Implicit finite difference in time-space domain with the helix transform

Spectral factorization is a method of creating causal filters which have causal inverses. I use spectral factorization of an implicit finite-difference stencil of the two-way wave equation approximation in order to model wave propagation by a sequence of deconvolutions. I deconvolve this filter’s coefficients with the wavefield propagating in a constant velocity medium using the helix approach. In comparison with explicit approximations, implicit approximations have unconditional stability, enabling the use of larger time steps during the modeling process. The advantages are both in reduced computation time, and in the extension and scalability to multiple dimensions enabled by the helix operator. INTRODUCTION Implicit finite difference is a widely used method in geophysical data processing, commonly utilized for the approximation of the differential wave equation when extrapolating wavefields. In comparison with explicit methods, implicit finite difference has unconditional numerical stability, thus enabling larger finite differencing steps during the computation. This is an attractive prospect, as the implication is a shorter processing time for wave extrapolation. However, implementing implicit finite difference in a multidimensional problem is not trivial. The method requires the solution of a sparse set of linear equations per each propagation step. The cost of solving these linear equations, and the computational complexity required, becomes unreasonable at anything greater than 2 dimensions. It is possible to split the solver so that only one dimension of the problem is computed at each propagation step, thus reducing the complexity and possibly the amount of resources required for the computation. However, this method may introduce azimuthal anisotropy to the solution if the actual differential equation being solved is non-separable. Two concepts combine to greatly aid us in this matter. The first is the helix approach, envisaged in Claerbout (1997). The helix effectively enables us to treat multidimensional problems as one dimensional problems. Specifically, it enables execution of multidimensional convolutions as 1-D convolutions, and likewise for deconvolutions. Convolution equates to polynomial multiplication, while deconvolution equates to polynomial division. The application of convolution or deconvolution to