Seismic numerical forward modeling; rays and wavesJean Virieux∗ and Gilles Lambare´†

∗ Institut des Sciences de la Terre, Universite´ Joseph Fourier Grenoble I & CNRS†CGG Veritas, Massy, France. E-mail: gilles.lambare@cggveritas.com

INTRODUCTION

Seismic waves propagate inside the Earth at large distances de-pending on the frequency content of waves. These waves sample the interior of the Earth and bring useful information once back

for non-dispersive waves. The phase is related to the travel- time T and the wavefront is now defined by a constant travel- time. Two fundamental equations can be deduced from elasto- dynamic equations. The first one is the Eikonal equation whichcan be written as

to the free surface one wishes to decipher. They provide when correctly interpreted the highest resolution one can expect from

(T )2(x) =1

c2(x), (3)

indirect geophysical methods of the interior of the Earth.

Except for specific targets as near-surface investigation or core- mantle boundary for which complex interpretations are required, recorded seismic traces follow a hierarchy between the duration of the source signal, the time lapses between different energetic phases over the total time of measurement.

This hierarchy allows one to consider that wavefronts are essen- tially preserved even if deformed when waves are propagating inside the heterogeneous Earth, keeping the energy concentrated in specific zones of the Earth at a given time. Migration velocity analysis is based on such assumption of reflected energy pre- served when recorded back at the free surface: pieces of hyper- bolae are detected from these locally preserved energetic pack- ets.

If wavefronts are preserved, orthogonal trajectories, known as rays, could also be considered although they cannot be observed directly as in optics.

In this presentation, we shall provide a quick review of the ray theory, the standard way of solving ray tracing by the method of characteristics, the wavefront tracking to keep under control the sampling of the medium before moving to static and dynamic Hamilton-Jacobi equations and related partial differential equa- tions (PDE) which can be solved in a robust way on fixed grids.

WAVES AND RAYS

The displacement, essentially the vertical component, recorded at the free surface could be written as the scalar quantity

d(x, t) = B(x, t)ei(x,t), (1)

where the position is denoted by x and the time by t. The ampli- tude B(x, t) deconvolved of the source signature S(t) is varying

where the wave speed is denoted by c(x). The second one definesthe amplitude through the transport equation as

2A(x) · T (x) + A(x)2 T (x) = 0. (4)

The eikonal and transport equations are fundamental ingredients of ray theory and highlight required properties as wavefront spa- tial coherence as well as amplitude conservation along ray tubes. Unfortunately, failures of such properties exist quite often in the Earth (interfaces with sharp discontinuity of media properties, shadow zones where no rays are entering, caustics where rays cross each other, strong velocity gradients and so on).

This approach, namely Geometrical optics (GO), is often con- sidered as more difficult to handle numerically than solving di- rectly partial differential equations of elastodynamics as long as one can afford the computation. We shall be concerned in this presentation by various strategies for finding solutions of GO in heterogeneous media.

TRACING RAYS AND PARAXIAL RAYS

The standard way of solving the Eikonal equation (3) is through the system of characteristics, which can be written as a non- linear system of first-order ordinary differential equations (ODE). We first introduce the slowness vector p = T (x) as well as thefunction H (x, p) = 1/2(p.p − u2(x)) for isotropic media where the expression u(x) is the inverse of the wavespeed c(x). Verify-ing the Eikonal equation means that the function H , often called the Hamiltonian, is equal to zero.

In a 3D medium, this system of tracing rays consists in seven equations connecting the position x, the slowness vector p and the travel time T along a ray, which is very similar to tracking dynamic trajectories of a particle in classical mechanics (Gold- stein, 1980). One can write

slowly as well as the phase (x, t), keeping the local coherenceof the wavefront defined by a constant phase. The oscillating fea-ture of the displacement is embedded into the use of the exponen-

dxi

d

=dpi

H pi

,

tial function. This ray ansatz promoted by Babic (1956); Karal and Keller (1959) could be found in many textbooks (Cˇ erveny , 2001; Chapman, 2004). In the frequency domain, this ansatz canbe written as

d(x, ) = A(x)ei T (x)S(), (2)

k

= − H , i = 1, 2, 3 (5)d xidT Hd

= pk p ,

where the variable is defined as d = c2(x)dT for the selected function H . The constant Hamiltonian equal to zero defines an

Rays and

Figure 2: Wavefront construction with a uniform ray density cri- terion in the smooth Marmousi model (Lambare´ et al., 1996). Upgoing rays have been eliminated.

WAVEFRONT TRACKING

Figure 1: Paraxial ray geometry: paraxial trajectory is sensitive only to properties on the central ray (parabolic extrapolation as seen on the slowness square section, string connection to the cen- tral ray for expressing the linear feature of paraxial equations). Paraxial ray moves, therefore, away from the associated neigh- bouring ray: we must consider that it stays infinitesimally close to the central ray.

hypersurface in the 6D phase space. Initial conditions could be either related to point conditions in the physical space or to plane conditions in the ajoint space (the complementary space when considering the phase space diminished by the physical space).

The evolution of a ray nearby a selected ray, called central ray, could be estimated through differential geometry (figure 1). De-noting a small perturbation yt = ( x, p) of the current ray position yt = (x, p), one can deduce a linear differential system from the non-linear system 5 as

Instead of solving these ODEs related to rays, one may consider sampling directly wavefronts. Solving directly the Eikonal equa- tion 3 for the first-arrival travel time turns out to be performed quite efficiently: finite-difference (FD) method has been pro- posed by Vidale (1988, 1990) and numerous more or less precise algorithms has been designed for both isotropic and anisotropic media (Podvin and Lecomte, 1991; Lecomte, 1993). Finite dif- ference techniques as the fast marching method (Sethian and Popovici, 1999) may also account for large velocity gradients. Specific techniques for tracking fastest wavefronts across inter- faces have been considered (Rawlinson and Sambridge, 2004). These techniques have been very attractive in many applications as first-arrival travel-time delays seismic tomography because they provide always a trajectory connecting the source and the receiver and, therefore, a synthetic travel times whatever is the precision on it. When inverting a hugh amount of data, few miscomputed travel times will not affect the tomographic inver- sion (Benz et al., 1996; Le Meur et al., 1997). These techniques have been also thoroughly used for 3D reflection seismic imag- ing (Gray and May, 1994). They are in essence formulated in an Eulerian framework where travel-time quantities are estimated atfixed nodes of the grid.

d ydT =

¸ pxH0 ppH0

−xxH0

−xpH0

, y, (6)

Considering other wavefronts behind the fastest one requires com-

where the Hamiltonian H has a subscript zero because it is eval- uated on the central ray regardless of the perturbation vector y. This linear system could be solved following the standard prop- agator formulation introduced by Gilbert and Backus (1966) in seismology (Farra and Madariaga, 1987), leading to the so-called paraxial ray tracing. Initial values of the perturbation vector y should be defined and one may consider a basis of six elementary trajectories from which any specific paraxial ray related to spe- cific initial conditions could be deduced. The amplitude could be deduced from this paraxial ray tracing, avoiding a specific numerical strategy for solving the transport equation 4.

These characteristics as rays sample the phase space with a nonuni- form resolution which is an intrinsic shortcoming of such ap- proach based on solving ordinary differential equations.

plex bookkeeping and interpolation in order to track the evo- lution of the desired wavefront, especially when folded. This strategy is the ray tracing by wavefronts where we perform ray tracings from an already evaluated wavefront (Vinje et al., 1993, 1996a,b). When ray density becomes too high, we may decimate rays and, when ray density becomes too low, we may increase the number. Adaptative techniques have been developed and paraxial information may help such strategy of fixing ray den- sity according to ray curvature in the phase space (Lucio et al., 1996) (figure 2). Of course, the entire medium is sampled which may be a quite intensive task compared to the 1D sampling per- formed by a single ray. It is also quite intensive compared to the efficiency of the first-arrival computations. This multi-values wavefront tracking is still formulated in an Lagrangian frame- work where one follows a wavefront evolution through rays re- gardless its

Rays and geometrical complexity: we are still solving ODEs.

Rays and Combining ray tracing and FD methods may lead to multi-valued travel-time estimations as suggested by Benamou (1996); Ab- grall and Benamou (1999). These authors have performed the link with the theory of viscosity solutions of Hamilton-Jacobi equations (Lions, 1982), although they have combined a mixture of Lagrangian and Eulerian frameworks leading to quite inten- sive computational challenges (Abgrall and Benamou, 1999).

HAMILTON-JACOBI EQUATIONS

An entirely Eulerian formulation for multi-valued wavefront re- construction could be recast into PDEs on fixed computational grids of the phase space. These PDEs are static Hamilton-Jacobi (HJ) equations which can be solved efficiently using various ap- proaches as level set methods (Osher et al., 2002; Qian and Le- ung, 2006) or fast phase space methods (Fomel and Sethian, 2002) or finite elements methods (Cheng and Shu, 2007).

In the phase space, one can write the static HJ equations as

H (x, T ) = 0, (7)

which is nothing than the Eikonal equation as p = T . Let us consider that the Hamiltonian is now equal to a given constant quantity E (set to zero for rays). We may introduce the function S = T − E and we could see that p = T = S at fixed . One can show that

Hamilton-Jacobi equation allows us to consider partial differen- tial equations which can be solved by various techniques, pro- viding multi-valued wavefronts. One expects more stable and robust estimations from these approaches if they turn out to be efficient in 3D geometries.

ACKNOWLEDGMENTS

This study is partially funded by the SEISCOPE consortium spon- sored by BP, CGG-VERITAS, ENI, EXXON-MOBIL, PETRO- BRAS, SAUDI ARAMCO, SHELL, STATOIL and TOTAL (htt p ://seiscope.oca.eu). Authors would like to thank Paul Williamson (TOTAL) and John Washbourne (CHEVRON) for active interac-tion and for the invitation to present this work.

S

+ H (x, S ) = 0, (8)

which is the time-dependent HJ equation for an Hamiltonian which does not depend explicitly in the evolutionary variable . Both static and time-dependent equations have been investigated by different researchers.

Combining fast marching methods and ordered upwind meth- ods, Fomel and Sethian (2002) have proposed an efficient tech- nique for solving the static HJ equation while Li et al. (2008) have concentrated on fast sweeping techniques and discontin- uous Galerkin methods. Various illustrations on 2D complex cases as the 2D Marmousi model (Qian and Leung, 2006) open roads for potential applications for seismic tomography and seis- mic imaging.

CONCLUSION

Geometrical optics estimate physical quantities as travel-times and amplitudes quite useful for the interpretation of seismic traces. Solving the ordinary differential equations related to rays as char- acteristics leads to a non-uniform sampling of the space. Wave- front evolution turns out to be quite efficient when considering first-arrival times as we solve partial differential equations in a fixed grid. The alternative strategy based on the wavefront tracking by controlling the local ray density allows considering complex wavefronts with foldings and shadows. This strategy is still governed by ordinary differential equations and tracking the complex evolution of a given wavefront is a difficult book- keeping. Embedding geometrical optics into the equivalent static

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