Wavefield extrapolation in pseudodepth domain

Introduction

Migrations based on wavefield extrapolation, both in depth and in time, are used extensively for imaging complex subsurface structures. Ray-based migrations such as Kirchhoff migration can not correctly image reflectors where multipathing occurs. In all the wave-equation based migrations, the seismic image is constructed directly from the wavefields by applying a proper imaging condition to the extrapolated wavefields. As a result, there is a constant demand of more accurate and more cost-effective means of wavefield extrapolation.

Wavefield extrapolation is commonly implemented by finite difference approximations using regularly spaced Cartesian meshes. It is popular because there is no need to interpolate wavefields between the computation domain and physical space. However, Cartesian coordinate frame is poor in several other aspects because it does not take into account the physics of the wavefield. For example, wavelength is varies in space due to velocity variation, and considering the regular spacing in conventional implementations, this could result in uneven spatial representation of wavefields. Specifically, we tend to undersample wavefields in layers with low velocities and oversample them in layers with high velocities. Another issue with the conventional Cartesian coordinate system arises with downward continuation in the presence of overturned events: the vertical extrapolation direction in a Cartesian mesh does not follow the direction of energy propagation.

Wavefield extrapolation in non-Cartesian coordinate frames has been address by several authors for different purposes. Among them, the Riemannian coordinate allows downward continuation with small dip angles due to the fact that the coordinates are more aligned with wave-propagation directions (Shragge, 2008; Sava and Fomel, 2005); tilted Cartesian coordinate allows imaging of steep dipping reflectors using downward continuation (Shan and Biondi, 2008).

Our goal is to develop and implement reverse-time migration using pseudodepth in the vertical direction, as opposed to the conventional depth. The pseudo depth is velocity dependent in a way that the wavelength remains constant in spite of the vertical velocity variation. This allows the vertical axis to be discretized with less number of samples, and thus speed up imaging. In this paper, we first develop the pseudodepth coordinate frame and the procedures needed to interpolate between pseudodepth and the Cartesian coordinates. Next, we derive the proper extrapolation operators for isotropic and anisotropic media in the new coordinate frame. Finally, we present examples of pseudodepth domain prestack depth migration and compare the cost and accuracy with Cartesian domain implementation.

Wavefield extrapolation in pseudodepth domain