High-order kernels for Riemannian wavefield extrapolation

Riemannian wavefield extrapolation is a technique for one-way extrapolation of acoustic waves. Riemannian wavefield extrapolation generalizes wavefield extrapolation by downward continuation by considering coordinate systems different from conventional Cartesian ones. Coordinate systems can conform with the extrapolated wavefield, with the velocity model or with the acquisition geometry. When coordinate systems conform with the propagated wavefield, extrapolation can be done accurately using low-order kernels. However, in complex media or in cases where the coordinate systems do not conform with the propagating wavefields, low order kernels are not accurate enough and need to be replaced by more accurate, higher-order kernels. Since Riemannian wavefield extrapolation is based on factorization of an acoustic wave-equation, higher-order kernels can be constructed using methods analogous to the one employed for factorization of the acoustic wave-equation in Cartesian coordinates. Thus, we can construct space-domain finite-differences as well as mixed-domain techniques for extrapolation. High-order Riemannian wavefield extrapolation kernels improve the accuracy of extrapolation, particularly when the Riemannian coordinate systems does not closely match the general direction of wave propagation. I N T R O D U C T I O N Riemannian wavefield extrapolation (Sava and Fomel 2005) generalizes solutions to the Helmholtz equation in Riemannian coordinate systems. Conventionally, the Helmholtz equation is solved in Cartesian coordinates which represent special cases of Riemannian coordinates. The main requirements imposed on the Riemannian coordinate systems are that they maintain orthogonality between the extrapolation coordinate and the other coordinates (2 in 3D, 1 in 2D). This requirement can be relaxed when using an even more general form of Riemannian wavefield extrapolation in non-orthogonal coordinates (Shragge 2007). In addition, it is desirable that the coordinate system does not triplicate, although numerical methods can stabilize extrapolation even in such situations (Sava and ∗E-mail: [email protected] Fomel 2005). Thus, wavefield extrapolation in Riemannian coordinates has the flexibility to be used in many applications where those basic conditions are fulfilled. Cartesian coordinate systems, including tilted coordinates, are special cases of Riemannian coordinate systems. Two straightforward applications of wave propagation in Riemannian coordinates are extrapolation in a coordinate system created by ray tracing in a smooth background velocity (Sava and Fomel 2005) and extrapolation with a coordinate system created by conformal mapping of a given geometry to a regular space, for example migration from topography (Shragge and Sava 2005). Coordinate systems created by ray tracing in a background medium often represent well wavefield propagation. In this context, we effectively split wave propagation effects into two parts: one part accounting for the general trend of wave propagation, which is incorporated into the coordinate system, and the other part accounting for the details of wavefield scattering C © 2007 European Association of Geoscientists & Engineers 49 50 P. Sava and S. Fomel due to rapid velocity variations. If the background medium is close to the real one, the wave-propagation can be properly described with low-order operators. However, if the background medium is far from the true one, the wavefield departs from the general direction of the coordinate system and the low-order extrapolators are not enough for accurate description of wave propagation. For a coordinate system describing a geometrical property of the medium (e.g. migration from topography), there is no guarantee that waves propagate in the direction of extrapolation. This situation is similar to that of Cartesian coordinates when waves propagate away from the vertical direction, except that conformal mapping gives us the flexibility to define any coordinates, as required by acquisition. In this case, too, low-order extrapolators are not enough for accurate description of wave propagation. Therefore, there is a need for higher-order Riemannian wavefield extrapolators in order to correctly handle waves propagating obliquely relative to the coordinate system. Usually, the high-order extrapolators are implemented as mixed operators, part in the Fourier domain using a reference medium, part in the space domain as a correction from the reference medium. Many methods have been developed for high-order extrapolation in Cartesian coordinates. In this paper, we explore some of those extrapolators in Riemannian coordinates, in particular high-order finite-differences solutions (Claerbout 1985) and methods from the pseudo-screen family (Huang, Fehler and Wu 1999) and Fourier finitedifferences family (Ristow and Ruhl 1994; Biondi 2002). In theory, any other high-order extrapolator developed in Cartesian coordinates can have a correspondent in Riemannian coordinates. In this paper, we implement the finite-differences portion of the high-order extrapolators with implicit methods. Such solutions are accurate and robust, but they face difficulties for 3D implementations because the finite-differences part cannot be solved by fast tridiagonal solvers any longer and require more complex and costlier approaches (Claerbout 1998; Rickett, Claerbout and Fomel 1998). The problem of 3D wavefield extrapolation is addressed in Cartesian coordinates either by splitting the one-way wave-equation along orthogonal directions (Ristow and Ruhl 1997) or by explicit numerical solutions (Hale 1991). Similar approaches can be employed for 3D Riemannian extrapolation. The explicit solution seems more appropriate, since splitting is difficult due to the mixed terms of the Riemannian equations. In this paper, we concentrate our attention on higher-order kernels implemented with implicit methods. R I E M A N N I A N WAV E F I E L D E X T R A P O L AT I O N Riemannian wavefield extrapolation (Sava and Fomel 2005) generalizes solutions to the Helmholtz equation of the acoustic wave-equation