RICE UNIVERSITY. 3 D First Arrival Traveltimes and Amplitudes via. Eikonal and Transport Finite Dierence Solvers. Maissa A.

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1 RICE UNIVERSITY 3 D First Arrival Traveltimes and Amplitudes via Eikonal and Transport Finite Dierence Solvers by Maissa A. Abd El-Mageed A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved, Thesis Committee: William W. Symes, Chairman Professor of Computational and Applied Mathematics Steven J. Cox Associate Professor of Computational and Applied Mathematics Colin A. Zelt Assistant Professor of Geology and Geophysics Houston, Texas July, 1996

2 3 D First Arrival Traveltimes and Amplitudes via Eikonal and Transport Finite Dierence Solvers Maissa A. Abd El-Mageed Abstract First arrival traveltimes and associated amplitudes are essential components in Kirchho migration and modeling. Traditionally they have been determined by ray tracing which have some diculties especially they are computationally intensive and not guaranteed to produce the minimum traveltime. Seismic traveltimes in three dimensions can be computed eciently and accurately using an essentially nonoscillatory (\ENO") Hamilton-Jacobi (HJ)second order scheme. The scheme can be implemented in fully vectorizable form. Several examples illustrate the eectiveness of this approach to traveltime computation. A similar accurate scheme is required to solve the transport equation for the amplitudes associated with the rst arrival traveltimes. A second-order Runge-Kutta upwind nite dierence scheme is constructed for this purpose. This scheme less accurately computes the amplitudes since the transport equation involves the traveltime Laplacian which must be evaluated using the output of the eikonal scheme. The error in is second order accurate, hence the approximation to the traveltime Laplacian is zeroth order accurate, and there is no reason to expect the traveltime Laplacian, hence the amplitude, to converge. One remedy to ensure that the traveltime Laplacian is suciently accurate to guarantee convergence, is to use a higher order scheme, say third order ENO upwind scheme to solve the eikonal equation. Preliminary numerical results are presented to demonstrate the third-order accuracy of the HJ-ENO numerical ux in spatial directions (x and y) and of TVD (Total variation diminishing) Runge-Kutta method in z-direction.

3 Acknowledgments I wish to thank the following people who have helped me and made the completion of this thesis possible. First of all, I owe a debt of thanks to my advisor Dr. William Symes who allocated considerable amount of his precious and valuable time to help and guide me through my research. I thank him for his eort and patience not just as a professor but also as a person. I will always remember Dr. Symes with lots of gratitude. I would also like to thank the other members of my committee, Dr. Steve Cox, and Dr. Colin Zelt, who have carefully read my thesis, and suggested helpful corrections. A special gratitude I owe to my colleagues in the Rice Inversion Project for their support and interest. In particular, Dr. Lucio Santos of the University of Campinas, Brazil, whose fruitful discussions helped me in my research. I wish also to express my great appreciation to Michael Pearlman for his technical support and for his attention. A most sincere gratitude goes to Dr. Amr Elbakry, and Dr. Mahmoud El-alem, from Alexandria University, Egypt, for their assistance, concern, and friendship. I must say a special thank you to the following friends who I consider a family at the Graduate House. Dr. Robert Patten, professor in the English Department at Rice University for helping me nd a suitable residence and for always encouraging me. Paul Uhlig and his wife Carmen for always being there with their support and caring. I would like to sincerely thank my mother-in-law for being a mother and taking care of my son while I was away. I am indebted to all members of my family who have provided much needed support. My twin sister, Maha, and my brother, Ehab, provided me with endless emotional support and helped take care of my children. I would like to dedicate my thesis to the following beloved persons. My late Mom and Dad who always unconditionally supported and encouraged me. My husband, Tarek, for always being so supportive, caring, and understanding. Thank you, Tarek, for our shared sacrice and commitment. My children, Loai and Maissah, have been, and will always be, the source of my joy and happiness.

4 Contents Abstract Acknowledgments List of Illustrations List of Tables ii iii vii ix 1 Introduction The point source solution of the eikonal equation : : : : : : : : : : : Formulation of the Eikonal solver : : : : : : : : : : : : : : : : : : : : Formulation of the Transport solver : : : : : : : : : : : : : : : : : : : Organization of the Thesis : : : : : : : : : : : : : : : : : : : : : : : : 5 2 Traveltime Calculation 6 3 Finite Dierence Scheme Stability of the Eikonal Equation : : : : : : : : : : : : : : : : : : : : Linearization of Eikonal Equation : : : : : : : : : : : : : : : : 12 4 Amplitude Calculation 16 5 Numerical Experiments ENO Numerical Examples : : : : : : : : : : : : : : : : : : : : : : : : RK Numerical Examples : : : : : : : : : : : : : : : : : : : : : : : : : 31 6 Initial Results For Third Order ENO Scheme Denition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Denition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Denition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : High Order Runge-Kutta Type Time Discretization : : : : : : : : : : ENO-HJ Numerical Flux : : : : : : : : : : : : : : : : : : : : : : : : : 39

5 v 6.6 Numerical Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 7 Conclusion 47 Bibliography 49 A Ray Tracer Method 52 B Third-Order TVD Runge-Kutta Method 56

6 Illustrations 2.1 Geometrical Considerations : : : : : : : : : : : : : : : : : : : : : : : The CFL Condition : : : : : : : : : : : : : : : : : : : : : : : : : : : : Horizontal slice of analytic & ENO traveltime contours at the bottom of the domain (depth 6km) with x = y = z =1. Dashed line = analytic traveltimes & solid line = nite dierence traveltimes. : : : : Horizontal slice of analytic & ENO traveltime contours at the bottom of the domain (depth 6km) with x = y = z = Dashed line = analytic traveltimes & solid line = nite dierence traveltimes. : : The rate of convergence of 3D ENO scheme : : : : : : : : : : : : : : velocity model at depth slice z=4km. x = y = z =.0625 : : : : : Lens model - Traveltime contours at depth slice z=4km with x = y = z =.0625 : : : : : : : : : : : : : : : : : : : : : : : : : : Lens model - ENO solution at depth slice z=4km. x = y = z =.0625 : : : : : : : : : : : : : : : : : : : : : : : : : : Lens model - The rate of convergence of 3D ENO Scheme : : : : : : : Horizontal slice of nite dierence & analytic amplitude contours at the bottom of the domain (depth 6km) with x = y = z =.125. dashed line = analytic amplitudes & solid line = nite dierence amplitudes. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Horizontal slice of nite dierence & analytic amplitude contours at the bottom of the domain (depth 6km) with x = y = z = dashed line = analytic amplitudes & solid line = nite dierence amplitudes. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The rate of convergence of 3D Runge-Kutta scheme. : : : : : : : : : The relation between the execution time & the step size. : : : : : : : 35

7 vii 6.1 The second and third order rate of convergence of TVD RK methods. dashed line is the second-order method & solid line is the third-order method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The second and third order rate of convergence of TVD RK methods. dashed line is the second-order method & solid line is the third-order method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The second and third order HJ numerical ux. dashed line is the second-order HJ numerical ux & solid line is the third-order HJ numerical ux : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 46 A.1 The Spherical Coordinates : : : : : : : : : : : : : : : : : : : : : : : : 55

8 Tables 5.1 The impact points & the relative errors for the coarser grid : : : : : : The impact points & the relative errors for the ner grid : : : : : : : Determination of Order of TVD Runge-Kutta Method : : : : : : : : Determination of Order of TVD Runge-Kutta Method : : : : : : : : Determination of Order of the HJ numerical ux : : : : : : : : : : : : 46

9 1 Chapter 1 Introduction In this work we are interested in the production of traveltimes and amplitudes by means of nite dierence solution of the eikonal and transport equations rather than ray tracing. Traveltime eld and amplitudes are useful in Kirchho migration and modeling where the Green functions depend on them between surface points and depth points. Traditionally, traveltimes and amplitudes have been calculated by ray tracing (e.g., see [6] and [3]). But ray tracing presents some diculties. Shadow zones can be quite large in regions of high velocity contrast or complex structure causing rays not to penetrate the shadow zones and so complete absence of traveltime data in the region. Also in case of velocities with discontinuous derivatives, approximations should be made or the velocity should be smoothed so they can be handled well by ray tracing. These diculties make interpolation cumbersome and computationally expensive. Also there is no guarantee with ray tracing methods, particularly the shooting method, that the computed traveltime is the minimum one amongst all travel paths connecting two points in the domain. However, it is possible that the minimum traveltime is only a local minimum in both of bending and shooting methods but not a global traveltime ([22]). All of these diculties introduced nite dierence methods for solving eikonal equation directly on regular grid which guarantee rst arrivaltime data, handle with velocities having discontinuous derivatives, rarely produce shadow zones, and more faster than ray tracing methods. Now, we would like to give a brief review of some work that has been done in this area. Reshef and Koslo in 1986 (see [14]) proposed using the eikonal equation to compute traveltime elds directly on a grid. Vidale in [25] also approximated the eikonal equation on three-dimensional numerical grid by nite-dierence extrapolation from point to point, the same technique which he was implemented for the two-dimensional case [24].Van Trier and Symes in [23] implemented an explicit rst order upwind method described by Engquist and Osher [4]. It solves the hyperbolic

10 2 conservation law that describes changes in the gradient components of the traveltime eld. Schneider in 1995 [18] has generalized this approach to three dimensions; also see his paper for many recent references. Another approach was using graph theory to calculate the shortest raypath between source and receiver (see, for instance, [11], [16], [17], and [12]). Harten et al. in [5] constructed ENO schemes which have global high order accuracy in smooth regions and use adaptive stencils, thus obtaining information from smooth regions if discontinuities exist. In this thesis, we propose a second order upwind computation of rst arrival traveltime in three dimensions in which we use the nite dierence scheme used by Symes et al. in two dimensions [21]. The scheme is one of a family of Essentially Nonoscillatory (\ENO") schemes introduced by Osher and Sethian in 1988 [13] for computing the solutions of Hamilton-Jacobi (HJ) initial-boundary value problems, of which the eikonal equation is a preeminent example. This family of schemes has three very attractive properties: (1) stable schemes of arbitrarily high order of accuracy exist.the high order of accuracy is achieved through upwinding by means of ENO approximation of derivatives by high order dierences; (2) versions exist in any dimension; (3) The ENO schemes are easy to program. Finally, another advantage that is very important specially for three dimensions is the vectorizability of the ENO nite-dierence scheme, which is not available in Vidale's scheme. Like other upwind nite dierence methods, beginning with Vidale's method [24], our ENO scheme computes only the rst-arrival traveltime because it is the unique viscosity (hence stable) solution to the eikonal equation [9]. We will assume that the velocity eld is smooth, to ensure that the truncation error of the eikonal solution is small, so that the nal result is accurate. Our scheme involves marching in spatial directions, in common with Vidale's and other similar schemes. To simplify the development of a rst version in three dimensions, we restricted our attention to downgoing propagation: we assume that all points at which we need accurate traveltimes are connected to the source by rays which have at all points a positive z velocity component. This restriction can be removed by more complex programming, or the use of polar coordinates as in [23] or [18]. As the examples in [18] clearly show, the downgoing restriction may not be a great drawback in Kirchho migration applications.

11 3 A second goal, is to present how to produce the amplitudes of the rst arrivals solving the transport equation without knowledge of the raypaths. For this, we discuss the second order upwind Runge-Kutta scheme to solve the transport equation. The transport equation involves the travel time Laplacian, which must be calculated using eikonal solution and since the error in is second order, the error in the approximation to the Laplacian is zeroth order. And since the error in is rst order at the boundary so the error in amplitudes is even worse at the boundary. But since the scheme is upwind and the boundary of the domain is in the downwind direction, so the interior of the domain is not aected by the error at the boundary. We present numerical results illustrating that away from the boundary the transport solution is reasonably accurate since there are oscillations throughout the solution and there is no evidence that we can get convergence. A remedy for these oscillations is to use a higher order upwind scheme for the eikonal equation, say a third order ENO scheme to get a third order accurate solution to the eikonal equation, and hence rst order convergence for the amplitudes. 1.1 The point source solution of the eikonal equation The eikonal equation in three dimensions is ( x ) 2 + ( y ) 2 + ( z ) 2 = s 2 (x; y; z) Where = (x; y; z) is the traveltime eld, and s(x; y; z) = v(x; y; z)?1 is the slowness model. Subscripts x,y and z denote partial derivatives with respect to x, y and z, respectively. The point source solution at (x s ; y s ; z s ) satises the additional condition j(x; y; z; x s ; y s ; z s )? s(x s ; y s ; z s )rj = O(r 2 ) where r = q (x? x s ) 2 + (y? y s ) 2 + (z? z s ) 2 The eikonal equation does not in general have global smooth solutions, even ignoring the singularity at the source point. Far from the source, typically more than one ray from the source passes over a given depth point. Therefore some selection

12 4 principle must be used to choose from amongst the many possible values a single value which may be viewed in some weak sense as the solution of the nonlinear partial dierential equation above. Lions in 1982 [9] showed that the viscosity solution of the eikonal equation is stable and therefore computable, and that it amounts to selecting the rst (least) arrival time at each point. 1.2 Formulation of the Eikonal solver We would like to solve the following eikonal equation by means of nite dierence ENO scheme: q z = s 2? ( x ) 2? ( y ) 2 We use the following notation k i;j is the discrete representation of traveltime eld and similarly s k i;j = v(x; y; z)?1 is the discrete representation of the slowness model. And we dene Similar operators are dened in y. D x k i;j = [ k i1;j? k i;j ]=x Thus, the two-dimensional rst order upwind ENO-HJ scheme is: k+1 i;j = k i;j + z H[]k i;j where the numerical Hamiltonian H[] k i;j (see [13]) is: H [] k i;j = (s k i;j) 2? max(d? x k i;j ; 0) 2? min(d + x k i;j ; 0) 2? max(d? y k i;j ; 0) 2? min(d + y k i;j ; 0) Throughout the following two chapters, we are going to show how to get higher order upwind ENO-HJ schemes under some simplications we have made to ensure that we satisfy the restriction we have mentioned above about our downgoing propagating algorithm. Also we are going to use the essentially non-oscillatory Runge-Kutta algorithm (see [19]) to step in z direction.

13 5 1.3 Formulation of the Transport solver We are going to solve the following Transport equation by using the second order upwind Runge-Kutta scheme : With initial conditions [1]: 2 5 : 5 a + a 5 2 = 0 a(x; y; z) = 1 4 r where r = q (x? x s ) 2 + (y? y s ) 2 + (z? z s ) 2 As we see, The transport equation involves the travel time Laplacian, which will be calculated using eikonal solution. we are going to make again some simplications related to the ones that we have done in the traveltime computation step. 1.4 Organization of the Thesis This thesis is built up as follows. In the next chapter we describe some geometrical considerations about the boundary value problems for the eikonal and transport equations. Chapter 3 contains a description of the nite dierence ENO method followed by studying the stability condition for the ENO scheme. Chapter 4 presents our method for computing amplitudes. In chapter 5 we present some numerical results. First we implement our scheme with constant velocity model and compare the results with the analytic solution and second, with piecewise analytic \slow lens" model and compare the results with the ray trace arrivaltime. We show that the rate of convergence of our scheme is quadratic as we claimed. In chapter 6 we present some initial results for the third order ENO scheme.finally we present our conclusions in chapter 7. In appendix A we review the relation between the eikonal equation and the ray equation and describe the ray tracer method used to calibrate the results obtained by the ENO nite dierence scheme. Appendix B contains a derivation of the third order explicit TVD Runge-Kutta method used to step in z direction.

14 6 Chapter 2 Traveltime Calculation The proposed algorithm attempts only to compute traveltimes at points which are connected to the source by rays with some takeo angles, so that all such rays have positive z velocity component, say. In this section we explain how to make this specialization in the context of the eikonal equation. Our algorithm does not work for turning rays. This means that for the traveltime (x; y; z; x s ; y s ; z s ) between the source point (x s ; y s ; z s ) and any point (x; y; z), > 0 at least when (z > z s). In other words, j sin j sin max < 1 where sin = is the angle between the ray and the vertical and max is the max angle between the ray and the vertical. Now, to eliminate this type of rays, we need to modify the point source initial condition for the traveltime and also the eikonal equation. First we choose a datum depth z d > z s, z d is an initial condition region of rst arrival traveltimes,and to compute point source traveltime elds, we assume that the velocity is constant in the volume surrounding the source location, and then compute the traveltime at each grid point on z d. We start the initialization of traveltime at z = z d within a cone whose apex is at the source point and whose base on z d. within the cone, the traveltime is calculated as = distance slowness, but outside the cone traveltime is approximated by a rst-order Taylor expansion around the radius of the cone base. The algorithm does not attempt to track high angle events, thus we need condition to limit the sine of incident angles to sin( max ). The steps of the initialization procedure are (see Figure (2.1)): 1. Compute the radius of the cone base (r) at z d : From gure (2.1) we have: r = h tan( max ), h = jz s? z d j 2. At any point (x; y), If R = q (x? x s ) 2 + (y? y s ) 2 r then

15 7 (xs,ys,zs) Zmin Omax h Zd (xs,ys,zd) r Figure 2.1 Geometrical Considerations (x; y) = distance q slowness, where distance = (x? x s ) 2 + (y? y s ) 2 + h 2 Otherwise, use rst-order Taylor expansion around r: (x; y) = exact (r) (R? Now, exact (r) = p r2 + h 2 v source = q h 2 (1 + tan 2 ( max )) v source = h v source cos max and since the medium has spherical symmetry, i.e. where depends only on the distance R from a xed point (x s ; y s ; z d ), so we can say (R) instead of (x; y) and expand (R) by rst-order Taylor formula around r: (R) = p R2 + h 2 v source p r2 + h 2 v source + (R? r) r v source p r2 + h 2 where, r v source p r2 + h 2 = h tan max v source h sec max = sin max v (r)

16 8 From the above, we get modif ied (x; y) = = p R2 + h 2 v source if R r h v source cos max + (R? r) sin max v source if R > r This modied traveltime ensures the maximum angle condition at the initial datum depth z d. To enforce it at depth, we modify the eikonal equation = v? (@ )2 also with the aim of limiting the angle to max. Note: z 2 = 1 v? (@ )2 = cos2 and also 2 v 2 z = cos 2 max. So, the following modication will satisfy our objective (Symes et al., v 2 = max[( 1 v? (@ )2 ); cos2 max ] v 2 This modied eikonal equation is satisfactory for computation of the traveltime eld, but not for the computation of the amplitude, since the transport equation involves the Laplacian of the traveltime and as the max function is not dierentiable produces discontinuity in the derivative and hence oscillations in the amplitude eld at the edge of the aperture. We will use another function bmax which is piecewise quadratic: = bmax[( 1 v? (@ )2 ); cos2 max ] (2.1) v 2 bmax(x; y) = y if x 0 = (y + x2 4y ) if 0 x 2y = x if x 2y

17 9 for x; y 0. In order to the computation be vectorized, bmax may be written as bmax(x; y) = min y + x2 4y! ; max(x; 2y)!

18 10 Chapter 3 Finite Dierence Scheme As we mentioned in chapter 1, we use a second-order method constructed by (Osher and Sethian, 1988) (see [13]) and implemented in 2-D by (Symes et al.,1994) (see [21]). We are going to show how to get explicit formula for ENO second order ux: Introduce basic nite dierence operators as follows. Let x = jx, z = kz, then the forward (+) and backward (-) divided dierence operators in x are D x k i;j = [ k i1;j? k i;j]=x where k i;j is the discrete representation of traveltime eld and similarly s k i;j = v(x; y; z)?1 is the discrete representation of the slowness model. Similar operators are dened in y and x = y. (Osher and Sethian, 1988) introduced a family of high order upwind explicit schemes. The high order of accuracy is achieved by means of Essentially Non Oscillatory (\ENO") approximation of derivatives by high order dierences. For example, the second order ENO correction of D x + is obtained by approximating the second derivative by: xm(d+ x D+ x ; D? x D+ ) x Note that we have neglected D? x D? x because it is purely in a dierent direction than D x +. Since from Taylor expansion we have: Thus D D+;2 x = D+ x? 1 2 xm(d+ x D+ x ; D? x D+ ) x

19 11 where Similarly to get D?;2 x. Here, we neglected D + x D+ x m(x; y) = x if jxj jyj and xy > 0 = y if jyj < jxj and xy > 0 = 0 if xy 0 D?;2 x = D? x xm(d? x D? x ; D? x D+ x ). Similar formulas can be obtained for D;2 y. The key idea is an adaptive stencil interpolation which automatically obtains information from the locally smoothest region, and hence yields a uniformly high order essentially non-oscillatory approximation for piecewise smooth functions. For a general order of accuracy, see [13]. From (2.1) we get the two-dimensional rst order upwind ENO-HJ scheme: = k i;j + z bmax k+1 i;j (s k i;j) 2? max(d? x k i;j ; 0) 2? min(d + x k i;j ; 0) 2? max(d? y k i;j ; 0) 2 1? min(d + y k i;j ; 0) 2 ; (s k i;j) 2 cos 2 2 max The main idea here is that we update in the upwind direction i.e. the directions in which the rays are owing. The scheme is fully vectorized since the choice is build on min and max operators. 3.1 Stability of the Eikonal Equation Because our scheme is an explicit one, it is conditionally stable. So, the Courant- Friedrichs-Lewy (\CFL") stability condition must be satised. To achieve that: each full step from kz to (k + 1)z may require several partial steps of length z local. The full step is always taken if it is stable. z local should not be too small relative to z, to avoid long internal loops. For smooth solutions to nonlinear problems, the numerical method can often be linearized to check the (\local") stability of linearized equation and results from the theory of linear nite dierence methods applied to obtain convergence results for

20 12 nonlinear problems. A very general theorem of this form is due to Strang [20] (see also x 5.6 of [15]). In the next sub-section, we linearize the eikonal equation and show how to get the (\CFL") stability condition Linearization of Eikonal Equation Start with the eikonal equation ( x ) 2 + ( y ) 2 + ( z ) 2 = s 2 (x; y; z) we get: q z = s 2? ( x ) 2? ( y ) 2 Let, H( x ; y ) =? q s 2? ( x ) 2? ( y ) 2 Thus, z + H( x ; y ) = 0 z = o z + o z x = o x + o x y = o y + o y H( o x + o x ; o y + o y ) = H( o x ; o y ) + o x H x ( o x ; o y ) + o y H y ( o x ; o y ) where H x ( o x ; o y ) =? o x o z & H y ( o x ; o y ) =? o y o z Thus, o z? o x o z o x? o y o z o y = 0 is the linearized equation of the nonlinear equation. Note: 0 represents the initial values at (x 0 ; y 0 ; z 0 ). The domain of dependence in the (x; y) plane of the linearized equation for a point P (X; Y; Z) is the circle (x? X) 2 + (y? Y ) 2 Z 2 which is cut from the (x; y) plane by the right circular cone with vertex at P, and axis parallel to the z-axis. This cone is called the characteristic cone.

21 13 From the two dimensional rst order ENO-HJ nite dierence scheme: The grid points which inuence the value of at the point P lie inside a pyramid which cuts out from the two initial planes z = 0, z two rhombuses as domains of dependence. If x; y & z! 0, with P remaining constant, the grid points which inuence the value of at P (X; Y; Z), which remains xed, continue to lie inside the above pyramid, and the domain of dependence of P is a rhombus cut out on the (x; y) plane by this pyramid. There is no loss of generality if we consider the grid point P to lie on the z-axis and so X=Y =0. To get the (\CFL") condition which is necessary for stability for our two-dimensional problem to be convergent (refer to Figure (3.1)): z x(i,j) nh x(i+n,j) x nh an z nh/sqrt(2) y(i,j-n) y an z<nh/sqrt(2) λ<1/sqrt(2)a Figure 3.1 The CFL Condition The (\CFL") condition requires that (see [8] and [10]):

22 14 The domain of dependence for the partial dierential equation the numerical domain of dependence. In our problem the domain of dependence of the linearized dierential equation is a circle with radius = az n = a nz, z n is the depth level n. We have a x is the velocity in x-direction, a y is the velocity in y-direction, and a = q a 2 x + a 2 y. And the numerical domain of dependence is a rhombus which has x i+n;j? x i;j = x i;j + nh x? x i;j = nx and y i;j+n? y i;j = y i;j + nh y? y i;j = ny. Let x = y = h thus the side of the triangle = nh. So, a n z n h= p 2 and hence 1 p 2 a where = z h. Thus the CFL condition necessary for stability is z h p 2 a 1 or equivalently, z h p 2 q(d upwind x 2 upwind 2 +D y ) D + z i.e.,cfl-step h p 2 where D upwind x q (D upwind x 1 D + z 2 upwind 2 +D y ) and D upwind y denote the upwind choice of D x and D y respectively. To complete the above discussion of sub-stepping, we like to mention that we implement a linear interpolation in velocity between regular grid levels to derive values of slowness at the sub-levels. Now, to get the two-dimension ENO-HJ second order scheme: 1. replace D x i;j k and D y i;j k with Dx ;2 i;j k and Dy ;2 i;j k in the two-dimension rst order upwind scheme 2. To step in z-direction, use second-order essentially non-oscillatory Runge-Kutta algorithm ( Shu and Osher, 1988 ): k+ 1 2 i;j = k i;j + z local H[] k i;j k+1 i;j where = 1 2 ( k i;j + k+ 1 2 i;j + z local H[] k+ 1 2 i;j )

23 15 H[] k i;j = bmax (s k i;j) 2? max(d 2;? x? max(d 2;? y k i;j ; 0) 2? k i;j ; 0) 2? min(d 2;+ x k i;j ; 0) 2 min(d 2;+ y k i;j ; 0) 2 ; (s k i;j) 2 cos 2 ( max ) Finally, there is an important test for inow of the ray at the boundaries of the cube. The sign of D x i;j and D y i;j must be checked to be sure that the upwind dierence operator at the boundary depends only on the values inside the cube. D + x i;j must be negative at the boundary of the left face of the cube, D? x i;j must be positive at the boundary of the right face of the cube, D + y i;j should be negative at the boundary of the rear end-face of the cube and D? y i;j should be positive at the boundary of the front end-face of the cube. These checks should be done at every down step. If any of these conditions does not occur, this means that the ray is directing into the cube at the boundary, because the gradient of is the ray velocity vector. This means that information outside the cube should be used, and so the computation should stop. 1 2

24 16 Chapter 4 Amplitude Calculation The objective is to compute the amplitudes of the rst arrivals in three dimensions given an arbitrary velocity eld without knowledge of the raypaths. The amplitude eld a solves the transport equation 2 5 : 5 a + a 5 2 = 0 Since the amplitude eld changes by several order of magnitudes, whereas its logarithm changes much less, it is easier to write the transport equation in the following @x @y 1 A log a =? (4.1) This form is advantageous in maintaining uniform accuracy for the logarithm in the presence of discretization and round-o error. Initial conditions at fz? z d g [1] are where r = a(x; y; z d ) = 1 4 r q (x? x s ) 2 + (y? y s ) 2 + (z d? z s ) In the traveltime calculation, there are times along rays not belonging to the cone are also calculated. To ensure that these rays do not contribute to the computed kirchho integral, we diminish the amplitudes assigned to them. As a consequence of the modied eikonal equation (2.1), we use a cuto function g, dened by g 1 if 0 else! 1 v? (@ )2 2 cos2 max v 2

25 17 We use this cuto function to modify the right hand side of the transport @z where, is the x? @z!? 1? ? 2 (1? g) A log a 2! This cuto function forces the amplitudes outside of the aperture to decay at least as quickly as those inside the aperture. The transport equation for the amplitude eld involves the traveltime Laplacian. To make good use of the eikonal solution z, it is preferable to write the traveltime Laplacian in a form involving only rst z derivatives. Dierentiation of the eikonal equation with respect to x,y and z, followed by some algebra, yields!?2! 5 2 = Second order approximations to the rst and second traveltime x, y derivatives, rst x derivative in y and the traveltime z derivative are obtained from the traveltime k i;j D x [] k = i;j D max 2;? x k i;j ; 0 + min 3 5 D 2;+ x k i;j 2! k i;j 1 k x? 2 2 i+1;j k + i;j k i?1;j Similar formulas can be obtained for k i;j 1 2 y (x ) k? i;j+1 ( x) k i;j?1

26 18 Where, x is the rst order forward dierence )k i;j H [] k i;j Substitute these approximations in the above Laplacian expression to get the second order approximation 5 2 L [] k i;j An upwind choice of derivatives of u = log a prevents the spurious oscillations occur if discontinuities in x and y exist, as such discontinuities occur only in the downwind direction. This suggests use of the upwind one-sided Runge-Kutta scheme. The Runge-Kutta method based on the Taylor series expression: u(x; y; z + z) = u(x; y; z) + z u z (x; y; z) + O(z) 2 (4.2) From (4.1) we can compute u x u x + y u y Runge-Kutta method is obtained from (4.2) and using second order one sided approximations of the derivatives. The second order one sided approximations of the rst derivatives are: D p x u i;j = 1 x (?3 2 u i;j + 2 u i+1;j D m x u i;j = 1 x (3 2 u i;j? 2 u i?1;j Similar formulas can be obtained for y.? 1 2 u i+2;j) u i?2;j) Set h k ; u ki i;j =? 1 H [] k i;j + max h max D x [] k i;j ; 0 h i D m x uk i;j + min D x [] k i;j ; 0 h i D p x uk i;j D y [] k i;j ; 0 [D m y uk i;j ] + min D y [] k i;j ; 0 [D p y uk i;j ]i? 1 L [] k i;j 2 H [] k i;j

27 19 The amplitude computation is completely vectorizable since the scheme is build on min and max operators. Then the Runge-Kutta scheme is u k+ 1 2 i;j = u k i;j + z local [; u] k i;j u k+1 i;j = 1 2 (uk i;j + u k+ 1 2 i;j + z local [; u] k+ 1 2 i;j ) Experimentally it is found that z h i.e., (D x + D y ) [] k i;j H[] k i;j < 0:5 H[] k i;j CF L? step < 0:5 hq 2 (Dx + D 2 y )[] k i;j provides conditional stability. where x = y = h.

28 20 Chapter 5 Numerical Experiments In this chapter we discuss the numerical results, obtained from applying both the Eikonal solver and Transport solver ( discussed in chapters 3 & 4) to two dierent velocity models. The experiments were performed in 32 bit arithmetic using a code implemented in Fortran. 5.1 ENO Numerical Examples In order to demonstrate accuracy and convergence for the ENO nite dierence scheme, the two experiments below are used. Analytic traveltime elds are also computed for the rst model and ray solutions are computed for the second model since it is dicult to get analytic solution for it. We take a sequence of step sizes which are halved and implement the method for every step size to get dierent errors corresponding to these steps. The errors should decrease quadratically with each successive decreasing of the step sizes. And then plot the relation between the step-size and the maximum relative error. The shape should be a parabola. To get the claimed second-order convergence, we need to illustrate the second-order convergence relationship: relative-error = (k analytic? F D k 1 =k analytic k 1 ) constant(x 2 +y 2 )=k analytic k 1 where the norm is the max-norm, analytic is the analytic solution, F D is the solution obtained by the ENO scheme and x&y are the step sizes in x and y directions respectively. We also dene the relative error used to show the quadratic convergence for the lens model: relative-error = j( int? ray?tracer )j= int

29 21 where ray?tracer is the solution got by the ray-tracer method at location (x; y) of the desired layer z f and int is the traveltime at the corresponding location (x; y) determined by interpolation within the ENO solution. Model 1: Constant Velocity In this model, the constant velocity is taken to be 1.0 m/ms and the domain of computation is 5 km 5 km 6 km. The point source is located at P s = (x s ; y s ; z s ) = (1,1, -4 ) km. We chose an aperture of 65 degrees. The analytic traveltime solution for this method is q d 2 (P s ; Q n i;j) n i;j = v where Q n i;j is the point ((i? 1)x; (j? 1)y; (n? 1)z) and d(p s ; Q n i;j) is the Euclidean distance between two points P s and Q n i;j. Figures (5.1 and 5.2) show the traveltime eld contours at the bottom of the domain for both the analytic solution and the ENO solution (with the same grid sizes). They are concentric semi-circles and there are no dierences between the analytic gures and the ENO gures. In gure (5.1), x = y = z = 1, while in gure (5.2), x = y = z = :0625. The coarse mesh gives less smooth results than the ne mesh does as should be expected. Figure (5.3) shows the rate of convergence of the ENO scheme. The shape is a parabola which conrms our claim that the rate of convergence is quadratic.

30 Figure 5.1 Horizontal slice of analytic & ENO traveltime contours at the bottom of the domain (depth 6km) with x = y = z =1. Dashed line = analytic traveltimes & solid line = nite dierence traveltimes.

31 Figure 5.2 Horizontal slice of analytic & ENO traveltime contours at the bottom of the domain (depth 6km) with x = y = z = Dashed line = analytic traveltimes & solid line = nite dierence traveltimes.

32 Max Relative Error(%) Step Size Figure 5.3 The rate of convergence of 3D ENO scheme

33 25 Model 2: Lens Model This model has a constant velocity medium (v = 1) with an embedded lens which is a slow velocity region. Figure (5.4 ) shows a plot of the velocity model for a layer inside the slow velocity region v y x > 3 Figure 5.4 velocity model at depth slice z=4km. x = y = z =.0625 The source is at (1 km, 1 km, -4 km ). Figure (5.5 ) is a rst arrival traveltime contour map of the ENO solution to the eikonal equation. The wavefronts propagate in concentric circles until they encounter the slow velocity region (lens), where the rays tend to bend toward the slow velocity region.

34 Figure 5.5 Lens model - Traveltime contours at depth slice z=4km with x = y = z =.0625

35 27 We can see from Figure (5.6 ) that the traveltime depicts a locus where rays would cross : \v" region. Rays traveling within the lens would bend toward the center, crossing and generating this locus t y x > 3 4 Figure 5.6 Lens model - ENO solution at depth slice z=4km. x = y = z =.0625 Now, to calibrate our ENO scheme, we used the ray-tracer method (described in Appendix A), where we select the desired layer (z f ) for which the ray arrive after time t to be inside the slow velocity region and we got the following results: For the coarse grid 5 km 5 km 7 km, the max relative error was equal to.0062 but for the ner grid , the max relative error was e-06. Figure (5.7) depicts

36 28 the claimed quadratic rate of convergence for the lens model.and tables (1 & 2) show the impact points of the rays ( = 0 and z f = 4) and the relative errors between the ray-tracer solution and the ENO solution. 7 x relative error step size Figure 5.7 Lens model - The rate of convergence of 3D ENO Scheme

37 29 Table 5.1 The impact points & the relative errors for the coarser grid (degrees) x(km) y(km) ray?tracer (s) int (s) relative-error e e e e e * * indicates the max relative error.

38 30 Table 5.2 The impact points & the relative errors for the ner grid (degrees) x(km) y(km) ray?tracer (s) int (s) relative-error e e e e e e e e e e e e e e e e e e-06 * e e e-06 * indicates the max relative error

39 RK Numerical Examples To give some idea of the accuracy found through solving the transport equation by means of second order Runge-Kutta scheme, we have carried out several tests using constant velocity, for which comparison with the analytic quantities is straightforward. We used Model(1) mentioned in section (5.1) where the velocity used is 1.0 m/ms and the domain of computation is 5 km 5 km 6 km. The point source is located at P s = (x s ; y s ; z s ) = (1,1, -4 ) km. We chose an aperture of 65 degrees. The analytic amplitude solution for this method is u n i;j = 1 4 d(p s ; Q n i;j) where Q n i;j is the point ((i? 1)x; (j? 1)y; (n? 1)z) and d(p s ; Q n i;j) is the Euclidean distance between two points P s and Q n. i;j We notice that, the worst error is occurred at the boundary of the solution and we attribute that to the traveltime Laplacian involved in the transport equation. To show that this error remains at the boundary and does not propagate and aect the solution in the middle of the domain: We disregard the boundary of the domain. By that we mean we delete two cells from each side of the domain. We get the following contours and relative error graphs which show that away from the boundary the solution is second order accurate. Figures (5.8 and 5.9) show the amplitude contours at the bottom of the domain for both the analytic and the nite dierence solutions. In gure (5.8), x = y = z = :125 and in gure (5.9), x = y = z = :0625. Figure (5.10) shows that the rate of convergence of transport solver is quadratic and asymptotically the shape is a parabola. Figure (5.11) shows that the time (in seconds) is a linear function in the step size.

40 Figure 5.8 Horizontal slice of nite dierence & analytic amplitude contours at the bottom of the domain (depth 6km) with x = y = z =.125. dashed line = analytic amplitudes & solid line = nite dierence amplitudes.

41 Figure 5.9 Horizontal slice of nite dierence & analytic amplitude contours at the bottom of the domain (depth 6km) with x = y = z = dashed line = analytic amplitudes & solid line = nite dierence amplitudes.

42 Max Relative Error Step Size Figure 5.10 The rate of convergence of 3D Runge-Kutta scheme.

43 Time(sec) # of Grid Points x 10 5 Figure 5.11 The relation between the execution time & the step size.

44 36 Chapter 6 Initial Results For Third Order ENO Scheme As we have noticed from the numerical results presented in chapter 5, there is no evidence of convergence for the traveltime Laplacian, hence the amplitude do not converge. One remedy is to use a more accurate eikonal solver, say of third order. In this chapter we discuss two issues associated with the third-order ENO upwind scheme, namely the third-order TVD Runge-Kutta method and the third-order ENO- HJ numerical ux. In the next sections we study these issues and their eect on the numerical behavior of the algorithm introduced in chapter Denition On a computational grid x j = jx, z n = nz, we use u n j to denote the computed approximation to the exact solution u(x j ; z n ). And if the local truncation error is dened as [7]: u(x j ; z n+1 )? u n+1 j = O(x r+1 ) (1) We say the scheme is rth order accurate. 6.2 Denition The total variation (TV) of a discrete scalar solution is dened by [8] T V (u) = X j ju j? u j?1 j (2) 6.3 Denition The numerical method u n+1 j if [8] = H(u n ; j) is called total variation diminishing (TVD) T V (u n+1 ) T V (u n ) (3)

45 37 for all grid functions u n 6.4 High Order Runge-Kutta Type Time Discretization Consider the following problem: u z + dx i=1 f i (u) xi = 0 (4) Here x = (x 1 ; x 2 ; :::; x d ) u(x; 0) = u 0 (x) (5) We use the abstract form u z = L(u) (6) instead of (4), where L is a spatial operator. Shu and Osher in [19] considered a TVD Runge-Kutta method for discretizing (6). They showed that the time discretization is TVD under suitable restriction on z, i.e. on the CFL number. Here we give a brief description of their method. Dene T (u) = (I + zl) (u) (7) Where T and L are nonlinear discrete operators. L is a rth order discrete approximate to the spatial operator L. i.e., L(u) = L(u) + O(x) r (8) (7) is the Euler forward approximation to (6), i.e. it is rst order in time approximations to u(:; t + t) And also dene a fully rth order approximation to the dierential equation (6) as u n+1 = H(u n ) (9) i.e., u(x; t n+1 )? H(u n ) j = O(x) r+1 (10) where u(x; t) is an exact smooth solution of (6).

46 38 (9) is called a time discretization rth order TVD if it satises (10) and T V (H(u)) T V (T (u)) (11) The general explicit Runge-Kutta method for (9) is dened to be([19]) u (i) = u (0) + z Xi?1 k=0 c ik L(u (k) ; z (0) + d k z); i = 1; ::::m (12) u (0) = u (n) ; u (m) = u (n+1) (13) where the spatial operator L depends explicitly on the depth z. and d k = k?1 X l=0 c kl (14) Now to analyze the nonlinear stability (TVD) of Runge-Kutta scheme. Assume (7) is TVD under a suitable CFL restriction 0 (15) We also need an approximation? ~ L to the spatial operator?l which satises ~T(u) = (I? t ~ L)(u) (16) ~L(u) = L(u) + O(x r ) (17) where (16) is TVD under the same CFL restriction (15). To get conditions for TVD, (12) is rewritten as follows: For ik 0, u (i) = Xi?1 k=0 Xi?1 k=0 [ ik u k + ik zl(u (k) ; z (0) + d k z)]; i = 1; ::::m (18) ik = 1, and ik = c ik? Xi?1 l=k+1 c lk il ik could be negative in (18). So to obtain TVD, we replace L by ~ L. (18) is TVD under the CFL restriction (see [19])

47 39 0 min i;k ik j ik j (19) The idea is to choose the ik and ik such that (18) is of highest order and the CF L condition is optimal. It's desirable also to reduce the number of negative ik 0 s so that the computational work involving ~ L is reduced. To do this, we can use a standard Runge-Kutta method and rewrite it in the form (18) to get ik and ik. Since most classical Runge-Kutta methods produce small CFL numbers and negative ik 0 s, we use (18) directly and follow the method in [7] to get the following optimal scheme (see Appendix B): u (1) = u (0) + zl(u (0) ; z(0)) u (2) = 3 4 u(0) u(1) zl(u(1) ; z (0) + z) (20) u (3) = 1 3 u(0) u(2) zl(u(2) ; z (0) z) CF L coecient = 1 Here CF L coecient means min i;k ( ik j ik j ) 6.5 ENO-HJ Numerical Flux From chapter 3, we write the second-order EN0-HJ numerical ux in 3?D as follows H[] k i;j = bmax (s k i;j) 2? max(d 2;? x k i;j ; 0) 2? min(d 2;+ x k i;j ; 0) 2? max(d 2;? y k i;j ; 0) 2? min(d 2;+ y k i;j ; 0) 2 ; (s k i;j) 2 cos 2 ( max ) oi 1 2 where D +;2 x = D+ x? 1 2 xm(d+ x D+ x ; D? x D+ x ) and We have D?;2 x = D? x xm(d? x D? x ; D? x D+ x ) D? x D+ x (x) 00 (x) + O(x 2 )

48 40 But since D + x D+ x (x) 00 (x + x) + O(x 2 ) 00 (x) + x 000 (x) + O(x 2 ) Thus 00 (x) = D + x D+ x (x)? x 000 (x) + O(x 2 ) Similarly D? x D? x (x) 00 (x? x) + O(x 2 ) 00 (x)? x 000 (x) + O(x 2 ) gives 00 (x) = D? x D? x (x) + x 000 (x) + O(x 2 ) From the above, we can rewrite D +;2 and D?;2 as follows D +;2 x = D+ x? 1 2 xm(d+ x D+ x? xd+ x D+ x D+ x ; D? x D+ x ) and D?;2 x = D? x xm(d? x D? x + xd? x D? x D? x ; D? x D+ x ) Similar formulas can be obtained for D ;2 y. We use D +;2 x and D?;2 x to get the third-order ENO correction of D x : D +;3 x D+;2 x? 1 6 (x)2 m(d + x D+ x D+ x ; D+ x D+ x D? x ; D+ x D? x D? ) x and i.e., where D?;3 x D?;2 x? 1 6 (x)2 m(d? x D? x D? x ; D+ x D+ x D? x ; D+ x D? x D? ) x m(x; y; z) = 8 >< >: m(x; y; z) = m (m(x; y); z) m(x; y) if jm(x; y)j jzj and m(x; y)z > 0 z if jzj < jm(x; y)j and m(x; y)z > 0 0 if m(x; y)z 0 Again, similar formulas can be obtained for D ;3 y.

49 41 Thus,we can write the third-order EN0-HJ numerical ux in 3? D as follows H[] k i;j = bmax (s k i;j) 2? max(d 3;? x k i;j ; 0) 2? min(d 3;+ x k i;j ; 0) 2? max(d 3;? y k i;j ; 0) 2? min(d 3;+ y k i;j ; 0) 2 ; (s k i;j) 2 cos 2 ( max ) oi Numerical Results We want to verify that both of the third-order TVD Runge-Kutta method and the third-order ENO-HJ numerical ux are indeed third order. We show the following rst two examples to verify the order of accuracy for the Runge-Kutta method and the latter example to verify the order of accuracy for the numerical ux: Example 1 We consider the initial value problem where y(0) =?1, 0 x 1. And the exact solution is y 0 = f(x; y) = y? x(x 2? 5 x + 4) y(x) = x 3? 2 x 2? exp(x) We compute the solution y h for h = 1, q = 0; 1; 2; 3; 4. If we assume that the exact 2q solution y(x) can be written as y(x) = y h (x) + C h R + O(h R+1 ) then we can estimate the order R of the method by 2 R = y? y h y? y h=2 We dene the error between the exact solution and the Runge-Kutta solution as follows: err = jy(x)? y h (x)j

50 42 And we dene also the relative-error as follows relative? error = jy(x)? y h(x)j jy(x)j In table (6.1), the errors and the values of R found by comparing y h and y h=2 for various values of h are listed for both the second-order TVD Runge-Kutta method (which we mentioned in chapter 3) and the third-order TVD Runge-Kutta method. Figure (6.1) shows the rate of convergence of the second-order TVD Runge-Kutta method and the rate of convergence of the third-order TVD Runge-Kutta method.