Implicit 3-D depth migration by wavefield extrapolation with helical boundary conditions

James Rickett, Jon Claerbout, and Sergey Fomel

james@sep.stanford.edu, jon@sep.stanford.edu, sergey@sep.stanford.edu

ABSTRACT

Wavefield extrapolation in the $(\omega-x)$ domain provides a tool for depth migration with strong lateral variations in velocity. Implicit formulations of depth extrapolation have several advantages over explicit methods. However, the simple 3-D extension of conventional 2-D wavefield extrapolation by implicit finite-differencing requires the inversion of a 2-D convolution matrix which is computationally difficult. In this paper, we solve the 45$^\circ$ wave equation with helical boundary conditions on one of the spatial axes. These boundary conditions reduce the 2-D convolution into an equivalent 1-D filter operation. We then factor this 1-D filter into causal and anti-causal parts using an extension of Kolmogoroff's spectral factorization method, and invert the convolution operator efficiently by 1-D recursive filtering. We include lateral variations in velocity by factoring spatially variable filters, and non-stationary deconvolution. The helical boundary conditions allow the 2-D convolution matrix to be inverted directly without the need for splitting approximations, with a cost that scales linearly with the size of the model space. Using this methodology, a whole range of implicit depth migrations may now be feasible in 3-D.