the start point, and the ray path is the solution of ODEs with the initial onditions. The ray arrives to some proximity of the destination, and the start diretion is then suessively refined, so that the ray path eventually inludes the destination point. Eigenray traing, however, is a boundary-value problem, rather than an initial-value problem. The boundary onditions are two endpoint loations, and the ray trajetory satisfies Fermat's priniple of least traveltime. In this study, we apply the non-linear Finite Element Analysis to find the least-time ray path. The ray trajetory is split into a number of three-nodal segments with quadrati interpolation of trajetory points, traveltime and other funtions between the nodes. For eah segment, we ompute the traveltime, and its first and seond derivatives with respet to the nodal loations. The loal derivatives (related to a single segment) are ombined into the global derivatives of the entire path. For the least time, the first derivatives vanish. Knowledge of the seond derivatives makes it possible to apply the Newton method for ray path optimization. 8th EAGE Conferene & Exhibition 8 - June 8, Copenhagen, Denmark

2 Introdution Conventional two-point ray traing in general 3D heterogeneous media is normally performed using the so-alled shooting method. In this method, a fan of rays is traed from a given starting point, where the rays arriving in the viinity of the target loation, with lose take-off angles, are used to onverge to the atual end point. By overing a wide range of take-off angles, multi-pathing stationary solutions an be found. However, in omplex geologial areas, the solution to the initial-value problem assoiated with the individual rays is highly sensitive to small hanges in the take-off angles at the starting point, resulting in shadow zones whih the numerial rays annot penetrate. The proposed Eigenray method (also known as the ray bending method) is a diret solution of the boundary-value problem, rather than the initial-value problem. The boundary onditions are the two endpoint loations, and the ray trajetory satisfies Fermat s priniple of stationary traveltime. This method is partiularly attrative for areas that involve sharp veloity variations or loal veloity anomalies. The ase of simulating head waves is an extreme example of Eigenrays providing stable solutions where onventional ray traing fails. Ray bending methods have been extensively studied. Westwood and Vidmar (987) applied the method to simulate the signals interating with a layered oean bottom. Waltham (988) studied models onsisting of onstant-veloity layers separated by urved interfaes and omputed ray paths whose traveltimes are stationary with respet to hanges in ray/interfae intersetion positions. The amplitude of the resulting event is related to the seond derivative of the traveltime with respet to hanges in the position of the ray/interfae intersetion. Moser (99) used this method to ompute the traveltime between the soure point and all points of a network. Moser et al. (99) improved the onventional ray bending approah by applying a) gradient searh methods and b) interpolation by beta-splines between the nodes. Farra (99) applied the Hamiltonian formulation to the ray bending approah. Shashidhar and Anand (995) solved the problem of three-dimensional Eigenray traing in an oean hannel. Bona et al. (9) demonstrated that Fermat s priniple of stationary traveltime holds for general heterogeneous anisotropi media using stohasti simulated annealing global searh methods. Reently, Sripanih and Fomel () presented an effiient algorithm for two-point seismi ray traing in layered media by means of the bending method, where the ray paths are disretized at the intersetion of the rays with the struture s interfaes. In this study, we demonstrate the power of applying this type of ray traing by using a non-linear spetral element method to effiiently find aurate least-time ray paths in omplex geologial areas, providing solutions beyond onventional ray traing. Model and Method Consider an initial ray trajetory that is divided into a number of multi-nodal segments (spetral finite elements) with a Legendre polynomial interpolation. Eah segment inludes two endpoints and a number of internal points plaed along the internal flow parameter aording to the Gauss-Lobatto quadrature. The quadrature also assoiates definite weights for all points of a segment. The simplest spetral element is a three-nodal segment whih we use to demonstrate the syntheti examples. The nodes are loated at, and + of the flow parameter. For eah segment, we ompute the traveltime and its first and seond derivatives with respet to the nodal loations. The loal derivatives (related to a single segment) are ombined into the global derivatives of the entire path. The loal derivatives are vetors of size 3n, and the loal seond derivatives are matries of the same size, where n is the number of segment nodes, inluding its endpoints. The global matrix of the seond derivatives has a band struture with a narrow band width 3n. This matrix is symmetri, and if the stationary traveltime is minimum (the most pratial ase), it is also positive-definite in the proximity of the stationary ray. For the least-time solution, the first derivatives vanish. Knowledge of the seond derivatives makes it possible to apply the Newton method for Eigenray optimization. The seond derivatives of the traveltime depend on the loations of the nodes, and thus the minimization problem is nonlinear. In addition, the seond derivatives may be used to ompute the geometrial spreading when the stationary ray path is found. We note that the least traveltime (or generally, the stationary traveltime) fully defines the ray path, but it still allows moving the nodes along the ray. Therefore, we apply an additional onstraint on the lengths between the suessive nodes. The nodes are loated more densely in regions where the veloity hanges rapidly. Thus, the onstraints are the ratios between the ar lengths onneting suessive nodes. There are, in general, two ways to implement the onstraints: stiff onstraints (e.g., applying Lagrangian multipliers method) and soft or relaxed 8th EAGE Conferene & Exhibition 8 - June 8, Copenhagen, Denmark

3 onstraints, adding a penalty term to the target funtion to be minimized. The soft onstraint method is simpler, does not lead to additional unknown parameters, does not inrease the bandwidth of the resolving matrix, and still provides exellent auray. Sine the onstraints only affet the loation of the nodes along the stationary ray, the omputational formulae are essentially simplified if the hords (i.e., the distanes between the suessive nodes) are applied instead of the ar lengths. Syntheti Examples Three examples of ray bending solutions are presented. Example : High-veloity half-spae under onstant veloity layer (head wave). Consider a D veloity model whose profile and gradient are shown in Figure, to the left. Eigenray Trae for Two Initial Guesses Eigenray Init Init 8 distane (km) Figure High-veloity half-spae under onstant veloity layer: Veloity model and stationary ray The smoothed veloity model is depth-dependent and desribed by, v z z v vo tanh t, () zo where vo km/s is the veloity of the homogeneous layer above the half-spae, v km/s is the differene between the veloity of the half-spae and that of the overlying layer; thus, the halfspae veloity is vh vo v km/s. Atually, neither the overlying layer nor the half-spae is homogeneous due to the transition zone. Parameter zt.5km is the mid-level of the transition zone, and parameter zo.km is the harateristi distane that shows the width of the transition zone. The offset h = km. Figure, to the right, shows the initial guesses and the stationary ray. We applied the initial guess twie: first with the floor depth km (whih is approximately the lower end of the transition zone), then with the over-estimated floor depth 5 km (red and blue lines, respetively). Both initial guesses onverge to a unique head wave solution (blak line). In the next example we show that in the ase of multi-arrivals, the final solution is sensitive to the initial guess. Example : One-way path in a medium with low-veloity ellipti anomaly. Consider a onstant bakground veloity with an ellipti anomaly region of a lower veloity, as shown in Figure. Loation of the ellipse enter is x, z, and the semi-axes of the ellipse are rx, r z. The veloity field is desribed by an analyti funtion, 8th EAGE Conferene & Exhibition 8 - June 8, Copenhagen, Denmark

4 x x z z v vx, z vb tanh A, A, () s rx rz where v b is the bakground veloity outside the ellipse, and v b v is the anomalous low veloity inside the ellipse. Negative v leads to anomalous high veloity inside the ellipse. Parameter s is the smoothing sale: the smaller s is, the sharper the veloity hange. For infinitesimal s, the veloity funtion beomes disontinuous. We aept the following parameters, v 5km/s, v 3km/s, x 5km, z 3km, r 3km, r km, s.. (3) b x z The soure is loated at the subsurfae point with zero horizontal oordinate and depth dz km, and the reeiver is on the surfae, with the one-way offset ds km. Three different initial guesses shown in Figure left, by green, blak and blue lines, lead to three different solutions shown in Figure right. The red ellipse is the ontour of the veloity anomaly. Traing with Low Veloity Ellipse 3 5 Contour Trae Trae Trae distane (km) Figure One-way path in a medium with low-veloity ellipti anomaly For rays and (blak and green lines), the resulting (minimum) traveltime is t.8s, and for ray 3 (blue line), the traveltime is t 3.79s. Rays and are symmetri solutions, where the ray bypasses the low-veloity inlusion and almost avoids penetration into the transition zone. These rays travel ompletely through the high-veloity bakground. For ray 3, refration ours twie, upon entry to and exit from the low-veloity ellipse. Thus, we deal with a multi-arrival ase, haraterized by three loal minima, two of whih are also global. Example 3: Veloity field with two ellipti anomalies. Consider a model that ombines slow- and high-veloity anomalies and fast half-spae. It an be analytially desribed by, v v v vx, z v tanh tanh h b A A tanh A3, () where x x z z x x z z z z A o, A, A 3 s r s z x r z rx r z d. (5) Parameters x and x are horizontal oordinates of entral points of ellipti anomalies, z is their ommon vertial oordinate, z o is the floor depth of the high-veloity half-spae. Parameters s and z d govern the width of the transition zones, vb km/s, v km/s, vh 3km/s, x km, x km, z km, r km, r km, z km, z.km, s., () x z o d 8th EAGE Conferene & Exhibition 8 - June 8, Copenhagen, Denmark

and the offset h = km. Figure 3 shows the veloity field (left) and the ray traing results (right). The distanes in the veloity field plot are in km, and veloity is in km/s.

5 and the offset h = km. Figure 3 shows the veloity field (left) and the ray traing results (right). The distanes in the veloity field plot are in km, and veloity is in km/s. Eigenray Traing with High & Low Veloity Ellipses & Fast Half-Spae 8 Trae Init Slow Fast Floor 8 distane (km) Figure 3 Ray traing in a medium with two ellipti anomalies and high-veloity half-spae In the right subplot, the blue and red ellipses show the low-veloity and high-veloity anomalies, respetively. The gray line is the loation of the half-spae floor midline. The green line is the initial guess, where the depth of the floor was deliberately overestimated, in order to demonstrate onvergene to a orret result even with an initial error. The blak line is the least-time ray path. Conlusions A ray bending algorithm, referred to as the Eigenray method, has been developed for two-point traing in a general 3D heterogeneous medium. The method uses spetral elements with Legendre polynomial interpolation between the nodes and Gauss-Lobatto quadrature. Additional soft onstraints related to the segment lengths govern the loations of the nodes along the stationary ray. Expliit expressions for the traveltime and its first and seond derivatives allow the implementation of the Newton method optimization, whih also yields the geometrial spreading. We demonstrated the attrativeness of the method in ases of loal low/high veloity anomalies and for solving the ray trajetory of head waves. For multi-arrivals, the final solution is obviously sensitive to the initial guess. A strategy of ombining the global ray shooting method with the proposed Eigenray method seems very attrative. Referenes Bona, A., Slawinski, M., and Smith, P. [9]. Ray traing by simulated annealing: Bending method. Geophysis, 7, no., T5 T3. Farra, V. [99]. Bending method revisited: a Hamiltonian approah. Geophysial Journal International, 9, no., Moser, T. [99]. Shortest path alulation of seismi rays, Geophysis, 5, no., Moser, T., Nolet, G., and Snieder, R. [99]. Ray bending revisited, Bulletin of the Seismologial Soiety of Ameria, 8, no., Shashidhar, N., and Anand, G. [995]. Eigenray traing in an oean using Fermat s priniple. Journal of Sound and Vibration, 8, no., 3-3. Sripanih, Y., and Fomel, S. []. Two-point seismi ray traing in layered media using bending, SEG Tehnial Program Expanded Abstrats, Waltham, D. [988]. Two-point ray traing using Fermat s priniple. Geophysial Journal International, 93, no. 3, Westwood, E., and Vidmar, P. [987]. Eigenray funding and time series simulation in a layeredbottom oean. The Journal of the Aoustial Soiety of Ameria, 8, no., th EAGE Conferene & Exhibition 8 - June 8, Copenhagen, Denmark

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ow.mit.edu 1.510 Introdution to Seismology Spring 008 For information about iting these materials or our Terms of Use, visit: http://ow.mit.edu/terms. 1.510 Leture Notes 3.3.007