Elastic wavefield directionality vectors

Introduction

When modeling elastic wave propagation, both pressure and shear wave energy is propagated. This enables a more physical reverse-time propagation of 3-C receiver data. However, the cross-correlation imaging condition (Claerbout, 1971) is not sensitive to the type of wave being modeled, and this can result in a migrated image containing many artifacts without physical meaning. Therefore, when imaging with elastically propagated wavefields, a method of separating the two wave types is required. For isotropic media, the Helmholtz separation (Morse and Feshbach, 1953) separates the wavefield into a scalar P amplitude field and a vector S amplitude field. Yan and Sava (2008) show how this separation can be used to image primary modes (PP or SS) and converted modes (PS or SP) separately, resulting in an image with a more physical meaning. Yan and Sava (2009) extend this method to VTI and TTI media. Though these are effective methods, they do not separate the actual displacement fields into P displacements and S displacements. Zhang and McMechan (2010) deal with this problem directly, developing an elastic wavefield decomposition into P and S displacements, while preserving the original physical units, phase, amplitude and vector characteristics of the modeled field. They show an implementation for both isotropic and anisotropic media.

One method of constructing angle domain common-image gatherss by wavefield methods is the extended imaging condition by Sava and Fomel (2003), which requires performing cross-correlations between source and receiver wavefields at several horizontal spatial lags. A transform is then applied to these correlated wavefields to create the angle gathers. An alternate ray-based method is presented by Koren et al. (2008), which involves shooting ray-pairs from the subsurface points at which angle gathers are required, and saving the travel times to the surface and the surface location of the ray's emergence. The trace data at those locations is then interpolated from nearby recorded seismic data, and is copied into the subsurface image point's angle gather, at the opening angle of the ray pair. This methodology also incorporates dip gathers, by considering various tilting angles of the plane of the ray pairs, in relation to the vertical direction.

The angle gather construction methodology I wish to implement is a conceptual hybrid of these wavefield and ray-based methods. In the isotropic case, the direction of P wave propagation is tangent to the particle displacement direction, while the S wave's propagation direction is perpendicular to the displacements. If the vector displacement field (which is calculated as part of elastic wavefield finite-difference algorithm) can be decomposed into P and S wave displacements, it follows that the direction of polarization of each wave mode can, in principle, be determined. Under the isotropic assumption, the wave propagation direction can be readily deduced from the polarization direction. This is true only if, at a certain model point at a particular time window, only one wave exists. If this is indeed the case, then the displacement vector will be along a certain line, e.g. - the wave will be linearly polarized. Such information constitutes an ``arrow'' in space and time, indicating the wave's propagation direction within a time window. In this respect, this arrow can be likened to a ray, since it points toward the propagation direction of the wavefront. Having a directionality determination capability for finite-difference wavefield propagation methods is an exciting prospect, since the consequence is an ability to construct angle gathers during wavefield propagation, without applying any additional transforms to the wavefields, as in the extended imaging condition by Sava and Fomel (2003). It also makes the expensive data gathering step required by the method developed in Koren et al. (2008) unnecessary.

The scope of this report is much more limited than actual angle gather construction for elastic wavefields, and my purpose here is only to show whether it is possible to acquire P and S wave polarization directions during wavefield propagation. Furthermore, I limit myself to an isotropic medium assumption. I decompose each wavefield into P and S displacements using Zhang and McMechan (2010)'s method, and deduce from the displacements the angle of the polarization direction of each wave mode to the vertical axis, at each point in the modeling space, and at each time. From these angles, it is possible to ascertain the wave propagation direction. Angle gathers can eventually be constructed by summing the source and receiver wave's propagation angle to the vertical. This will constitute the ``opening angle''. The value of the angle gather at the particular opening angle will be the cross-correlation product of the source and receiver wavefields at the imaging point.

Elastic wavefield directionality vectors